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Integrability and attractors
Chaos, Solitons & Fractals Vol. 4, No. 11, pp. 1969-I984, 1.994 Elsevier Science Ltd Printed in Great Britain.
Pergamon
0960-0779(94)00160-X
Integrability and Attractors JOSEPH L. McCAULEY
Physics Department, University of Houston, Houston, TX 77204-5506, USA (Received 16 June 1994) Abstract--We discuss global and local integrability and also attractors from a physicist's standpoint. A local Lie method is used to form an analytic and geometric picture of integrability via conservation laws. The starting point is Lie's proof that two-dimensional flows, even dissipative ones, have a conservation law and so are integrable. Through conservation laws, integrable higher-dimensional flows reduce to the two-dimensional case. We consider a class of models defined by the EulerLagrange equations for a set of nonintegrable velocities (nonholonomic coordinates). The destruction of the global conservation laws of this conservative system by the inclusion of linear damping and constant external driving leads to self-confinement and attractors in phase space. In particular, the Lorenz model belongs to this class and can be seen as a damped, driven symmetric top.
1. INTEGRABLE FLOWS IN PHASE SPACE
We study integrability by following the consequences of an elementary theorem by Sophus Lie. According to Bell [1], Lie showed that every one-parameter infinitesimal transformation acting on two variables defines a one-parameter group, and that any such group has an invariant. This means that two-dimensional flows in phase space, whether conservative or dissipative, have a conservation law. This fact is not pointed out in standard texts on differential equations or mechanics. To understand Lie's theorem, it is only necessary to establish the language of phase flows along with a dictionary to translate back and forth between the language of groups [2] and the language of differential equations [3]. The discussion that follows is heuristic in the spirit of Lie [4] and Klein [5] (and in the spirit of physics) rather than rigorous in the spirit of modern abstract mathematics [6]. By a flow, we mean an autonomous system of n (generally coupled and nonlinear) differential equations dxi --
V/(x1
.....
dt
Xn)
(1)
where the solutions are finite at all finite times (any and all singularities are confined to the imaginary time axis [3]). Phase space is the n-dimensional space indicated by the local coordinates (xl . . . . , x,). By a conservative flow we mean one that preserves the volume element dff2 = dxa • •. dx, in phase space, so that n
°
V" V = z Oxi = O. i=10xi
(2)
1970
J.L. McCAuLEV
In an arbitrary conservative system, the volume element is the only quantity that is left invariant by the flow. That is, in a completely arbitrary conservative flow there are generally no conservation laws that are point functions. Conservation laws are functions of the phase space variables and the time, Gg(x 1. . . . . x n, t) = constant, that satisfy d Gi
3G i
-- V " V G i -b - -
dt
= O.
~t
(3)
By a nonconservative or 'dissipative' system we m e a n any system where V . V fails to vanish identically over all of phase space. A typical dissipative system has no invariant at all, neither the phase volume element nor a solution of the partial differential equation (3). The integration of the flow (1) studied as a collection of infinitesimal transformations (~Xi ~- V/(X1. . . . .
Xn)f~t
(4)
defines a one p a r a m e t e r continuous group of transformations Xn(to) , t -- to)
Xi(t ) = lPi(Xl(tO) . . . . .
(5)
from coordinates (xl(to) . . . . . xn(to)) to ( x l ( t ) . . . . . x~(t)) as the group p a r a m e t e r is varied from to to t. In the language of differential equations, (5) is the solution at a later time t and (x1(to) . . . . . xn(to)) is the initial condition at to. The time t is a canonical group p a r a m e t e r , which means that it obeys an additive parameter-combination law. Noncanonical group p a r a m e t e r s obey nonlinear p a r a m e t e r - c o m b i n a t i o n laws. For a one p a r a m e t e r transformation on two variables, a canonical p a r a m e t e r can always be found by integration [7]. In terms of the canonical parameter, the infinitesimal transformations of the group are defined by the differential equations dxl -
Vl(xl, x2)
dt dx2 -
dt
g2(xl, x2)
(6)
which also describe a flow in the (x~, x2) phase space. Lie's t h e o r e m is now easy to demonstrate heuristically. By rewriting the flow equations in the form dt = d x l / V 1 = dx2/V2
(7)
we can obtain the time-independent differential form V2(x 1, x 2 ) d x l -- VI(X 1, x2)dx2 = 0.
(8)
E v e r y differential form in only two variables is either integrable or else has an integrating factor M ( X l , x2) that makes it integrable, so that we can integrate the differential form M ( V 2 dXl - VI dx2) = d G
(8b)
to obtain G ( X a , x 2 ) = C, which is a conservation law. For example, this means that even two-dimensional Newtonian flows where the mechanical energy E is not constant have a conservation law. Before continuing, we pause to state the condition under which the conservation law for a dissipative system exists, which is simply the condition that an integrating factor M exists. We can rewrite the differential form 1/2 dXl - V1 dx2 = 0 as the one-dimensional equation dx2
V2(Xl, x2) -
dXl
-
Vl(xl, x2)
F ( x l , x2)
(9)
Integrability and attractors
1971
whose solution satisfies the integral equation
f(s, x2(s))ds.
x2 = X2o +
(9b)
Xlo
According to Picard [8], this equation can be solved by the method of successive approximations so long as F satisfies a Lipshitz condition. A sufficient condition is that that ratio of the components of the velocity in phase V2/V1 must be at least once continuously differentiable with respect to the variable x~. This requires that we avoid integrating over regions that contain equilibria or other zeros of Vs. The resulting solution of the integral equation yields x2 as a function of xl. A qualitatively similar result follows from solving the conservation law G(xl, x2) = C to find x2 = f(x~, C), which is just the statement that the conservation law determines x2 as a function of Xl and C. In particular, differentiating the conservation law once yields
dG = 3G dx~ + 3G dx2 = 0, 3xl
(10)
9x2
and comparison with the differential form V2 dxl - V1 dx2
=
0 then shows that
/
- 3G/3---G-G=
V2/V1.
(lla)
3Xl/3X2 This last equation is satisfied only if
V 1 : - M 3G
and 172 = M olG, (llb) 3x2 ~Xl which guarantees the existence of the integrating factor M. In other words, the existence of the conservation law follows from satisfying Picard's conditions for a unique solution of the differential equation (9). This guarantees, by differentiation, that the integrating factor M exists. Except in a few special cases, there is no systematic way to construct the integrating factor M(x~, x2) even though it is guaranteed formally to exist. Notice that the condition that the flow is phase volume preserving, ~V 2
3V 1 -
9x2
- - ,
(llc)
3Xl
is just the condition that the differential form V2(xl, x 2 ) d x l - Vl(Xl, x2)dx2 = 0 is exact without any need for an integrating factor. This means that the velocity field of a conservative two-dimensional flow can always be obtained from a stream function G:
V = (V1, V2) : ( ~x29G'9~1.
(12)
By inspection, the conservative system is Hamiltonian with Hamiltonian G and canonically conjugate variables (q, p ) = (x2, &), but the collapse of an integrable conservative flow into canonical form is peculiar to two dimensions and generally is not true for integrable conservative flows in higher dimensions: in particular, every three-dimensional integrable flow is noncanonical although it may still be Hamiltonian on a Poisson manifold that is not symplectic [10]. As an example of a conservation law for a dissipative system, consider the damped simple harmonic oscillator de' + /3:~ + wax = 0.
(13)
1972 With x I
J.L. McCAULEY =
X and x 2
-~-
dx/dt the corresponding differential form (fix2 + a~zxl)dxa + x2 dx2 = 0
(14)
is not exact but is guaranteed to have an integrating factor M that yields a time-independent conservation law G(x, dx/dt) = constant via the condition dG -
dt
M((13x2 + O~Xl)~l + x2~2) = 0.
(15)
The conservation law for this specific case was originally discovered and constructed by Palmore et al. [9]. In all cases, the conservation law G(x, d x / d t ) = C is simply the equation that describes that family of streamlines: different constants C correspond to different streamlines within the same family. For the damped oscillator, the streamlines are a family of spirals. Lie's result explains qualitatively why two-dimensional dissipative as well as conservative flows have conservation laws: the condition C --- G ( X l o , x2o ) fixes the constant C in terms of the two initial conditions Xio at time t = to and is simply the equation of the family of streamlines of a flow, G(Xl, x2) = C, in the phase plane. Each streamline corresponds to a different value of the constant C. For a dissipative system, the conservation law is singular at a source or sink and this nonanalyticity gives rise to nontrivially different constants C for different streamlines [10]. If we can solve G(Xl, x2) = C to find the streamline equations in the form xz = f2(Xl, C) and/or xl = fl(x2, C), which requires that the implicit function theorem [11] is satisfied, then we can rewrite at least one of the differential equations in the form dG =0
dt = dxi/vi(xi, C).
(16)
The problem is therefore 'integrable', meaning that the complete solution can be reduced to two independent integrations. The complete solution therefore has the form
G(x 1, x2) = C t + Ci = gi(xi, C),
(17)
where C~ is a constant that is fixed by to and Xio. The following universal geometric interpretation is due to Lie and is based upon the construction of a coordinate transformation that exhibits integrability in its basic geometric form. In all that follows (and that precedes), we assume that the implicit function theorem [11] is satisfied so that the necessary manipulations can be carried out. Using the conservation law in the form C = G(xl, x2), we can write t + C1 = ga(xl, C) = F(xl, x2) (or t + C2 = H(Xl,X2)). Next, we define a new pair of coordinates ( u , v ) by the transformation
u = G(xa, xz) v = F(xl, x2).
(18)
In general, this tranformation preserves nothing but integrability. In the (u, v) system of coordinates, the differential equations have the form
du/dt = 0 dv/dt = 1.
(19)
Integrability and attractors
1973
Every two-dimensional flow is, therefore, universally equivalent by the coordinate transformation (18) to a translation at constant velocity: u=C
(20)
O = t "[- C 1 .
Integrable systems are, therefore, systems where the effects of interactions, the effects of binding and scattering, can be transformed away by a special choice of coordinates. That choice of coordinates is determined by the conservation laws than dictate integrability. In other words, the solution of an integrable system can be written in such a form to make it appear qualitatively as if there would be no interactions at all. That form is universally a constant velocity translation. Whether it is a translation, in the original (xl, x2) coordinates, on a spiral, a limit cycle or a hyperbola is completely irrelevant, as the equivalence is of a topologic rather than metric nature. Note that the transformation (18) maps sinks into the point at infinity, so that the flow into a sink is 'straightened out' into a collection of parallel streamlines. Again, the only thing that is generally preserved by this transformation is integrability itself. With a three-dimensional flow dt -
dx 1
dx2 - -- -
dx 3
V1
V2
V3
,
(21)
and where each component V~ of the vector field V depends upon all three variables (Xl, x2, x3), we are no longer guaranteed integrability by Lie's theorem. However, if the flow has one t-independent conservation law then by necessity it has two and the motion is integrable: let G1(Xl, x2, x3) = C1 an invariant of the flow (21). G1 therefore satisfies the first order linear partial differential equation dG
-
V.VG
=
0.
(22)
dt The differential equations (21) generate the characteristic curves (or just the characteristics) of the partial differential equation and are simply the curves along which G is constant. V ' V G 1 = 0 means that VG1 is perpendicular to V. By Lie's theorem, a second conservation law G2 must also exist, because with one conserved quantity G1 the flow (21) is confined to a two-dimensional subspace and is therefore integrable, G2 is just the equation of the streamlines on the surface G1 = constant (the roles played by G1 and G2 are generally interchangeable). The conserved quantity G 2 also must satisfy (22) so that V is perpendicular to both VGa and VG2. From this it follows that MV = VG1 × VG2
(23)
where M ( x ) is Jacobi's multiplier. Note that M = 1 for a conservative flow. An example of an integrable conservative flow is provided by Euler's top, which satisfies L1 = L 2 L 3 ( 1 / I 3 - 1/12) L2 = L I L 3 ( 1 / I 1 -
1/13)
L3 = L 2 L l ( 1 / I 2 - 1/11)
(24a)
where L and I denote angular m o m e n t u m and moment of inertia in the body-fixed frame. The two conserved quantities are simply the kinetic energy and angular m o m e n t u m
1974
J . L . McCAULEY
squared. With a change of variables these equations can be written as
~f= ayz = - bxz 2 = cxy,
(24b)
which we shall refer to in our discussion of the L o r e n z m o d e l in Part 3 below. If the t h r e e - d i m e n s i o n a l flow (21) in integrable, then the two invariants Gi(xl, x2, x3) = Ci must d e t e r m i n e two variables xi = f ( x k , C1, C2) in terms of the third, which we d e n o t e as xk. This can be substituted into dt = dxk/Vk to yield
dt = dx~/vk(xk, C~, C2).
(25)
t + D = gk(xk, CI, C2) = F ( x I , X2, X3) ,
(26)
F r o m this we obtain
where the last substitution follows f r o m using the two and C2. Following Lie and transforming to new Gl(Xl, x2, x3), v = G2(Xl, x2, x3), and w = F(Xl, x2, x3), the f o r m of a constant velocity translation along a single tt = C
1
o =C
2
w = t + D.
conservation laws to eliminate Ca variables ( u , v , w,) where u = we obtain the solution of (21) in axis:
(27)
A g a i n , integrability universally m e a n s that interactions can be t r a n s f o r m e d away: the flow is formally equivalent to a pure translation at constant speed. T h e formal generalization to flows in n dimensions is easy a l t h o u g h to m a k e it all completely rigorous is not: the use of the implicit function t h e o r e m only leads to the patching t o g e t h e r of coordinatized local n e i g h b o r h o o d s [11] where the flow in each n e i g h b o r h o o d is integrable. W e n o w formally generalize o u r heuristic definition of integrability: we shall call a system of n differential equations dx i
-
V/(xl . . . . .
x.)
(1)
dt integrable, or ' c o m p l e t e l y integrable', if the c o m p l e t e solution can be o b t a i n e d by p e r f o r m i n g n i n d e p e n d e n t integrations of n different exact differentials. T h e idea is that n - 1 of those exact differentials determine n - 1 t i m e - i n d e p e n d e n t conservation laws G1 . . . . . Gn-l. If there are n - 1 t - i n d e p e n d e n t conservation laws (or n t - d e p e n d e n t ones) then the flow is confined to a two-dimensional subspace of the n-dimensional phase space. T h e main point is that the n - 1 conservation laws must be well-enough b e h a v e d that n - 1 o f the variables (xi . . . . xn-i) are d e t e r m i n e d by the remaining nth variable x~ and the n - 1 constants Ci t h r o u g h the n - 1 conservation laws in o r d e r that the c o m p l e t e solution is expressible as n i n d e p e n d e n t integrations. T o state what one m e a n s by 'well-enough b e h a v e d ' , h o w e v e r , is difficult [11]. To be m o r e explicit, it must be true that the conservation laws ' d e t e r m i n e ' n - 1 of the variables x~ so that the n - 1 conservation laws can themselves be rewritten in the f o r m xj = fj(xn, CA. . . . . C, a). In this event, n - 1 of the coordinates xt are d e t e r m i n e d by the nth c o o r d i n a t e x~ (it does not m a t t e r which c o o r d i n a t e we call xn) along with the n - 1 constants of the m o t i o n C1 . . . . Cn_~. By the substitution of these last n - 1 equations into the e q u a t i o n of m o t i o n for the nth variable dxn/dt = Vn(xx . . . . xn), this remaining unsolved differential e q u a t i o n is r e d u c e d to the
Integrability and attractors
1975
one-dimensional form d x n / d t = vn(xn, C1 . . . . . Cn_l) and can therefore be solved by integrating the exact differential dt = d x n / v . . The complete solution of the system of equations then follows (i) from inverting the result of this last integration to find x n ( t ) = • n(t, C1 . . . . . C~-1), and (ii) by substituting this result back into the n - 1 equations Xj = fl(Xn, C 1. . . . . C n _ l ) and find x j ( t ) = dPj(t, C 1 , . . . , Cn-1) for j = 1 . . . . . n - 1. A system that can be solved by the reduction to n independent integrations is called 'integrable', or 'completely integrable'. Regardless of the details of the system of differential equations, complete integrability means having n - 1 conservation laws for a noncanonical system (one needs only n / 2 of them, where n is even, to demonstrate integrability for a canonical system). More generally, if a noncanonical system is integrable then there are n - 1 time-independent or n time-dependent conservation laws. In what follow, we discuss the case of integrable noncanonical flows with n - 1 time independent conservation laws. In the time-independent case, the conservation laws will have the form G i ( x l . . . . . x~) -- Ci where Ci is constant along a streamline because d G i / d t = O. Before the turn of the century it was not clear that this is not the most general form that a solution can have. Nonintegrability in the form of deterministic chaos was first discovered by Poincar6 in the same era. There are, therefore, two points to emphasize. The first is that 'nonintegrable' does not mean 'not solvable': most systems that are solvable by Picard's iterative method (constructed in the era of Lie and Poincar6) or by power series (combined with analytic continuation) do not have solutions that are expressible in the highly restricted form that defines 'integrability'. In other words, the so-called integrable flows are only a subclass of the much larger class of solvable flows. In Picard's method, which reduces to the power series method for autonomous systems, one must perform n coupled integrations infinitely many times, whereas in an integrable system the form of the solution is expressible in terms of n independent integrals by the use of conservation laws. In a nonintegrable system, there are fewer than n - 1 conservation laws so that there is no way to write the solution in the form of n independent integrals. Consider next the perturbation an integrable flow V by adding onto V a 'small' vector field ¢6V so that the resulting flow is nonintegrable. The original integrable flow is confined to a two-dimensional surface. If that flow is uniformly bounded for all positive times, then the surface (if orientable) is closed (for Euler's top, the closed surface is a sphere or an ellipse). In what follows we consider only integrable flows with uniformly bounded motion. Under a small nonintegrable perturbation, do all of the closed 2-surfaces disappear immediately or do some of them survive the perturbation? The analysis of this question leads to the study of attractors in phase space, to the idea of local integrability and then to nonintegrability. Global integrability is important for understanding canonical Hamiltonian systems, but the conditions for global integrability are generally not met by most dissipative systems whenever n/> 3, even for completely nonchaotic motions, For example, the Lorenz model Yc = o ( y - x ) = px - y - xz = -/3z + xy
(28)
has enough time-dependent conservation laws to be globally integrable only for the three very special sets of parameter values: (o, /3, p) = (1/2, 1, 0), (1, 2, 1/9), and (1/3, 1, p is arbitrary) [12]. In spite of the lack of global integrability, this model has only stable, uniformly bounded motions for r < o (0 + / 3 + 3)/(0 - / 3 - 1) whenever o - / 3 + 1 > 0. This is a much larger parameter range for the occurrence of stable motions (global motions
1976
J . L . McCAULEY
with nonpositive L y a p u n o v exponents) than can be explained on the basis of global integrability alone. It is interesting to ask why some nonintegrable systems are chaotic while others are not. It turns out that local integrability due to a one or two-dimensional attractor that is a differentiable curve or surface is enough to keep the motion regular, and so we review next how self-confinement and attractors arise through nonintegrable perturbations of integrable systems with bounded regular motion. 2. LOCAL INTEGRABILITY The idea of the foliation of the phase space of globally integrable systems with bounded motion by closed surfaces is a useful idea. If we use Euler's top as an example of global integrability, then we can understand how global conservation laws can 'foliate' a phase space: for each different value of the angular m o m e n t u m , the top's motion is confined to a sphere in the three-dimensional angular m o m e n t u m phase space (kinetic energy conservation then yields the streamlines on each sphere). In the three-dimensional phase space these spheres are all concentric with each other and there are no streamlines that are not confined to one of the spheres. The system of spheres is said to 'foliate' the phase space, which can be thought to geometrically as consisting of the concentric system of infinitely thin spherical shells. For the one-dimensional simple harmonic oscillator, the phase space is foliated by concentric ellipses, which are one-dimensional tori. Our first question is: when nonconservative terms are added to such a system, are all of the original foliations destroyed immediately or do some of them survive? We shall see that, with the right combination of damping and driving, one of the foliations can survive in distorted form. We now illustrate this by means of a Newtonian example. Consider the system .~ _~_ e ( X 2 ..[_ ~ 2 __ 1)X "~- X = 0 ,
(29)
which we can think of as a particle of unit mass attached to a spring with unit force constant, but subject also to a nonlinear dissipative force. In what follows, we take e > 0. The energy E = (22 + xZ)/2 changes according to
d E / d t = - e ~ 2 ( x 2 + 22 - 1).
(30)
F r o m Newton's law it follows that /~ = f ~ , where the nonconservative force is given by f = __E.~(X 2 + ~ 2 _ 1 ) . This force represents contributions from both external driving and dissipation, and the effect that dominates depends upon where you look in the phase plane. To see this, note that there is an energy flow through the system corresponding to positive damping (represented by a net dissipation) whenever r 2 = x 2 + 2 z > 1, but the system gains energy in the region r 2 < 1. Therefore, the phase plane is divided into two distinct regions of energy gain and energy loss by a circle at r = 1, because /~ = 0 on that circle. On the unit circle, we have a steady state where energy gain is balanced by dissipation so that the energy E is maintained constant. By inspection, the unit circle x 2 + .~2 = 1 is a solution of (1-28), and this solution is a stable limit cycle, a one-dimensional attractor. H e r e , all foliations of the conservative system with e = 0 are destroyed but one: only the circle at r = 1 survives. The torus at at r = 1 survives precisely because periodicity is maintained through energy balance there, and only there. H e r e , one of the original foliations survives perfectly. In the general case, for example in the case of the van der Pohl oscillator, one original torus survives the perturbation as a limit cycle, but only in distorted form. H e r e , we have a one-dimensional periodic attractor that is defined by a local, but not global, conservation law: energy balance is maintained on the attractor. In general, in the case of the van der Pohl oscillator for example, energy balance is not maintained pointwise
Integrability and attractors
1977
on the limit cycle but only on the average over one complete cycle of the periodic motion. On the limit cycle, local integrability follows from the periodicity of the solution. The generalization to flows with arbitrary limit cycles is, therefore, called the method of averaging [13], or the energy-balance method. In general, a limit cycle of a set of dissipative nonlinear equations is a periodic solution that divides the phase plane into regions of net average energy gain and energy loss only in an average way. For Newtonian flows the generalization of the above example is easy, and we state the analysis for the case of a single particle with energy E = (dx/dt)2/2 + V(x), although one can make a more general statement by starting with a Hamiltonian flow and perturbing it dissipatively. With x = x and y = dx/dt, a planar Newtonian flow is defined by 2=y
:9 = - V ' ( x ) + f(x, y)
(31)
so that dE/dt = fy. Suppose that f changes signs in an open region of phase space consistent with the requirement of a periodic solution. If we integrate around any closed curve C we get I .t+T
E(T) - E(O) = - 1
y(s)f(x(s), y(s))ds,
(32)
J t
and if C is a limit cycle, a periodic solution of the Newtonian flow (31) with period T, then this requires that E(T) = E(0), which means that t+ T
ft
y(s)f(x(s), y(s))ds = 0
(33)
on C. Stability would require E(T) - E(0) > 0 for any curve C' enclosed by C, and also that E(T) - E(0) < 0 for any curve C" that encloses C. For weekly dissipative Newtonian flows, this leads to a practical method for locating limit cycles approximately [13]. Let f(x, y ) = - e h ( x , y) where s is small. In this case, we require a potential V(x) such that, for some range of energies E m i n < E < E . . . . the motion of the conservative system (the e = 0 system) is bounded and is therefore periodic with period x.
= 2fi :(
dx
1,2(E - V ( x ) )
(34) 1/2'
where Xl and x2 are the turning points of V(x). To lowest order in e, we expect that the limit cycle is approximately, very roughly, given by one of these periodic solutions (we assume implicitly that it is possible to describe the limit cycle approximately from the standpoint of the original conservative solution by using perturbation theory). When an integrable conservative system with periodic orbits is perturbed weakly by nonlinear dissipative/driving terms that are able to satisfy energy balance over a single definite, repeatable time interval r, then a limit cycle is born through the destruction of global periodicity, leaving behind one particular distorted periodic orbit of the original conservative system as an attractor or repeller. The motion on the limit cycle is trivially integrable (local integrability) although the motion off of the attractor is generally nonintegrable but regular whenever n/> 3. In a nonlinear dissipative system that is far from an integrable conservative one, a limit cycle can be born from a certain kind of bifurcation as a control parameter is varied: a stable limit cycle can follow from the loss of stability of a spiral sink through a H o p f bifurcation. To show that this scenario also fits into the way of thinking based upon
1978
J . L . McCAULE¥
destruction of foliations, we consider again the flow (31) but with a control p a r a m e t e r /t where 2=y 5~ = ( ~ _
x 2 _
y2)y
_
x.
(35)
The origin is an equilibrium point, so consider the linear flow 2=y
2 =#Y -x
(36a)
near the origin. The eigenvalues, for - 2
(36b)
when /~ = 0 the origin is elliptic and the phase plane of the linear flow is foliated by 1-tori If /~ < 0, all tori are destroyed and the origin is a sink due to overdamping: the only r e m n a n t of the tori is the spiraling motion into the sink. When /~ > 0, the motion spirals out of a source at the origin, but in the full nonlinear system (35) a 1-torus at r = ~//~ survives as a stable limit cycle that attracts all other trajectories. A s / ~ passes through 0 we have a H o p f bifurcation that describes the change of stability from spiraling flow into a sink to spiraling flow out of a source and onto the limit cycle. Regardless of the details of how a limit cycle is born, it is always a closed streamline where the flow is 'locally integrable' because of periodicity: any closed curve is topologically equivalent to a circle, and so there is a continuous transformation that takes us from the dynamics on the limit cycle to dynamics on a circle where the differential equations have the explicitly integrable form ?=0 q~ = (o.
(37)
We have seen how local integrability follows as a consequence of an average maintenance of a local conservation law in a Newtonian system: the net energy is conserved for each cycle of the motion by means of average energy balance along the limit cycle. In two dimensions, and therfore in all of the examples considered above, even if the energy is not conserved the motion in the phase plane is by necessity integrable: the required conservation law is just the streamline equation as we have shown earlier in Section 1. H o w e v e r , when n ~> 3 the motion away from an attractor (repeller) is generally nonintegrable and only the motion on the attractor (repeller) can be integrable. Therefore, we have introduced our way of thinking about attractors via destruction of foliations from a convenient but superficial standpoint. In the next section, we show how the over-simplified point of view presented above for two-dimensional Newtonian flows can be correctly generalized and leads to the idea of trapping regions in phase space.
3. D A M P E D D R I V E N E U L E R - L A G R A N G E D Y N A M I C S
We want to arrive at the ideas of trapping and attractors in phase space in a general way within a class of Newtonian models. In order to study self-confinement and attractors of a u t o n o m o u s flows in three dimensions, or in any odd-dimensional phase space, we cannot start with Newtonian particle mechanics: there, except for time-dependent external driving (which we ignore here), the phase space dimension is always even.
Integrability and attractors
1979
Even if one is not a physicist, the method of averaging described in the last section gives a practical motivation for studying Newtonian flows. There, we have seen that it is useful to have a quantity that is conserved in the absence of friction and driving, but that can be balanced on the average over some region of phase space in the presence of combined dissipation and driving. This led to the idea of an attractor with dimension one, a periodic solution called a limit cycle that is a ghost of the destroyed foliations of the original conservative system• Lorenz started explicitly from this viewpoint in Section 2 of his very beautiful 1963 paper [14]. In what follows, we start with a class of conservative models that has two conservation laws that are broken by the introduction of dissipation and driving in order deduce that there is a finite region in phase space where every trajectory eventually becomes trapped. What follows is a generalization of Section 2 of Lorenz's paper that is motivated by the class of systems studied in ref. [15]. There, phase spaces of both even and odd dimension are possible, depending upon the complexity of the rigid body• For example, one can consider several rigid bodies linked together to form a single system, like a wheel and axle system with steering. A Newtonian starting point for the self-confinement of a damped driven nonlinear flow to a definite finite region in an n-dimensional phase space is provided by the E u l e r Lagrange equations [15] dJx + cx~ dt J~
- Nx,
(38)
which describe a conservative dynamical system in the n-dimensional (J1 . . . . . Jn) phase space (phase-volume preserving flow). H e r e , the inertia matrix I is assumed to be diagonal and constant and the c's are the structure constants of some Lie algebra (of rigid body systems more complicated than a top, for example) and so must obey the antisymmetry ~, ~. condition c,,, =--c~,~. The symbols J, 1, and N can, but need not, denote the angular momentum, moment of inertia, and torque or a rigid body problem: in the discussion that follows, all that we require is the mathematical structure of the E u l e r - L a g r a n g e equations (38): those equations may represent rigid body motions or some other physically motivated form of mathematical modeling. We also consider only the case where the vector N is a constant vector in what follows. That is, we study an autonomous system (38) with constant driving terms N;~ in the n-dimensional (J1 . . . . . Jn) phase space. Whenever the structure constants do not all vanish then the Ji are nonintegrable velocities in the original Newtonian system [15]. There are always at least two point functions that are left invariant by (38) whenever N~ = 0, and the study of the nonconservation of either of those quantities in the presence of damping can lead to the deduction of a trapping region in the (J1 . . . . . Jn) phase space. A trapping region is more general than the idea of an attractor: it is a region within which any and all trajectories become self-confined after a long enough finite time (the trapping time is trajectory-dependent). If there is an attractor, then it lies within the trapping region but the trapping region has a larger dimension than the attractor due to the phase-volume contraction law that reflects dissipation. For example, the trapping region always has dimension n, but fixed points and limit cycles have dimension 0 and 1, respectively. If n = 3 and N~ = 0 then the conservative flow (38) is integrable, but this feature is not essential in what follows and we assume that n t> 3. To obtain a model that can have an attractor of dimension greater than zero, we must generalize the E u l e r - L a g r a n g e dynamics (38) by adding damping (while keeping constant external driving), which we take here to be linear: •
.
__ J~Jv d J z + b z u J . + cZ.,, = N~. dt l;.
(39)
1980
J . L . McCAULEY
Since V-V-
3)° _ trb, (40) 9Jo if we require that t r b > 0 then we expect to find that very large scale motions will be damped, which is a necessary condition for an attractor to exist somewhere in the phase space. To have an attractor that is not a zero-dimensional sink, then the energy input must dominate dissipation locally at small scales to m a k e the smallest scale motions unstable. That is, to have an attractor the large scale motions must be unstable against further growth but the motions at small enough scales must also be unstable against further decay: these circumstances lead to attractors of dimension higher than zero in phase space, and it is helpful to use a quantity that is conserved whenever b = 0 in order formulate this notion quantitatively. We show show next that the conservative flow with N~ = 0 always has two invariants, so that the n = 3 flow without forcing is always integrable. The conservative flow (38) with N~ = 0 always leaves the kinetic energy T - (J12/I1 + J22/I2 + . . . + JnE/In)/2 invariant. The p r o o f of a second conserved quantity C is really only a disguised form of, the p r o o f that a Lie algebra has at least one Casimir o p e r a t o r [16]. This conserved quantity is a generalization of the square of the angular momentum. We can show directly by differentiation that the quantity
C = gP°Jflo
(41)
is left invariant by the undriven conservative flow dJx
v J~Jv
+ cz¢~,,
- 0,
(42)
dt where by gVO we m e a n the inverse of the matrix t~ ~ g~v = c~aCv/~.
(43)
H a m m e r m e s h states that the matrix g provides a metric for defining inner products in linear representations of Lie algebras. F r o m the differential equations (42) we obtain
dC_ dt
2gooC'p, jflvjo
(44)
I,
so that dC/dt = 0 follows if the rank three tensor gV°c~oo is antisymmetric under exchange of the two indices a and v. In fact, this tensor is antisymmetric under exchange of any two of the three indices (o, ~, v) [16]. Therefore, dC/dt = 0 and C is left invariant by the conservative flow (42). We have two conservation laws that describe two ellipsoids in phase space, or two hyperellipsoids if n > 3. D e n o t e the kinetic energy ellipsoid by E1 and the 'angular m o m e n t u m ' ellipsoid described by the conserved quantity C by E2. In the conservative flow (where b = 0), the invariant curves (the solutions of (42)) lie in intersections of E 1 and E2 and both these ellipsoids foliate the phase space. In this case, each streamline of the flow is self-confined to an n - 2-dimensional hypersurface in the n-dimensional phase space. When b :~ 0, then (even if N = 0) the foliations of phase space are destroyed (invariant curves of (39) generally cannot lie on the intersections of Ea with Ez). Next, we ask what happens when both driving and dissipation are included. Consider the surface E2 for large values of J, and assume that t r b > 0. Think of any point on E2 as an initial condition for the flow so that the ellipsoid changes rigidly in size
Integrability and attractors
1981
as the single point on E 2 follows the streamline: then for large enough J we expect that dC/dt < 0, while E 2 contracts rigidly according to the law dC
.lg-lbj + NJ. (45) dt Since this expression is not positive definite, we cannot conclude that C decreases eventually to zero as t goes to infinity. In fact, in order to get a trapping region that contains an attractor with dimension greater than zero, the action of the matrix g - l b combined with that of the constant vector N in (45) must be such as to divide the phase space into regions of net dissipation at for large enough J and regions of net energy input at small enough J. Let us assume that the matrix b has been constructed so that this is the case. Then a s E 2 contracts rigidly while one point on E2 follows a streamline, E2 will eventually intersect a third surface E3 that is defined by the condition that dC/dt -- 0 at all points along the intersection. Inside the surface E3, dC/dt > 0 and so any trajectory that enters E 3 tends to be repelled outward again. If these expectations are born out, then we can define the trapping region optimally as the smallest ellipsoid EEc that contains the (still unknown) surface E 3. Every streamline must enter the n-dimensional region Ezc in finite time, and once having entered cannot escape again (this is self-trapping, or self-confinement). For one set of parameter values, we also expect to find at least one attracting point set somewhere within the ellipsoidal trapping region E2c. To find E3 and E2c explicitly, we follow Lorenz [14] and define next a vector e with components ev that solve the linear equations -
(b' + b')e = N,
(46)
where b' = g - i b . In this case, direct comparison with (45) yields dC
- ~b'e - ( f ' Z - e ) b ' ( J -
e).
(47)
dt This means that
dC/dt
= 0 on the surface E3 defined by ~b'e = ( J - e ) b ' ( J -
e).
(48)
To get a trapping region (a closed surface E3) it is sufficient to have a positive definite matrix b', which explicitly guarantees that t r b ' > 0 and also makes E3 an ellipsoid or hyperellipsoid. If follows from (47) that dC/dt < 0 outside the ellipsoid E3 (attraction toward E3) but dC/dt > 0 inside the ellipsoid E3 (expulsion from E3), but the surface of the ellipsoid E3 usually does not contain an invariant curve (invariant set, or just 'solution') of the damped-driven flow (39). We can define the trapping region Ezc to be the smallest ellipsoid E2 that completely contains the smaller ellipsoid E 3. Note that if N~ = 0 in equations (39), then a set of equations with finite, constant values of N~ can always be obtained by a translation L[ = Li + ai in phase space. To give an example, we return to a specific class of damped driven E u l e r - L a g r a n g e flows. If n = 3 and c ~ = e~u~, which represents the motion of a rigid body with three moments of inertia in three dimensions, then C = J~ + J~ + J~ is just the square of the angular momentum. More generally, we can assume that n/> 3 but still require that c~u~J~J~J~= 0 (this condition defines the class of models considered in Section 2 of Lorenz's paper). In this case, it still follows that C = J~ + . . . + J2n, and so the trapping region Ezc is the smallest hyper-sphere, the sphere with radius ~/C~, that contains the entire ellipsoid E3. We are now going to show that the Lorenz model falls into this category. The Lorenz model is normally derived from the continuum equations of hydrodynamics
1982
J . L . MCCAULEY
and energy transport via a mode expansion and truncation, but we are going to show that the Lorenz model can also be understood as a damped, driven symmetric top. The Lorenz model is defined by the flow = o(y = px =
-
-/3z
x) y +
-
xz
(28)
xy.
Normally, one throws away nonlinear terms in order to identify a linear system. Here, we note that a conservative Euler top flow is obtained by deleting all linear terms in (28), yielding 2=0 .F = - - X Z
(49)
= xy.
This flow represents a torque-free symmetric top with 11 = 12 ~ 13. For example, if we write the equations of Euler's top in the form = ayz p
=
- bxz
2
= cxy,
(24b)
then x 2 + y2 + z 2 = constant and with a = 0, b = c = 1 (symmetric top with 11 = I 2 4= 13), and linear damping/driving added on corresponding to the matrix b =
1 0
-
,
(50)
then we get exactly the Lorenz model. The Lorenz model is, therefore, formally equivalent to a certain damped and driven symmetric top, and to get a trapping region in phase space we need only t r b = o + / 3 + l > 0 along with the condition that o, /3, and p are all positive. The latter condition is specifically needed in order to get the ellipsoid E3, and the trapping region is, therefore, the smallest sphere that contains that ellipsoid. In general, truncations to finitely many modes of the equations of fluid mechanics and energy transport give rise to equations of motion of the form (39) of the damped E u l e r - L a g r a n g e equations of rigid body motion [17]. Presumably, the coincidence reflects rotational symmetry in the original hydrodynamics equations. Because training in differential equations normally concentrates strongly on linear systems, writers on nonlinear dynamics tend to emphasize that the addition of nonlinear terms to linear equations can lead to complicated behavior, even to deterministic chaos. However, a viewpoint that takes linear differential equations as the starting point for discussions of nonlinear dynamics is not very useful for physics. Some writers on economics even express the misconception that linearity and equilibrium are the main methods of physics [18]. To the contrary, most of the standard textbook problems of classical mechanics are conservative and integrable but are nonlinear. A better starting point for studying differential equations would therefore begin with the study of nonlinear integrable systems, or at least with nonlinear conservative ones [10], which reflects the emphasis in differential equations at the turn of the century before the time of the linear Schr6dinger equation. Here, we have approached nonlinear dynamics from that standpoint and have
Integrability and attractors
1983
shown that the addition of linear damping to nonlinear conservative systems can lead to self-confinement. As is well known, the trapping regions of such systems can contain chaotic attractors [14]. In pure mathematics, a writer is not self-constrained to restrict his studies to models that are motivated by physics, although dynamical systems theory grew out of physics and even Poincar6 seems to have been motivated in large part by problems from physics. Newton's laws of motion (and their relativistic and quantal generalizations) are the only known universal laws of motion. The empirically-based law of inertia discovered by Galileo and Descartes is the foundation for Newton's universal laws of motion, but there is no equivalent of either Galilean invariance of Newtonian universality in economics or sociology, which is why those and other similar fields have not been reduced to mathematics in any useful way that is consistent with the wide range of their empirical observations. Those fields have not given birth to a Kepler, a Galileo, or a Newton, and no amount of computer simulation based upon arbitrary models from dynamical systems theory, probability theory, or any combination thereof can alleviate the fact. So far, only physics and physics-related fields have gone beyond the stage of the epicycles of scholasticism. Descartes' D r e a m [19] to the contrary, it is not at all evident that the modern equivalent of epicycles, purely empirical modeling without any fundamental underlying laws of motion, can be eliminated from fields like economics, sociology and psychology. Descartes, who advocated the mathematical formulation of all observable phenomena, was motivated by the law of inertia and the action-reaction principle, which he apparently understood independently of Galileo and Newton [20]. Primarily, he used the law of inertia and the action-reaction principle to construct a form of neoscholasticism whereby a purely theoretical discovery of truth was thought to be possible without the aid of adequate careful observation and measurement. Some of his ideas about mechanics were right but most of them were wrong. The belief that sociological phenomena like economics can be explained by mathematics by using computers, but without adequate universal and precise laws that account for a wide range of observations without juggling a lot of arbitrary parameters freely, reflects Descartes' D r e a m but also resembles the medieval state of science before the law of inertia and Kepler's laws were discovered, and before the Newtonian synthesis. Epicycles by any other name are still epicycles. Acknowledgement--I am grateful to Egil Andersen for introducing me to the historical survey of Lie's ideas in Bell's interesting work [1]. I am also grateful to Julian Palmore for introducing me to ref. [9], where a systematic approach to return maps is established.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
E. T. Bell, The Development of Mathematics. McGraw-Hill, New York (1945). P. J. Olver, Applications of Lie Groups to Differential Equations. Springer, New York (1993). V. I. Arnol'd, Mathematical Methods of Classical Mechanics. Springer, Berlin (1978). S. Lie, Forhand. Vid.-Sels. Christiania 198 (1874); 1 (1875). F. Klein, Vorlesungen iiber hOhere Geometrie, Dr. Aufl. Springer, Berlin (1968). J. E. Marsden, Lectures on Mechanics. Cambridge University Press, Cambridge (1992). J. L. McCauley, Chaos, Dynamics, and Fractals: an Algorithmic Approach to Deterministic Chaos. Cambridge University Press, Cambridge (1993). E. L. Ince, Ordinary Differential Equations. Dover (1956). S. A. Burns and J. I. Palmore, Physica D37, 83-90 (1989). J. L. McCauley, Classical Dynamics: Flows, Transformations, lntegrability and Chaos. Cambridge University Press, Cambridge (1995). (To be published) B, A. Dubrovin, A. T. Fomenko and S. P. Novikov Modern Geometry--Methods and Applications. Part 1. The Geometry of Surfaces, Transformation Groups, and Fields, transl, by R. G. Burns, Chapters 2 & 5. Springer, New York (1984). M. Tabor, Chaos and Integrability in Nonlinear Dynamics. Wiley, New York (1989).
1984
J . L . McCAuLEY
13. D. W. Jordan and P. Smith, Introduction to Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford (1977). 14. E. N. Lorenz Deterministic nonperiodic flow in University in Chaos, edited by P. Cvitanovic. Adam Hi|ger Ltd., Bristol (1984). 15. J. L. McCauley, Nonintegrable velocities and nonholonomic coordinates in dynamics, preprint (1994). 16. M. Hammermesh, Group Theory. Addison-Wesley, Reading, MA (1962). 17. M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics. Springer, New York (1987). 18. E. E. Peters, Chaos and Order in the Capital Markets. Wiley, New York (1991). 19. P. J. Davis and R. Hersh, Descartes' Dream: The World According to Mathematics. Harcourt Brace Janovich, San Diego (1986). 20. R. Dugas, Mechanics in the Seventeenth Century, From the Scholastic Antecedents to Classical Thought, transl. from French by Freda Jacquot. Central Book Co., New York (1958).
Pergamon
0960-0779(94)00160-X
Integrability and Attractors JOSEPH L. McCAULEY
Physics Department, University of Houston, Houston, TX 77204-5506, USA (Received 16 June 1994) Abstract--We discuss global and local integrability and also attractors from a physicist's standpoint. A local Lie method is used to form an analytic and geometric picture of integrability via conservation laws. The starting point is Lie's proof that two-dimensional flows, even dissipative ones, have a conservation law and so are integrable. Through conservation laws, integrable higher-dimensional flows reduce to the two-dimensional case. We consider a class of models defined by the EulerLagrange equations for a set of nonintegrable velocities (nonholonomic coordinates). The destruction of the global conservation laws of this conservative system by the inclusion of linear damping and constant external driving leads to self-confinement and attractors in phase space. In particular, the Lorenz model belongs to this class and can be seen as a damped, driven symmetric top.
1. INTEGRABLE FLOWS IN PHASE SPACE
We study integrability by following the consequences of an elementary theorem by Sophus Lie. According to Bell [1], Lie showed that every one-parameter infinitesimal transformation acting on two variables defines a one-parameter group, and that any such group has an invariant. This means that two-dimensional flows in phase space, whether conservative or dissipative, have a conservation law. This fact is not pointed out in standard texts on differential equations or mechanics. To understand Lie's theorem, it is only necessary to establish the language of phase flows along with a dictionary to translate back and forth between the language of groups [2] and the language of differential equations [3]. The discussion that follows is heuristic in the spirit of Lie [4] and Klein [5] (and in the spirit of physics) rather than rigorous in the spirit of modern abstract mathematics [6]. By a flow, we mean an autonomous system of n (generally coupled and nonlinear) differential equations dxi --
V/(x1
.....
dt
Xn)
(1)
where the solutions are finite at all finite times (any and all singularities are confined to the imaginary time axis [3]). Phase space is the n-dimensional space indicated by the local coordinates (xl . . . . , x,). By a conservative flow we mean one that preserves the volume element dff2 = dxa • •. dx, in phase space, so that n
°
V" V = z Oxi = O. i=10xi
(2)
1970
J.L. McCAuLEV
In an arbitrary conservative system, the volume element is the only quantity that is left invariant by the flow. That is, in a completely arbitrary conservative flow there are generally no conservation laws that are point functions. Conservation laws are functions of the phase space variables and the time, Gg(x 1. . . . . x n, t) = constant, that satisfy d Gi
3G i
-- V " V G i -b - -
dt
= O.
~t
(3)
By a nonconservative or 'dissipative' system we m e a n any system where V . V fails to vanish identically over all of phase space. A typical dissipative system has no invariant at all, neither the phase volume element nor a solution of the partial differential equation (3). The integration of the flow (1) studied as a collection of infinitesimal transformations (~Xi ~- V/(X1. . . . .
Xn)f~t
(4)
defines a one p a r a m e t e r continuous group of transformations Xn(to) , t -- to)
Xi(t ) = lPi(Xl(tO) . . . . .
(5)
from coordinates (xl(to) . . . . . xn(to)) to ( x l ( t ) . . . . . x~(t)) as the group p a r a m e t e r is varied from to to t. In the language of differential equations, (5) is the solution at a later time t and (x1(to) . . . . . xn(to)) is the initial condition at to. The time t is a canonical group p a r a m e t e r , which means that it obeys an additive parameter-combination law. Noncanonical group p a r a m e t e r s obey nonlinear p a r a m e t e r - c o m b i n a t i o n laws. For a one p a r a m e t e r transformation on two variables, a canonical p a r a m e t e r can always be found by integration [7]. In terms of the canonical parameter, the infinitesimal transformations of the group are defined by the differential equations dxl -
Vl(xl, x2)
dt dx2 -
dt
g2(xl, x2)
(6)
which also describe a flow in the (x~, x2) phase space. Lie's t h e o r e m is now easy to demonstrate heuristically. By rewriting the flow equations in the form dt = d x l / V 1 = dx2/V2
(7)
we can obtain the time-independent differential form V2(x 1, x 2 ) d x l -- VI(X 1, x2)dx2 = 0.
(8)
E v e r y differential form in only two variables is either integrable or else has an integrating factor M ( X l , x2) that makes it integrable, so that we can integrate the differential form M ( V 2 dXl - VI dx2) = d G
(8b)
to obtain G ( X a , x 2 ) = C, which is a conservation law. For example, this means that even two-dimensional Newtonian flows where the mechanical energy E is not constant have a conservation law. Before continuing, we pause to state the condition under which the conservation law for a dissipative system exists, which is simply the condition that an integrating factor M exists. We can rewrite the differential form 1/2 dXl - V1 dx2 = 0 as the one-dimensional equation dx2
V2(Xl, x2) -
dXl
-
Vl(xl, x2)
F ( x l , x2)
(9)
Integrability and attractors
1971
whose solution satisfies the integral equation
f(s, x2(s))ds.
x2 = X2o +
(9b)
Xlo
According to Picard [8], this equation can be solved by the method of successive approximations so long as F satisfies a Lipshitz condition. A sufficient condition is that that ratio of the components of the velocity in phase V2/V1 must be at least once continuously differentiable with respect to the variable x~. This requires that we avoid integrating over regions that contain equilibria or other zeros of Vs. The resulting solution of the integral equation yields x2 as a function of xl. A qualitatively similar result follows from solving the conservation law G(xl, x2) = C to find x2 = f(x~, C), which is just the statement that the conservation law determines x2 as a function of Xl and C. In particular, differentiating the conservation law once yields
dG = 3G dx~ + 3G dx2 = 0, 3xl
(10)
9x2
and comparison with the differential form V2 dxl - V1 dx2
=
0 then shows that
/
- 3G/3---G-G=
V2/V1.
(lla)
3Xl/3X2 This last equation is satisfied only if
V 1 : - M 3G
and 172 = M olG, (llb) 3x2 ~Xl which guarantees the existence of the integrating factor M. In other words, the existence of the conservation law follows from satisfying Picard's conditions for a unique solution of the differential equation (9). This guarantees, by differentiation, that the integrating factor M exists. Except in a few special cases, there is no systematic way to construct the integrating factor M(x~, x2) even though it is guaranteed formally to exist. Notice that the condition that the flow is phase volume preserving, ~V 2
3V 1 -
9x2
- - ,
(llc)
3Xl
is just the condition that the differential form V2(xl, x 2 ) d x l - Vl(Xl, x2)dx2 = 0 is exact without any need for an integrating factor. This means that the velocity field of a conservative two-dimensional flow can always be obtained from a stream function G:
V = (V1, V2) : ( ~x29G'9~1.
(12)
By inspection, the conservative system is Hamiltonian with Hamiltonian G and canonically conjugate variables (q, p ) = (x2, &), but the collapse of an integrable conservative flow into canonical form is peculiar to two dimensions and generally is not true for integrable conservative flows in higher dimensions: in particular, every three-dimensional integrable flow is noncanonical although it may still be Hamiltonian on a Poisson manifold that is not symplectic [10]. As an example of a conservation law for a dissipative system, consider the damped simple harmonic oscillator de' + /3:~ + wax = 0.
(13)
1972 With x I
J.L. McCAULEY =
X and x 2
-~-
dx/dt the corresponding differential form (fix2 + a~zxl)dxa + x2 dx2 = 0
(14)
is not exact but is guaranteed to have an integrating factor M that yields a time-independent conservation law G(x, dx/dt) = constant via the condition dG -
dt
M((13x2 + O~Xl)~l + x2~2) = 0.
(15)
The conservation law for this specific case was originally discovered and constructed by Palmore et al. [9]. In all cases, the conservation law G(x, d x / d t ) = C is simply the equation that describes that family of streamlines: different constants C correspond to different streamlines within the same family. For the damped oscillator, the streamlines are a family of spirals. Lie's result explains qualitatively why two-dimensional dissipative as well as conservative flows have conservation laws: the condition C --- G ( X l o , x2o ) fixes the constant C in terms of the two initial conditions Xio at time t = to and is simply the equation of the family of streamlines of a flow, G(Xl, x2) = C, in the phase plane. Each streamline corresponds to a different value of the constant C. For a dissipative system, the conservation law is singular at a source or sink and this nonanalyticity gives rise to nontrivially different constants C for different streamlines [10]. If we can solve G(Xl, x2) = C to find the streamline equations in the form xz = f2(Xl, C) and/or xl = fl(x2, C), which requires that the implicit function theorem [11] is satisfied, then we can rewrite at least one of the differential equations in the form dG =0
dt = dxi/vi(xi, C).
(16)
The problem is therefore 'integrable', meaning that the complete solution can be reduced to two independent integrations. The complete solution therefore has the form
G(x 1, x2) = C t + Ci = gi(xi, C),
(17)
where C~ is a constant that is fixed by to and Xio. The following universal geometric interpretation is due to Lie and is based upon the construction of a coordinate transformation that exhibits integrability in its basic geometric form. In all that follows (and that precedes), we assume that the implicit function theorem [11] is satisfied so that the necessary manipulations can be carried out. Using the conservation law in the form C = G(xl, x2), we can write t + C1 = ga(xl, C) = F(xl, x2) (or t + C2 = H(Xl,X2)). Next, we define a new pair of coordinates ( u , v ) by the transformation
u = G(xa, xz) v = F(xl, x2).
(18)
In general, this tranformation preserves nothing but integrability. In the (u, v) system of coordinates, the differential equations have the form
du/dt = 0 dv/dt = 1.
(19)
Integrability and attractors
1973
Every two-dimensional flow is, therefore, universally equivalent by the coordinate transformation (18) to a translation at constant velocity: u=C
(20)
O = t "[- C 1 .
Integrable systems are, therefore, systems where the effects of interactions, the effects of binding and scattering, can be transformed away by a special choice of coordinates. That choice of coordinates is determined by the conservation laws than dictate integrability. In other words, the solution of an integrable system can be written in such a form to make it appear qualitatively as if there would be no interactions at all. That form is universally a constant velocity translation. Whether it is a translation, in the original (xl, x2) coordinates, on a spiral, a limit cycle or a hyperbola is completely irrelevant, as the equivalence is of a topologic rather than metric nature. Note that the transformation (18) maps sinks into the point at infinity, so that the flow into a sink is 'straightened out' into a collection of parallel streamlines. Again, the only thing that is generally preserved by this transformation is integrability itself. With a three-dimensional flow dt -
dx 1
dx2 - -- -
dx 3
V1
V2
V3
,
(21)
and where each component V~ of the vector field V depends upon all three variables (Xl, x2, x3), we are no longer guaranteed integrability by Lie's theorem. However, if the flow has one t-independent conservation law then by necessity it has two and the motion is integrable: let G1(Xl, x2, x3) = C1 an invariant of the flow (21). G1 therefore satisfies the first order linear partial differential equation dG
-
V.VG
=
0.
(22)
dt The differential equations (21) generate the characteristic curves (or just the characteristics) of the partial differential equation and are simply the curves along which G is constant. V ' V G 1 = 0 means that VG1 is perpendicular to V. By Lie's theorem, a second conservation law G2 must also exist, because with one conserved quantity G1 the flow (21) is confined to a two-dimensional subspace and is therefore integrable, G2 is just the equation of the streamlines on the surface G1 = constant (the roles played by G1 and G2 are generally interchangeable). The conserved quantity G 2 also must satisfy (22) so that V is perpendicular to both VGa and VG2. From this it follows that MV = VG1 × VG2
(23)
where M ( x ) is Jacobi's multiplier. Note that M = 1 for a conservative flow. An example of an integrable conservative flow is provided by Euler's top, which satisfies L1 = L 2 L 3 ( 1 / I 3 - 1/12) L2 = L I L 3 ( 1 / I 1 -
1/13)
L3 = L 2 L l ( 1 / I 2 - 1/11)
(24a)
where L and I denote angular m o m e n t u m and moment of inertia in the body-fixed frame. The two conserved quantities are simply the kinetic energy and angular m o m e n t u m
1974
J . L . McCAULEY
squared. With a change of variables these equations can be written as
~f= ayz = - bxz 2 = cxy,
(24b)
which we shall refer to in our discussion of the L o r e n z m o d e l in Part 3 below. If the t h r e e - d i m e n s i o n a l flow (21) in integrable, then the two invariants Gi(xl, x2, x3) = Ci must d e t e r m i n e two variables xi = f ( x k , C1, C2) in terms of the third, which we d e n o t e as xk. This can be substituted into dt = dxk/Vk to yield
dt = dx~/vk(xk, C~, C2).
(25)
t + D = gk(xk, CI, C2) = F ( x I , X2, X3) ,
(26)
F r o m this we obtain
where the last substitution follows f r o m using the two and C2. Following Lie and transforming to new Gl(Xl, x2, x3), v = G2(Xl, x2, x3), and w = F(Xl, x2, x3), the f o r m of a constant velocity translation along a single tt = C
1
o =C
2
w = t + D.
conservation laws to eliminate Ca variables ( u , v , w,) where u = we obtain the solution of (21) in axis:
(27)
A g a i n , integrability universally m e a n s that interactions can be t r a n s f o r m e d away: the flow is formally equivalent to a pure translation at constant speed. T h e formal generalization to flows in n dimensions is easy a l t h o u g h to m a k e it all completely rigorous is not: the use of the implicit function t h e o r e m only leads to the patching t o g e t h e r of coordinatized local n e i g h b o r h o o d s [11] where the flow in each n e i g h b o r h o o d is integrable. W e n o w formally generalize o u r heuristic definition of integrability: we shall call a system of n differential equations dx i
-
V/(xl . . . . .
x.)
(1)
dt integrable, or ' c o m p l e t e l y integrable', if the c o m p l e t e solution can be o b t a i n e d by p e r f o r m i n g n i n d e p e n d e n t integrations of n different exact differentials. T h e idea is that n - 1 of those exact differentials determine n - 1 t i m e - i n d e p e n d e n t conservation laws G1 . . . . . Gn-l. If there are n - 1 t - i n d e p e n d e n t conservation laws (or n t - d e p e n d e n t ones) then the flow is confined to a two-dimensional subspace of the n-dimensional phase space. T h e main point is that the n - 1 conservation laws must be well-enough b e h a v e d that n - 1 o f the variables (xi . . . . xn-i) are d e t e r m i n e d by the remaining nth variable x~ and the n - 1 constants Ci t h r o u g h the n - 1 conservation laws in o r d e r that the c o m p l e t e solution is expressible as n i n d e p e n d e n t integrations. T o state what one m e a n s by 'well-enough b e h a v e d ' , h o w e v e r , is difficult [11]. To be m o r e explicit, it must be true that the conservation laws ' d e t e r m i n e ' n - 1 of the variables x~ so that the n - 1 conservation laws can themselves be rewritten in the f o r m xj = fj(xn, CA. . . . . C, a). In this event, n - 1 of the coordinates xt are d e t e r m i n e d by the nth c o o r d i n a t e x~ (it does not m a t t e r which c o o r d i n a t e we call xn) along with the n - 1 constants of the m o t i o n C1 . . . . Cn_~. By the substitution of these last n - 1 equations into the e q u a t i o n of m o t i o n for the nth variable dxn/dt = Vn(xx . . . . xn), this remaining unsolved differential e q u a t i o n is r e d u c e d to the
Integrability and attractors
1975
one-dimensional form d x n / d t = vn(xn, C1 . . . . . Cn_l) and can therefore be solved by integrating the exact differential dt = d x n / v . . The complete solution of the system of equations then follows (i) from inverting the result of this last integration to find x n ( t ) = • n(t, C1 . . . . . C~-1), and (ii) by substituting this result back into the n - 1 equations Xj = fl(Xn, C 1. . . . . C n _ l ) and find x j ( t ) = dPj(t, C 1 , . . . , Cn-1) for j = 1 . . . . . n - 1. A system that can be solved by the reduction to n independent integrations is called 'integrable', or 'completely integrable'. Regardless of the details of the system of differential equations, complete integrability means having n - 1 conservation laws for a noncanonical system (one needs only n / 2 of them, where n is even, to demonstrate integrability for a canonical system). More generally, if a noncanonical system is integrable then there are n - 1 time-independent or n time-dependent conservation laws. In what follow, we discuss the case of integrable noncanonical flows with n - 1 time independent conservation laws. In the time-independent case, the conservation laws will have the form G i ( x l . . . . . x~) -- Ci where Ci is constant along a streamline because d G i / d t = O. Before the turn of the century it was not clear that this is not the most general form that a solution can have. Nonintegrability in the form of deterministic chaos was first discovered by Poincar6 in the same era. There are, therefore, two points to emphasize. The first is that 'nonintegrable' does not mean 'not solvable': most systems that are solvable by Picard's iterative method (constructed in the era of Lie and Poincar6) or by power series (combined with analytic continuation) do not have solutions that are expressible in the highly restricted form that defines 'integrability'. In other words, the so-called integrable flows are only a subclass of the much larger class of solvable flows. In Picard's method, which reduces to the power series method for autonomous systems, one must perform n coupled integrations infinitely many times, whereas in an integrable system the form of the solution is expressible in terms of n independent integrals by the use of conservation laws. In a nonintegrable system, there are fewer than n - 1 conservation laws so that there is no way to write the solution in the form of n independent integrals. Consider next the perturbation an integrable flow V by adding onto V a 'small' vector field ¢6V so that the resulting flow is nonintegrable. The original integrable flow is confined to a two-dimensional surface. If that flow is uniformly bounded for all positive times, then the surface (if orientable) is closed (for Euler's top, the closed surface is a sphere or an ellipse). In what follows we consider only integrable flows with uniformly bounded motion. Under a small nonintegrable perturbation, do all of the closed 2-surfaces disappear immediately or do some of them survive the perturbation? The analysis of this question leads to the study of attractors in phase space, to the idea of local integrability and then to nonintegrability. Global integrability is important for understanding canonical Hamiltonian systems, but the conditions for global integrability are generally not met by most dissipative systems whenever n/> 3, even for completely nonchaotic motions, For example, the Lorenz model Yc = o ( y - x ) = px - y - xz = -/3z + xy
(28)
has enough time-dependent conservation laws to be globally integrable only for the three very special sets of parameter values: (o, /3, p) = (1/2, 1, 0), (1, 2, 1/9), and (1/3, 1, p is arbitrary) [12]. In spite of the lack of global integrability, this model has only stable, uniformly bounded motions for r < o (0 + / 3 + 3)/(0 - / 3 - 1) whenever o - / 3 + 1 > 0. This is a much larger parameter range for the occurrence of stable motions (global motions
1976
J . L . McCAULEY
with nonpositive L y a p u n o v exponents) than can be explained on the basis of global integrability alone. It is interesting to ask why some nonintegrable systems are chaotic while others are not. It turns out that local integrability due to a one or two-dimensional attractor that is a differentiable curve or surface is enough to keep the motion regular, and so we review next how self-confinement and attractors arise through nonintegrable perturbations of integrable systems with bounded regular motion. 2. LOCAL INTEGRABILITY The idea of the foliation of the phase space of globally integrable systems with bounded motion by closed surfaces is a useful idea. If we use Euler's top as an example of global integrability, then we can understand how global conservation laws can 'foliate' a phase space: for each different value of the angular m o m e n t u m , the top's motion is confined to a sphere in the three-dimensional angular m o m e n t u m phase space (kinetic energy conservation then yields the streamlines on each sphere). In the three-dimensional phase space these spheres are all concentric with each other and there are no streamlines that are not confined to one of the spheres. The system of spheres is said to 'foliate' the phase space, which can be thought to geometrically as consisting of the concentric system of infinitely thin spherical shells. For the one-dimensional simple harmonic oscillator, the phase space is foliated by concentric ellipses, which are one-dimensional tori. Our first question is: when nonconservative terms are added to such a system, are all of the original foliations destroyed immediately or do some of them survive? We shall see that, with the right combination of damping and driving, one of the foliations can survive in distorted form. We now illustrate this by means of a Newtonian example. Consider the system .~ _~_ e ( X 2 ..[_ ~ 2 __ 1)X "~- X = 0 ,
(29)
which we can think of as a particle of unit mass attached to a spring with unit force constant, but subject also to a nonlinear dissipative force. In what follows, we take e > 0. The energy E = (22 + xZ)/2 changes according to
d E / d t = - e ~ 2 ( x 2 + 22 - 1).
(30)
F r o m Newton's law it follows that /~ = f ~ , where the nonconservative force is given by f = __E.~(X 2 + ~ 2 _ 1 ) . This force represents contributions from both external driving and dissipation, and the effect that dominates depends upon where you look in the phase plane. To see this, note that there is an energy flow through the system corresponding to positive damping (represented by a net dissipation) whenever r 2 = x 2 + 2 z > 1, but the system gains energy in the region r 2 < 1. Therefore, the phase plane is divided into two distinct regions of energy gain and energy loss by a circle at r = 1, because /~ = 0 on that circle. On the unit circle, we have a steady state where energy gain is balanced by dissipation so that the energy E is maintained constant. By inspection, the unit circle x 2 + .~2 = 1 is a solution of (1-28), and this solution is a stable limit cycle, a one-dimensional attractor. H e r e , all foliations of the conservative system with e = 0 are destroyed but one: only the circle at r = 1 survives. The torus at at r = 1 survives precisely because periodicity is maintained through energy balance there, and only there. H e r e , one of the original foliations survives perfectly. In the general case, for example in the case of the van der Pohl oscillator, one original torus survives the perturbation as a limit cycle, but only in distorted form. H e r e , we have a one-dimensional periodic attractor that is defined by a local, but not global, conservation law: energy balance is maintained on the attractor. In general, in the case of the van der Pohl oscillator for example, energy balance is not maintained pointwise
Integrability and attractors
1977
on the limit cycle but only on the average over one complete cycle of the periodic motion. On the limit cycle, local integrability follows from the periodicity of the solution. The generalization to flows with arbitrary limit cycles is, therefore, called the method of averaging [13], or the energy-balance method. In general, a limit cycle of a set of dissipative nonlinear equations is a periodic solution that divides the phase plane into regions of net average energy gain and energy loss only in an average way. For Newtonian flows the generalization of the above example is easy, and we state the analysis for the case of a single particle with energy E = (dx/dt)2/2 + V(x), although one can make a more general statement by starting with a Hamiltonian flow and perturbing it dissipatively. With x = x and y = dx/dt, a planar Newtonian flow is defined by 2=y
:9 = - V ' ( x ) + f(x, y)
(31)
so that dE/dt = fy. Suppose that f changes signs in an open region of phase space consistent with the requirement of a periodic solution. If we integrate around any closed curve C we get I .t+T
E(T) - E(O) = - 1
y(s)f(x(s), y(s))ds,
(32)
J t
and if C is a limit cycle, a periodic solution of the Newtonian flow (31) with period T, then this requires that E(T) = E(0), which means that t+ T
ft
y(s)f(x(s), y(s))ds = 0
(33)
on C. Stability would require E(T) - E(0) > 0 for any curve C' enclosed by C, and also that E(T) - E(0) < 0 for any curve C" that encloses C. For weekly dissipative Newtonian flows, this leads to a practical method for locating limit cycles approximately [13]. Let f(x, y ) = - e h ( x , y) where s is small. In this case, we require a potential V(x) such that, for some range of energies E m i n < E < E . . . . the motion of the conservative system (the e = 0 system) is bounded and is therefore periodic with period x.
= 2fi :(
dx
1,2(E - V ( x ) )
(34) 1/2'
where Xl and x2 are the turning points of V(x). To lowest order in e, we expect that the limit cycle is approximately, very roughly, given by one of these periodic solutions (we assume implicitly that it is possible to describe the limit cycle approximately from the standpoint of the original conservative solution by using perturbation theory). When an integrable conservative system with periodic orbits is perturbed weakly by nonlinear dissipative/driving terms that are able to satisfy energy balance over a single definite, repeatable time interval r, then a limit cycle is born through the destruction of global periodicity, leaving behind one particular distorted periodic orbit of the original conservative system as an attractor or repeller. The motion on the limit cycle is trivially integrable (local integrability) although the motion off of the attractor is generally nonintegrable but regular whenever n/> 3. In a nonlinear dissipative system that is far from an integrable conservative one, a limit cycle can be born from a certain kind of bifurcation as a control parameter is varied: a stable limit cycle can follow from the loss of stability of a spiral sink through a H o p f bifurcation. To show that this scenario also fits into the way of thinking based upon
1978
J . L . McCAULE¥
destruction of foliations, we consider again the flow (31) but with a control p a r a m e t e r /t where 2=y 5~ = ( ~ _
x 2 _
y2)y
_
x.
(35)
The origin is an equilibrium point, so consider the linear flow 2=y
2 =#Y -x
(36a)
near the origin. The eigenvalues, for - 2
(36b)
when /~ = 0 the origin is elliptic and the phase plane of the linear flow is foliated by 1-tori If /~ < 0, all tori are destroyed and the origin is a sink due to overdamping: the only r e m n a n t of the tori is the spiraling motion into the sink. When /~ > 0, the motion spirals out of a source at the origin, but in the full nonlinear system (35) a 1-torus at r = ~//~ survives as a stable limit cycle that attracts all other trajectories. A s / ~ passes through 0 we have a H o p f bifurcation that describes the change of stability from spiraling flow into a sink to spiraling flow out of a source and onto the limit cycle. Regardless of the details of how a limit cycle is born, it is always a closed streamline where the flow is 'locally integrable' because of periodicity: any closed curve is topologically equivalent to a circle, and so there is a continuous transformation that takes us from the dynamics on the limit cycle to dynamics on a circle where the differential equations have the explicitly integrable form ?=0 q~ = (o.
(37)
We have seen how local integrability follows as a consequence of an average maintenance of a local conservation law in a Newtonian system: the net energy is conserved for each cycle of the motion by means of average energy balance along the limit cycle. In two dimensions, and therfore in all of the examples considered above, even if the energy is not conserved the motion in the phase plane is by necessity integrable: the required conservation law is just the streamline equation as we have shown earlier in Section 1. H o w e v e r , when n ~> 3 the motion away from an attractor (repeller) is generally nonintegrable and only the motion on the attractor (repeller) can be integrable. Therefore, we have introduced our way of thinking about attractors via destruction of foliations from a convenient but superficial standpoint. In the next section, we show how the over-simplified point of view presented above for two-dimensional Newtonian flows can be correctly generalized and leads to the idea of trapping regions in phase space.
3. D A M P E D D R I V E N E U L E R - L A G R A N G E D Y N A M I C S
We want to arrive at the ideas of trapping and attractors in phase space in a general way within a class of Newtonian models. In order to study self-confinement and attractors of a u t o n o m o u s flows in three dimensions, or in any odd-dimensional phase space, we cannot start with Newtonian particle mechanics: there, except for time-dependent external driving (which we ignore here), the phase space dimension is always even.
Integrability and attractors
1979
Even if one is not a physicist, the method of averaging described in the last section gives a practical motivation for studying Newtonian flows. There, we have seen that it is useful to have a quantity that is conserved in the absence of friction and driving, but that can be balanced on the average over some region of phase space in the presence of combined dissipation and driving. This led to the idea of an attractor with dimension one, a periodic solution called a limit cycle that is a ghost of the destroyed foliations of the original conservative system• Lorenz started explicitly from this viewpoint in Section 2 of his very beautiful 1963 paper [14]. In what follows, we start with a class of conservative models that has two conservation laws that are broken by the introduction of dissipation and driving in order deduce that there is a finite region in phase space where every trajectory eventually becomes trapped. What follows is a generalization of Section 2 of Lorenz's paper that is motivated by the class of systems studied in ref. [15]. There, phase spaces of both even and odd dimension are possible, depending upon the complexity of the rigid body• For example, one can consider several rigid bodies linked together to form a single system, like a wheel and axle system with steering. A Newtonian starting point for the self-confinement of a damped driven nonlinear flow to a definite finite region in an n-dimensional phase space is provided by the E u l e r Lagrange equations [15] dJx + cx~ dt J~
- Nx,
(38)
which describe a conservative dynamical system in the n-dimensional (J1 . . . . . Jn) phase space (phase-volume preserving flow). H e r e , the inertia matrix I is assumed to be diagonal and constant and the c's are the structure constants of some Lie algebra (of rigid body systems more complicated than a top, for example) and so must obey the antisymmetry ~, ~. condition c,,, =--c~,~. The symbols J, 1, and N can, but need not, denote the angular momentum, moment of inertia, and torque or a rigid body problem: in the discussion that follows, all that we require is the mathematical structure of the E u l e r - L a g r a n g e equations (38): those equations may represent rigid body motions or some other physically motivated form of mathematical modeling. We also consider only the case where the vector N is a constant vector in what follows. That is, we study an autonomous system (38) with constant driving terms N;~ in the n-dimensional (J1 . . . . . Jn) phase space. Whenever the structure constants do not all vanish then the Ji are nonintegrable velocities in the original Newtonian system [15]. There are always at least two point functions that are left invariant by (38) whenever N~ = 0, and the study of the nonconservation of either of those quantities in the presence of damping can lead to the deduction of a trapping region in the (J1 . . . . . Jn) phase space. A trapping region is more general than the idea of an attractor: it is a region within which any and all trajectories become self-confined after a long enough finite time (the trapping time is trajectory-dependent). If there is an attractor, then it lies within the trapping region but the trapping region has a larger dimension than the attractor due to the phase-volume contraction law that reflects dissipation. For example, the trapping region always has dimension n, but fixed points and limit cycles have dimension 0 and 1, respectively. If n = 3 and N~ = 0 then the conservative flow (38) is integrable, but this feature is not essential in what follows and we assume that n t> 3. To obtain a model that can have an attractor of dimension greater than zero, we must generalize the E u l e r - L a g r a n g e dynamics (38) by adding damping (while keeping constant external driving), which we take here to be linear: •
.
__ J~Jv d J z + b z u J . + cZ.,, = N~. dt l;.
(39)
1980
J . L . McCAULEY
Since V-V-
3)° _ trb, (40) 9Jo if we require that t r b > 0 then we expect to find that very large scale motions will be damped, which is a necessary condition for an attractor to exist somewhere in the phase space. To have an attractor that is not a zero-dimensional sink, then the energy input must dominate dissipation locally at small scales to m a k e the smallest scale motions unstable. That is, to have an attractor the large scale motions must be unstable against further growth but the motions at small enough scales must also be unstable against further decay: these circumstances lead to attractors of dimension higher than zero in phase space, and it is helpful to use a quantity that is conserved whenever b = 0 in order formulate this notion quantitatively. We show show next that the conservative flow with N~ = 0 always has two invariants, so that the n = 3 flow without forcing is always integrable. The conservative flow (38) with N~ = 0 always leaves the kinetic energy T - (J12/I1 + J22/I2 + . . . + JnE/In)/2 invariant. The p r o o f of a second conserved quantity C is really only a disguised form of, the p r o o f that a Lie algebra has at least one Casimir o p e r a t o r [16]. This conserved quantity is a generalization of the square of the angular momentum. We can show directly by differentiation that the quantity
C = gP°Jflo
(41)
is left invariant by the undriven conservative flow dJx
v J~Jv
+ cz¢~,,
- 0,
(42)
dt where by gVO we m e a n the inverse of the matrix t~ ~ g~v = c~aCv/~.
(43)
H a m m e r m e s h states that the matrix g provides a metric for defining inner products in linear representations of Lie algebras. F r o m the differential equations (42) we obtain
dC_ dt
2gooC'p, jflvjo
(44)
I,
so that dC/dt = 0 follows if the rank three tensor gV°c~oo is antisymmetric under exchange of the two indices a and v. In fact, this tensor is antisymmetric under exchange of any two of the three indices (o, ~, v) [16]. Therefore, dC/dt = 0 and C is left invariant by the conservative flow (42). We have two conservation laws that describe two ellipsoids in phase space, or two hyperellipsoids if n > 3. D e n o t e the kinetic energy ellipsoid by E1 and the 'angular m o m e n t u m ' ellipsoid described by the conserved quantity C by E2. In the conservative flow (where b = 0), the invariant curves (the solutions of (42)) lie in intersections of E 1 and E2 and both these ellipsoids foliate the phase space. In this case, each streamline of the flow is self-confined to an n - 2-dimensional hypersurface in the n-dimensional phase space. When b :~ 0, then (even if N = 0) the foliations of phase space are destroyed (invariant curves of (39) generally cannot lie on the intersections of Ea with Ez). Next, we ask what happens when both driving and dissipation are included. Consider the surface E2 for large values of J, and assume that t r b > 0. Think of any point on E2 as an initial condition for the flow so that the ellipsoid changes rigidly in size
Integrability and attractors
1981
as the single point on E 2 follows the streamline: then for large enough J we expect that dC/dt < 0, while E 2 contracts rigidly according to the law dC
.lg-lbj + NJ. (45) dt Since this expression is not positive definite, we cannot conclude that C decreases eventually to zero as t goes to infinity. In fact, in order to get a trapping region that contains an attractor with dimension greater than zero, the action of the matrix g - l b combined with that of the constant vector N in (45) must be such as to divide the phase space into regions of net dissipation at for large enough J and regions of net energy input at small enough J. Let us assume that the matrix b has been constructed so that this is the case. Then a s E 2 contracts rigidly while one point on E2 follows a streamline, E2 will eventually intersect a third surface E3 that is defined by the condition that dC/dt -- 0 at all points along the intersection. Inside the surface E3, dC/dt > 0 and so any trajectory that enters E 3 tends to be repelled outward again. If these expectations are born out, then we can define the trapping region optimally as the smallest ellipsoid EEc that contains the (still unknown) surface E 3. Every streamline must enter the n-dimensional region Ezc in finite time, and once having entered cannot escape again (this is self-trapping, or self-confinement). For one set of parameter values, we also expect to find at least one attracting point set somewhere within the ellipsoidal trapping region E2c. To find E3 and E2c explicitly, we follow Lorenz [14] and define next a vector e with components ev that solve the linear equations -
(b' + b')e = N,
(46)
where b' = g - i b . In this case, direct comparison with (45) yields dC
- ~b'e - ( f ' Z - e ) b ' ( J -
e).
(47)
dt This means that
dC/dt
= 0 on the surface E3 defined by ~b'e = ( J - e ) b ' ( J -
e).
(48)
To get a trapping region (a closed surface E3) it is sufficient to have a positive definite matrix b', which explicitly guarantees that t r b ' > 0 and also makes E3 an ellipsoid or hyperellipsoid. If follows from (47) that dC/dt < 0 outside the ellipsoid E3 (attraction toward E3) but dC/dt > 0 inside the ellipsoid E3 (expulsion from E3), but the surface of the ellipsoid E3 usually does not contain an invariant curve (invariant set, or just 'solution') of the damped-driven flow (39). We can define the trapping region Ezc to be the smallest ellipsoid E2 that completely contains the smaller ellipsoid E 3. Note that if N~ = 0 in equations (39), then a set of equations with finite, constant values of N~ can always be obtained by a translation L[ = Li + ai in phase space. To give an example, we return to a specific class of damped driven E u l e r - L a g r a n g e flows. If n = 3 and c ~ = e~u~, which represents the motion of a rigid body with three moments of inertia in three dimensions, then C = J~ + J~ + J~ is just the square of the angular momentum. More generally, we can assume that n/> 3 but still require that c~u~J~J~J~= 0 (this condition defines the class of models considered in Section 2 of Lorenz's paper). In this case, it still follows that C = J~ + . . . + J2n, and so the trapping region Ezc is the smallest hyper-sphere, the sphere with radius ~/C~, that contains the entire ellipsoid E3. We are now going to show that the Lorenz model falls into this category. The Lorenz model is normally derived from the continuum equations of hydrodynamics
1982
J . L . MCCAULEY
and energy transport via a mode expansion and truncation, but we are going to show that the Lorenz model can also be understood as a damped, driven symmetric top. The Lorenz model is defined by the flow = o(y = px =
-
-/3z
x) y +
-
xz
(28)
xy.
Normally, one throws away nonlinear terms in order to identify a linear system. Here, we note that a conservative Euler top flow is obtained by deleting all linear terms in (28), yielding 2=0 .F = - - X Z
(49)
= xy.
This flow represents a torque-free symmetric top with 11 = 12 ~ 13. For example, if we write the equations of Euler's top in the form = ayz p
=
- bxz
2
= cxy,
(24b)
then x 2 + y2 + z 2 = constant and with a = 0, b = c = 1 (symmetric top with 11 = I 2 4= 13), and linear damping/driving added on corresponding to the matrix b =
1 0
-
,
(50)
then we get exactly the Lorenz model. The Lorenz model is, therefore, formally equivalent to a certain damped and driven symmetric top, and to get a trapping region in phase space we need only t r b = o + / 3 + l > 0 along with the condition that o, /3, and p are all positive. The latter condition is specifically needed in order to get the ellipsoid E3, and the trapping region is, therefore, the smallest sphere that contains that ellipsoid. In general, truncations to finitely many modes of the equations of fluid mechanics and energy transport give rise to equations of motion of the form (39) of the damped E u l e r - L a g r a n g e equations of rigid body motion [17]. Presumably, the coincidence reflects rotational symmetry in the original hydrodynamics equations. Because training in differential equations normally concentrates strongly on linear systems, writers on nonlinear dynamics tend to emphasize that the addition of nonlinear terms to linear equations can lead to complicated behavior, even to deterministic chaos. However, a viewpoint that takes linear differential equations as the starting point for discussions of nonlinear dynamics is not very useful for physics. Some writers on economics even express the misconception that linearity and equilibrium are the main methods of physics [18]. To the contrary, most of the standard textbook problems of classical mechanics are conservative and integrable but are nonlinear. A better starting point for studying differential equations would therefore begin with the study of nonlinear integrable systems, or at least with nonlinear conservative ones [10], which reflects the emphasis in differential equations at the turn of the century before the time of the linear Schr6dinger equation. Here, we have approached nonlinear dynamics from that standpoint and have
Integrability and attractors
1983
shown that the addition of linear damping to nonlinear conservative systems can lead to self-confinement. As is well known, the trapping regions of such systems can contain chaotic attractors [14]. In pure mathematics, a writer is not self-constrained to restrict his studies to models that are motivated by physics, although dynamical systems theory grew out of physics and even Poincar6 seems to have been motivated in large part by problems from physics. Newton's laws of motion (and their relativistic and quantal generalizations) are the only known universal laws of motion. The empirically-based law of inertia discovered by Galileo and Descartes is the foundation for Newton's universal laws of motion, but there is no equivalent of either Galilean invariance of Newtonian universality in economics or sociology, which is why those and other similar fields have not been reduced to mathematics in any useful way that is consistent with the wide range of their empirical observations. Those fields have not given birth to a Kepler, a Galileo, or a Newton, and no amount of computer simulation based upon arbitrary models from dynamical systems theory, probability theory, or any combination thereof can alleviate the fact. So far, only physics and physics-related fields have gone beyond the stage of the epicycles of scholasticism. Descartes' D r e a m [19] to the contrary, it is not at all evident that the modern equivalent of epicycles, purely empirical modeling without any fundamental underlying laws of motion, can be eliminated from fields like economics, sociology and psychology. Descartes, who advocated the mathematical formulation of all observable phenomena, was motivated by the law of inertia and the action-reaction principle, which he apparently understood independently of Galileo and Newton [20]. Primarily, he used the law of inertia and the action-reaction principle to construct a form of neoscholasticism whereby a purely theoretical discovery of truth was thought to be possible without the aid of adequate careful observation and measurement. Some of his ideas about mechanics were right but most of them were wrong. The belief that sociological phenomena like economics can be explained by mathematics by using computers, but without adequate universal and precise laws that account for a wide range of observations without juggling a lot of arbitrary parameters freely, reflects Descartes' D r e a m but also resembles the medieval state of science before the law of inertia and Kepler's laws were discovered, and before the Newtonian synthesis. Epicycles by any other name are still epicycles. Acknowledgement--I am grateful to Egil Andersen for introducing me to the historical survey of Lie's ideas in Bell's interesting work [1]. I am also grateful to Julian Palmore for introducing me to ref. [9], where a systematic approach to return maps is established.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
E. T. Bell, The Development of Mathematics. McGraw-Hill, New York (1945). P. J. Olver, Applications of Lie Groups to Differential Equations. Springer, New York (1993). V. I. Arnol'd, Mathematical Methods of Classical Mechanics. Springer, Berlin (1978). S. Lie, Forhand. Vid.-Sels. Christiania 198 (1874); 1 (1875). F. Klein, Vorlesungen iiber hOhere Geometrie, Dr. Aufl. Springer, Berlin (1968). J. E. Marsden, Lectures on Mechanics. Cambridge University Press, Cambridge (1992). J. L. McCauley, Chaos, Dynamics, and Fractals: an Algorithmic Approach to Deterministic Chaos. Cambridge University Press, Cambridge (1993). E. L. Ince, Ordinary Differential Equations. Dover (1956). S. A. Burns and J. I. Palmore, Physica D37, 83-90 (1989). J. L. McCauley, Classical Dynamics: Flows, Transformations, lntegrability and Chaos. Cambridge University Press, Cambridge (1995). (To be published) B, A. Dubrovin, A. T. Fomenko and S. P. Novikov Modern Geometry--Methods and Applications. Part 1. The Geometry of Surfaces, Transformation Groups, and Fields, transl, by R. G. Burns, Chapters 2 & 5. Springer, New York (1984). M. Tabor, Chaos and Integrability in Nonlinear Dynamics. Wiley, New York (1989).
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13. D. W. Jordan and P. Smith, Introduction to Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford (1977). 14. E. N. Lorenz Deterministic nonperiodic flow in University in Chaos, edited by P. Cvitanovic. Adam Hi|ger Ltd., Bristol (1984). 15. J. L. McCauley, Nonintegrable velocities and nonholonomic coordinates in dynamics, preprint (1994). 16. M. Hammermesh, Group Theory. Addison-Wesley, Reading, MA (1962). 17. M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics. Springer, New York (1987). 18. E. E. Peters, Chaos and Order in the Capital Markets. Wiley, New York (1991). 19. P. J. Davis and R. Hersh, Descartes' Dream: The World According to Mathematics. Harcourt Brace Janovich, San Diego (1986). 20. R. Dugas, Mechanics in the Seventeenth Century, From the Scholastic Antecedents to Classical Thought, transl. from French by Freda Jacquot. Central Book Co., New York (1958).