Dynamics of numerical approximations

li N O g l ~ - ~

Dynamics of Numerical Approximations Jack K. Hale

School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332

ABSTRACT Assuming that a differential equation has a hyperbolic invariant set, we discuss the existence of a similar invariant set for the map defined by a numerical approximation, as well as the closeness of these sets in the CLtopology. Particular emphasis is given to equilibrium points, periodic orbits and invariant tori for the differential equation. We present also a result on the preservation of structural stability for gradient systems. © Elsevier Science Inc., 1998

1.

INTRODUCTION

Suppose t h a t A is a parameter in a Banach space, [h[ < 8, where ~ is a fixed constant. Also, suppose that, for each such A, T~(t), t >/0, is a CUsemigroup on a Banach space X with a global attractor ~ ; t h a t is, ~4~ is a compact, invariant set ( T A ( t ) ~ = ~ for all t) and, for any bounded set B ~ X, we have lim dist( T~(t) B, ~ )

= 0.

t--~o0

One of the basic problems in dynamical systems is to determine how the set ~¢A depends upon A and how the dynamics on ~4~ depend upon A. T h e simplest type of result is one t h a t ignores completely the dynamics on ~ , and is the assertion t h a t ~ is upper semicontinuous in A at some point A0; t h a t is, lira dist(~¢~,~¢~ 0 .) . . . .

=0.

A--*A o

APPLIED MA THEMATICSAND COMPUTATION89:5-15 (1998) © Elsevier Science Inc., 1998 655 Avenue of the Americas, New York, NY 10010

0096-3003/98/$19.00 PII S0096-3003(96)00257-3

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J . K . HALE

Upper semicontinuity will hold if T~(t)y ---) T%(t)y uniformly on bounded sets of [0, oo) × X and there is a bounded set B c X such that ~ c B for [A[ < 8. The proof is a simple consequence of the stability of the attractor ~ 0 (see, for example, Hale [1]. If we desire more detailed properties, we must take into consideration some of the properties of the flow on the attractor. For example, if the semigroup is gradient, then the eo-limit set of each orbit belongs to the set E~ of equilibrium points. If we suppose, in addition to the above hypotheses, that each equilibrium point is hyperbolic, then it is possible to show that the attractors are lower semicontinuous; that is lim dist(~o,~]...... = 0 A ~ Ao

(see Hale and Raugel [2]). The proof of the lower semicontinuity in this case uses the fact that ~¢~ = ( J , ~ E W2(~p) and the continuity with respect to A of the unstable manifold W~(~) of the equilibrium point ¢. If we have both upper and lower semicontinuity of ~ at A0, then we have Hausdorff continuity at A0, and the size of the attractors do not change appreciably if we do not deviate very much from A0. If we want to know that the flow on the attractors are topologically equivalent for the gradient case, then it is necessary to discuss the transversality of the stable and unstable manifolds. Thus, we need to know something about the Cl-closeness of the global unstable manifolds. The verification of the above hypotheses is not too difficult for the case when the semigroup depends in a nice way upon A. However, if the perturbed semigroup is obtained from the time discretization of an evolutionary differential equation (or the time a n d / o r the space discretization of a partial differential equation), then the situation is much more complicated. The abstract setup for such discretizations is the following. Suppose that we have a family of Banach spaces X~ with X~ ~ X 0 and continuous projection operators P~ : X 0 --* X~ such that lim distx0(V , X~) = 0 Vv ~ X 0. A--*0

If we suppose also that T~(t) P~ y --) T0(t) y uniformly on bounded sets of [0, ~) × X0, the same results as mentioned above hold true (see, for example, Hale, Lin, and Raugel [3], Hale and Rangel [2]). In the case of time discretizations, we obtain a discrete dynamical system T~ on X 0 and we take P~ = /, X~ = X0, and we compare the attractors in X 0. For space discretizations of PDE, the spaces X A ~ X 0 and, in fact, they

Dynamics of Numerical Approximations

7

will be finite dimensional subspaces of X 0. The attractors are still to be compared in X 0. If we consider dynamical systems that are not gradient, then it is necessary to discuss more complicated invariant sets and their stable and unstable sets. For time discretizations for ODE, much attention has been given to the situation where the hyperbolic invariant set for the ODE is a closed curve. It is known that there is a hyperbolic invariant closed curve for the map defined by the discretization (for references, see Stuart [4]). For parabolic equations, see Alouges and Debussche [5]. The proofs of such results have been given by considering directly the map defined by the difference approximation. Our objective in this note is to show that a recent result of Fiedler and Scheurle [6] for ordinary differential equations, together with the classical theory of invariant manifolds of ODE, permit us to obtain directly the known results about the behavior of equilibrium points and periodic orbits under discretization in time. As we will see, our remarks are elementary and relate only to bringing together known results. However, this information should be known to people working in numerics because it also allows the consideration of more complicated invariant sets as well as global comparisons of the dynamics. 2 AN ODE FOR A TIME DISCRETIZATION In this section, we state a general result of Fiedler and Scheurle [6] which asserts that the map defined by a one-step difference approximation to an ODE coincides with the Poincar@ map of a rapidly oscillating periodic ODE that is close to the original ODE. Suppose that f ~ Ck(R ~, ~ ~), k ~> 1, and consider the ODE k = f(x).

(2.1)

Let F(t, x) be the solution of (2.1) satisfying F(0, x) = x and suppose that there are an integer p/> 1, a continuous function C: [0, ~) --+ [0, oo) and a one-step difference approximation of step size h, X +l = ¢ ( h , x 0 ,

0 < h < 4,

(2.2)

such that Iq)(h, x) - F( h, x)l <~ C(Ixl)h p+I.

(2.3)

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J . K . HALE

Fiedler and Scheurle [6] have proved the following natural but nontrivial assertion.

THEOREM 2.1. There exists a smooth function g( h, "r, x) ~ ~ 4, periodic in .r of period 1, such that, if G( t, s, h, x), G( s, s, h, x) = x, is the solution operator of the equation

(t)

2 = f( x) + hPg h,-~, x ,

(2.4)

a(h,0, h, x) = ¢ ( h , x).

(2.5)

then

The mapping G(h, 0, h, • ) : R ~ -~ ]~ d is the Poincar~ m a p (equivalently, the period map) for the periodic system (2.4) corresponding to the initial time cr = 0. Relation (2.5) says that the Poincar~ map for (2.4) is the same as the map (2.2) defined by the difference approximation. Fiedler and Scheurle [6] used Theorem 2.1 to assist in the discussion of the following problem. Suppose that (2.1) has an homoclinic orbit to a hyperbolic saddle point. If (2.1) is approximated by a difference equation (2.2), the map q ) ( h , - ) will have a hyperbolic fixed point close to the hyperbolic saddle of (2.1) if h is small. If we introduce another parameter,/~, into the vector field in a generic way, then we also expect to determine /~ = tL(h) in such a way that there is an orbit for the m a p that is homoclinic to the fixed point. Furthermore, the stable and unstable manifolds of the fixed point should be transversal. Fiedler and Scheurle [6] discuss in detail the nature of this tranversaiity. In the next section, we indicate some other applications of this theorem. 3.

STABLE, UNSTABLE MANIFOLDS, S T R U C T U R A L S T A B I L I T Y

From Theorem 2.1, we see t h a t the following relation holds for some continuous function C 1 : [0, oo) -~ [0, ~):

IF(h, x) - G(h,O, h, x)lk <~ Cl(IXl)h p+I

Yx~

n d,

(3.1)

where the subscript k designates that we measure the norm in the C ktopology.

Dynamics of Numerical Approximations

9

Let x0 be an hyperbolic equilibrium point of (2.1) with stable and unstable manifolds W~(x0), WU(x0). From (3.1), there is a fixed point x3 of the map G(h, O, h, • ), x oh --* x o as h --* 0, x0h is hyperbolic with stable and unstable manifolds W~(x0h), Wh"(x0h) that are Ck-close to Ws(Xo), W~(Xo). This is not a trivial result to prove for ODE (2.4), but it is well known (see, for example, Chow and Hale [7], Irwin [8]). As a consequence of this known theory, we obtain the same remark for the difference approximation O(h, x) by (2.5). Let us now assume that the system (2.1) is a gradient system with a global attractor, and there is an open set U with ~ U smooth such that the attractor belongs to U and the flow defined by (2.1) is transversal to $ U. The existence of U follows by the consideration of the converse theorems for the existence of Liapunov functions. Recall that (2.1) is structurally stable in the set U if there is a neighborhood V of f in the Cl-topology on U such that, for any / ~ V, there is a homeomorphism of U into itself that maps orbits of f onto orbits of / a n d preserves the sense of direction in time. For a gradient system, (2.1) is structurally stable if all equilibrium points are hyperbolic and the stable and unstable manifolds intersect transversally (Palis [9], Palls and Smale [10]). If (2.1) is gradient and has a global attractor as above, and we consider the perturbed equation (2.4), then it is not too difficult to prove that the only minimal sets of the Poincar~ map are fixed points and the stable and unstable manifolds intersect transversally if h is sufficiently small. With the same proof as in Palis [9], Palis and Smale [10], we see that the map ~ ( h , x) is topologically equivalent to the map F(h, x). We emphasize this by stating the following result.

THEOREM 3.1. Suppose that (2.1) is a gradient system for which there is a global attractor. Then there is an open set U containing the attractor such that, if (2.1) is structurally stable on U, then, on U, the flow defined by the map F( h, • ) is topologically equivalent to the flow defined by the difference approximation O( h, • ). 4.

PRESERVATION OF INVARIANT SETS

THEOREM 4.1. Suppose that k >1 2 and (2.1) has a hyperbolic periodic orbit F with stable manifold Ws(F) and unstable manifold Wu(F). Then there is an h o > 0 such that, for 0 < h <~ h0, the map ¢P( h, • ) in (2.2) has an invariant closed curve F h that is hyperbolic with stable manifold W~(Fh) ,

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J . K . HALE

unstable manifold Wd(Fh) , and these manifolds are Ck-l-close to the corresponding ones for (2.1). The existence of the closed curve F h and the fact that it converges to F in the C a- l-topology has been known for some time (see Stuart [4] for references). The proofs involved working directly with the mapping O(h, • ). We present another proof which is based on the result of Fiedler and Scheurle [6] in Section 2 and the classical theory of invariant manifolds. The first remark is that the flow in the neighborhood of the periodic orbit F of (2.1) is more easily studied by the consideration of a rotating coordinate system

x = p(O) + B ( O ) r ,

(4.1)

where I" = { p(0), 0 ~ •} and p(t) is a periodic solution of (2.1) of period to. In (4.1), r is a d - 1-vector lying in the normal plane to F at p(0), and B(0) is a column orthogonal d X ( d - 1) matrix, periodic in 0 of period to. This type of coordinate system was first introduced by Urabe [11] and may be found in Hale [12]. In the new variables 8, r, equation (2.1) in a neighborhood of F is equivalent to the system

(}= 1 + O(8, r) ~= Cr+ R(O, r),

(4.2)

where the eigenvalues of the (d - 1) X ( d - 1) matrix C have nonzero real parts (this is the reflection of the hyperbolicity of F), the function O and R are C k- 1-functions, periodic in 0 of period ~o, with O, R, ~ R / 4 r vanishing at r = 0 . If we apply this same transformation to the equation (2.4), we obtain

= 1 + 0 ( 8 , r) + hP01

= Cr+R(8,

r)+hPRI

(t) (t)

h,-~, 8, r (4.3)

h,-~,8, r .

If we let t = h~ and let ' denote the differentiation with respect to T, then (4.3) is equivalent to the system

0' = h q- h(~(0, r) q- hp+lO1(h, T, 8, r) (4.4) r' = h C r + hR( 8, r) + h p+ I RI( h, T, 8, r).

Dynamics of Numerical Approximations

11

Equation (4.4) is now in the form in which we can apply the classical theory of invariant manifolds (see, for example, Bogoliubov and Mitropolskii [13] or Hale [12]) to conclude that there is an invariant manifold given by

o,

=

o,

o e R}

where r*(7, 0, h) is periodic in r of period 1 and periodic in 0 of period o~. Furthermore, r*(~, O, h) -* 0 in the C k- Ltopology as h --) 0. If we return to the original coordinate system through (4.1) and use Theorem 2.1, then this implies the existence of the invariant curve F h of the Poincar~ map of (2.4). Invariant manifold theory applied to (4.4) also gives the assertions concerning the stable and unstable manifolds, and the theorem is proved. A more general result is obtained in a similar way. For example, suppose that (2.1) has an invariant curve F which is given by F = { x ~ R a : x = u ( 0 ) , 0 ~ 0 < ~o}, where u ~ Ck(R, Rd), k > / 2 , u ( 0 + ~o)= u(0) for all 0 ~ R. The curve F is Ck-diffeomorphic to a circle but the curve does not need to represent a periodic orbit. It is invariant so that, if it is not periodic, then the only minimal invariant sets on [" are equilibrium points. It is again possible to introduce a moving orthonormal coordinate system for (2.1) in a neighborhood of [" to obtain an equivalent system for (2.4) of the form

(t) (4.5)

c ( o ) r + R(o,

+ hPR1

O,

If the function g(O) has no zeros, then the curve [" is a periodic orbit. If it has zeros, then there are equilibrium points on F. In spite of this, we can prove a result very similar to Theorem 4.1. The only hypothesis that is necessary is to suppose that the invariant curve I" is normally hyperbolic; that is, no Floquet multiplier of the linear equation dr d--O = C(O) r

(4.6)

is on the unit circle in the complex plane, and the rate of expansion or contraction of the flow on I" is not as large as that away from F. In the case of a periodic orbit, this is always satisfied by simply assuming that F is

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J . K . HALE

hyperbolic. If there axe equilibrium points on F, then the rate of approach to an equilibrium or rate of expansion from an equilibrium is not too large relative to the Floquet multipliers of (4.6). We summarize these remarks in

THEOREM 4.2. Suppose that k >1 2 and (2.1) has a normally hyperbolic C2-invariant closed curve F. Then there is an h o > 0 such that, for 0 < h < h 0, the map ~P(h,. ) has a normally hyperbolic invariant closed curve Fh which satisfies the properties stated in Theorem 4.1. The idea of using Theorem 2.1 together with the theory of invariant manifolds leads to much more general results. In fact, suppose that (2.1) has an invariant manifold F 2 that is Ck-diffeomorphic to a torus T 2. Then it is possible (see, for example, Henry [14, p.300ffl) to introduce a coordinate system in a neighborhood of F2 of the type x=~b+B(~)r,

~be F2,lr[ < 3,

(4.T)

such that the differential equation (2.4) is equivalent to the system

¢=g(~b)

+~(~b,r)

(t)

+ hP~l h , - ~ , ~ b , r (4.8)

with ~(~b, 0) = 0, R(~b, 0) = 0, OR(~b, O ) / d r = O. The coordinate ~b is two dimensional and can be considered as two angles that are used to describe F 2, and the coordinate r has dimensional d - 2 and is in the normal plane to F 2. If we think of the problem this way, then the functions above are periodic in the two vectors ~b and t. The flow on F 2 is described by the differential equation = g(~b).

(4.9)

The linear flow near F 2 is given by the solutions of the equation

/~ = C( ~b(t)) r, where ~b(t) is a solution of (4.9).

(4.10)

Dynamics of Numerical Approximations

13

The invariant set F 2 of (2.1) is said to be normally hyperbolic if (4.10) has an exponential dichotomy with all of the constants in the dichotomy being independent of ~(0), and the rate of contraction or expansion of the solutions of (4.9) does not exceed the rates of expansion and contraction of the exponential dichotomy. If we assume that all of these conditions are satisfied, then Theorem 4.2 can be generalized to obtain the following result.

THEOREM 4.3. Suppose that k >>.2 and (2.1) has a normally hyperbolic C2-invariant manifold [,2 as above. Then there is an h o > 0 such that, for 0 < h ~< h 0, the map ¢P(h, • ) has a normally hyperbolic invariant manifold F~ that satisfies the properties stated in Theorem 4.1. The proof of this result can be supplied by using the general results of Henry [14, p.275hh]. We remark that the flow on F 2 could contain equilibrium points and periodic orbits as minimal sets. Whether or not this occurs depends upon the form of the function g(~b). If we suppose there are no equilibrium points on F e, then we can define the rotation number p(g) of the flow defined by (4.9). If this number is irrational, then the theorem of Denjoy says that the flow on F 2 is the parallel flow. In this case, it is possible to choose the original coordinates ~b so that g(~b) = /2, where I S is the two dimensional identity matrix. If we assume that p(g) is rational and all solutions on F 2 are periodic, then we can choose the coordinates so that the same property is true. In these situations, the linear equation becomes

~= C( t + qJo, t + Oo)r.

(4.11)

The assumption that (4.11) has an exponential dichotomy with all of the constants in the dichotomy being independent of ~0, 00 is nontrivial to verify since the coefficient matrix in (4.11) is in general almost periodic. Much more general results can be given about hyperbolic invariant sets for (2.1) by either using the more general theory in Henry [14] on invariant manifolds for continuous flows or the results of Hirsch, Pugh and Shub [15] by exploiting the fact that the Poincar~ map and the solution operator of (2.1) satisfy (3.1) in the Ck-topology. 5.

OPEN PROBLEMS

Assuming that a system of quasilinear parabolic equations has either a hyperbolic equilibrium or a hyperbolic periodic orbit, Alouges and Debussche [5], [16] have shown that the map obtained by time discretization

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J . K . HALE

has either a hyperbolic fixed point or any hyperbolic closed curve that is close to the one for the continuous system. If we had an extension of the result of Fiedler and Scheurle [6] to parabolic equations, then we could obtain these results as well as all of the ones mentioned previously by employing the transformation and invariant manifold theory of Henry [14] that was developed to apply to parabolic equations. Such an extension should be possible because the resolvent of the linear operator involving the second order spatial derivatives is compact. When we discuss PDE, we also must consider spatial discretization. Assuming that there is a hyperbolic equilibrium point for the continuous system, the standard types of spatial discretization lead to the conclusion that the resulting ODE has a hyperbolic equilibrium point whose stable and unstable manifolds are close. A proof can be given based on the variation of constants formula and the closeness of the linear approximations. For the case of a periodic orbit, one can use the methods in Henry [14] on invariant manifolds. For linearly dampled hyperbolic systems, almost nothing seems to be available. For the case of a hyperbolic equilibrium, the methods in Hale and Raugel [17] on thin domains should be applicable. The case of a hyperbolic periodic orbit seems to be an interesting open problem. Other interesting problems occur even for ODE when the time discretization involves variable step size. Assuming that the ODE has a global attractor and for a one-step process, Kloeden and Schmalfuss have recently shown the existence of a cocycle attractor and upper semicontinuity of the attractors (article to be published). The proof uses ideas from skew product flows. This method needs to be further exploited to obtain more detailed properties of the flows; in particular, the discussion of hyperbolic equilibrium points and periodic orbits. It also would be interesting in this latter situation to understand if there is some nonautonomous ODE that is in some sense equivalent to the discretization. REFERENCES 1 J.K. Hale, Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., 1988. 2 J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann, Math. Pura AppL 154:281-326 (1989). 3 J. K. Hale, X.-B Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp. 50:89-123 (1988). 4 A Stuart, Numerical analysis of dynamical systems, Acta Numeriea (1974). 5 F. Alouges and A. Debussche, On the discretization of a partial differential

Dynamics of Numerical Approximations

6 7 8 9 10 11 12 13 14 15 16

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equation in the neighborhood of a period orbit, Numer. Matl~, 65:143-175 (1993). B. Fiedler and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and "invisible chaos", Mew. Am. Mat]~ Soc. 570 (1996). S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, 1982. M. C. Irwin, On the stable manifold theorem, Bull London Mat]~ Soc., 2:196-198 (1970). J. Palis, On Morse-Smale dynamical systems, Topology, 8:385-405 (1969). J. Palis and S. Smale, Structural stability theorems, Global Analysis, Proc. Syrup. Pure Math. 14 (1970). M. Urabe, Nonlinear Autonomous Oscillation~ Academic Press, 1967. J.K. Hale, Ordinary Differential Equation~ Krieger, 1980. N.N. Bogoliubov and Y. A. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillation~ Gordon and Breach, 1961. D. Henry, Geometric Theory of Semilinear Parabolic Equation~ Lect. Notes Math. 840, Springer-Verlag (1981). M.W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifold~ Lect. Notes Math. 583, Springer-Verlag (1977). F. Alouges and A. Debussche, On the qualitative behavior of the orbits of a parabolic partial differential equation in the neighborhood of a hyperbolic fixed point, Numer. FuncL Anal and Optimi~, 12:253-269 (1991). J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures AppL, 71:33-95 (1992).