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Adiabatic chaos
Volume 128, number
ADIABATIC Stephen
6,7
PHYSICS
LETTERS
A
11 April 1988
CHAOS
WIGGINS
Applied Mechanics. 104-44, Califorma Institute Received 16 September 1987; revised manuscript Communicated by D.D. Holm
of Technology, Pasadena, CA 91125, USA received
25 January
1988: accepted
for publication
5 February
1988
We consider single degree of freedom hamiltonian systems depending periodically on a parameter which varies slowly in time. We show how a recent extension of Melnikov’s method coupled with a theorem of Arnold implies that in some systems of this type the action of open sets of orbits may remain close to their initial values for all time yet the orbits evolve in a chaotic manner. We illustrate the results with the example of a simple pendulum whose base is slowly accelerated.
The adiabatic invariance of the action in single degree of freedom hamiltonian systems containing a slowly varying parameter is important in many applications. However, most classical results along these lines depend crucially on the fact that the frequency of the motion is nonzero. This condition is violated on separatrices. Recently, there have been some results concerning motion across separatrices in such systems, see refs. [ 1,2]. These studies have concentrated on determining the change in the action of a specific orbit as the separatrix is crossed. In this Letter we show how recently developed global perturbation techniques (see ref. [ 31) along the lines of Melnikov [4] allow one to determine that the motion near separatrices in such systems where the parameter varies slowly, but periodically, in time may be chaotic due to the presence of Smale horseshoes (note: the results of Tennyson et al. and Hannay allow a more general time dependence). The difference in our methods and the original Melnikov [ 41 technique can be succinctly stated in the context of nonlinear oscillations. Melnikov’s method applies to single degree of freedom oscillators forced periodically with an excitation having 0( 6) amplitude and 0( 1) frequency. Our method applies to single degree of freedom oscillators forced periodically with an excitation having 0 ( 1) amplitude and 0 ( t ) frequency. Thus our method should give information concerning chaotic dynamics in an entirely new class of problems. Coupling these results with a theorem 0375-9601/88/S (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
of Arnold [ 51 we conclude that in some situations the action of orbits near separatrices may not wander far from its initial value yet the orbit evolves in a chaotic manner. We illustrate the theory with an example of a pendulum whose base is oscillated slowly and periodically in time. We remark that simultaneously Escande [ 61 has considered the possibility of chaos in the periodic case and has obtained similar results although with different methods. Also, Cary et al. [ 7 ] have studied similar problems arising in the context of plasma physics. We consider an analytic hamiltonian function, H=H(x, y; z),z=wt, periodic in z with period T which defines the following hamiltonian vector field:
dqx,y;z), )I=-g(x)y;z)) dY I-=tCO
(lb
and we will refer to the following parametrized ily of systems as the unperturbed system
fam-
i=
qx, aY
i=o.
y; z),
li= -
+;(x,y;z)
)
(110
We have the following assumption on the structure of the phase space of the unperturbed system. Assumption. Vz E (0, T] ( 1 ). has a hyperbolic fixed B.V.
339
Volume
128, number
PHYSICS
6.7
point y(z) which is connected to itself by a homoclinic orbit qt: (t). Moreover we assume that y(z) and q; ( t ) are differentiable (at least C’ ) VZE (0, T] See fig. 1 for an illustration of the homoclinic structure of the unperturbed system ( l)(). Thus the two-dimensional stable and unstable manifolds of y(z), denoted W’( u(i) ) and W’(v(z) ), respectively, coincide along l-s
u qi;(t). z-(0./l
We now turn our attention to the perturbed system ( 1 )c By invariant manifold theory (see ref. [ 81 or ref. [ 9 ] ) for t sufficiently small v(z) becomes a hyperbolic periodic orbit, denoted yt( z) = Y(Z) + 0 ( t ), of period 2~1~10 having two-dimensional stable and unstable manifolds denoted W’( ye(z) ) and W” (ye(z) ), respectively. which need not intersect as in the unperturbed system. In ref. [ 31 it is shown that (up to a normalization constant) the O(t) term in the Taylor expansion of the distance between W’(yt(z)) and W”(y,(z)) is given by
LETTERS
then for t sufficiently small W ‘(y,(z)) and W” ( yt( z) ) intersect transversely. Proqf See ref. [ 3 1. We refer to M(Z) as the adiabatic Melnikov,function. Recall from the Smale-Birkhoff homoclinic theorem (see ref. [ lo] ) that the transversal intersection of W’( ye( 3) ) and W” ( y,( z) ) implies the existence of Smale horseshoes and their attendant chaotic dynamics. Thus (2 ) allows one to explicitly determine the existence of chaos for systems such as ( 1)‘. We now want to show how our techniques are complementary to results of Arnold [ 51 concerning the infinite time adiabatic invariance of the action in ( 1),. In an open set U c IR’ bounded away from r suppose that the unperturbed system (1 )(, contains a one parameter family of closed orbits for each I’E (0, T] (note: this situation is typical of planar hamiltonian systems). Now using the hamiltonian structure of ( 1 ),], a z-dependent canonical transformation to action-angle variables [ 111 can be made in U. Then in action-angle variables ( 1 )(, is written as i=o.
(2) and the following theorem is proved Theorem I. Suppose there exists z= FE (0, T] such that
(2) d+#O, ,’
I I .April 1988
A
&sz(I,z),
of the phase space of
( 1 Jo.
(3)
Thus the structure of the unperturbed phase space in U consists of a family of invariant 2-tori with 0 and -7 being the angular variables on the tori. Arnold defines an adiabatic invariant of the system ( l)E to be a function J(x, y; Z) which changes O(t) on a time scale 0( l/t). It is a standard result that the action is an adiabatic invariant for ( 1 )L [ 111. however Arnold proves that the action not only experiences small changes on finite time intervals. but also that it only changes O(E) on infinite time intervals. More precisely we have the following theorem: Theorem 2 [ 51. Suppose (3) satisfies the hypotheses of the KAM theorem (see ref. [ 12 ] ) . Consider the action. I. of the orbits of ( 1 ), in U. Then for any 6> 0 there exists an e,,(6) >O such that for O
Fig. 1. Geometry
i=o.
<6
forall
--co
The theorem is proved using the techniques of RAM theory [ 121 and is based on the fact that in U the invariant 2-tori of the unperturbed system (3) having “sufficiently irrational” rotation numbers are
Volume 128, number
PHYSICS
6,7
LETTERS
11April 1988
A
preserved for the perturbed system ( 1),, thus these codimension one invariant surfaces form obstacles preventing orbits starting between neighboring nonresonant tori from escaping. The estimates needed to prove the theorem break down as I is approached, thus no conclusions can be made concerning orbits of (1 )c in arbitrarily small neighborhoods of r. However, this is precisely the region where our version of Melnikov’s theory may be applied. The adiabatic Melnikov function gives us information concerning the behavior of I- in the perturbed system and indicates if and how I breaks UP.
We illustrate the theory with an example. Consider a simple planar pendulum whose base is subjected to a slow periodic vertical acceleration. The equation of motion for this system is given by (see ref. [ 13 ] ). 8=21, &-(l-cusinz)sin0,
i=eW,
(4)
where CX>0 represents the amplitude of the vertical oscillation. The unperturbed system is given by B=v,
ti=-(l-cwsinz)sin&
and is hamiltonian
i=O,
with hamiltonian
(5) function
given
by H(B,v;z)=$v2-(l-crsinz)cos0.
(6)
The unperturbed system has a hyperbolic fixed point at (&G)=(rt, 0)=(-n, 0) foreachzc(0, 2x] provided CY< 1. These fixed points are connected to themselves by pairs of homoclinic trajectories given
Fig. 2. Phase space of (5). fixed z.
4ao
cos z (9)
=JiGZz’
So we see that at z=z=krt/2, k= + 1, _+2, ... (9) is zero with dM(zQ a!, w)/dz#O for o#O, O
by (0,’ (l)> 6
(t))
=(+-2sin-‘[tanh(Jmt)], ?cZ,/msech(dmt)). From
(2 ), the adiabatic
(7) Melnikov
function
is given
by M’(z;
Y>w) rn
=
Using
[cuotcoszu$(t)sint?$(t)]dt. s -CL(7), we compute
(8) and obtain
(8)
Fig. 3. PoincarC map of (4).
341
Volume
128. number
6.7
PHYSICS
tion of the 19-&z phase space on which a Poincart map is constructed. The Poincare map is constructed by defining images of points on the cross-section to be their points of first return to the cross-section (fig. 3 ). This is only defined for the perturbed system and occurs after time 27r/t0. Now in the unperturbed system for fixed z, the frequencies of the periodic orbits both inside and outside the homoclinic orbits decrease to zero monotonically as the homoclinic orbits are approached. Thus by Arnold’s theorem, for t sufficiently small, we can find a KAM torus .qinside and close to the homoclinic orbits and KAM tori .“i; and .&outside and close to the homoclinic orbits which serve to create an invariant set in a neighborhood of the homoclinic orbits, i.e., points starting in the region bounded by ,z .c and 4forever remain in this region. However, this region also contains Smale horseshoes associated with the homoclinic orbits. Thus, although the action of orbits starting in this region remains close to its initial value, the orbits nevertheless evolve chaotically due to the presence of the Smale horseshoes. We close by remarking that our methods have recently been generalized to multidegree of freedom systems undergoing multiply periodic excitations with weak damping, see ref. [ 3 1.
342
LETTERS
A
I I April 1988
References [ I] J.L. Tennyson, J.R. Gary. and D.F. Escande. Phys. Rev. Lett. 56 (1986) 2117. [2] J.H. Hannay, J. Phys. .4 19 (1986) L1067. [3] S. Wiggins. SIAM J. Appl. Math.. April 1988. to be published: Global bifurcations and chaos-analytical methods, Springer Applied Mathematical Sciences Series (Springer. Berlin). to be published. [4] V.K. Melnikov. Trans. Moscow Math. Sot. 12 ( 1963) I. [5] V.I. Arnold, Soy. Math. Dokl. 3 (1962) 136. [6] D.F. Escande. Proc. Int. Workshop on Plasma theory and nonlinear and turbulent processes in physics, Kiev, 1987. [7] J.R. Gary. D.L. Bruhwiler, P. Rusu and R.T. Skodje, Bull. Am. Phys. Sot. 32 (1987) 1861. [ 8] M.W. Hirsch. CC. Pugh and M. Shub, Springer lecture notes m mathematics, Vol. 583. Invariant manifolds (Springer, Berlin, 1977). [9 N. Fenichel. Indiana Univ. Math. J. 2 I ( I97 I ) 193. and P.J. Holmes. Nonlinear oscillattons. ]I0 J. Guckenheimer dynamical systems and bifurcations of vector fields (Sprtnger. Berlin. 1983). V.I. Arnold. Mathematical methods of classical mechantcs ]ll (Sprtnger, Berlin. 1977). ]I2 V.I. -\rnold, Russ. Math. Surv. I8 ( 1963) 9. [l3 A.H. Nayfeh and D.T. Mook. Nonlinear oscillations (Wiley. New York, 1979).
ADIABATIC Stephen
6,7
PHYSICS
LETTERS
A
11 April 1988
CHAOS
WIGGINS
Applied Mechanics. 104-44, Califorma Institute Received 16 September 1987; revised manuscript Communicated by D.D. Holm
of Technology, Pasadena, CA 91125, USA received
25 January
1988: accepted
for publication
5 February
1988
We consider single degree of freedom hamiltonian systems depending periodically on a parameter which varies slowly in time. We show how a recent extension of Melnikov’s method coupled with a theorem of Arnold implies that in some systems of this type the action of open sets of orbits may remain close to their initial values for all time yet the orbits evolve in a chaotic manner. We illustrate the results with the example of a simple pendulum whose base is slowly accelerated.
The adiabatic invariance of the action in single degree of freedom hamiltonian systems containing a slowly varying parameter is important in many applications. However, most classical results along these lines depend crucially on the fact that the frequency of the motion is nonzero. This condition is violated on separatrices. Recently, there have been some results concerning motion across separatrices in such systems, see refs. [ 1,2]. These studies have concentrated on determining the change in the action of a specific orbit as the separatrix is crossed. In this Letter we show how recently developed global perturbation techniques (see ref. [ 31) along the lines of Melnikov [4] allow one to determine that the motion near separatrices in such systems where the parameter varies slowly, but periodically, in time may be chaotic due to the presence of Smale horseshoes (note: the results of Tennyson et al. and Hannay allow a more general time dependence). The difference in our methods and the original Melnikov [ 41 technique can be succinctly stated in the context of nonlinear oscillations. Melnikov’s method applies to single degree of freedom oscillators forced periodically with an excitation having 0( 6) amplitude and 0( 1) frequency. Our method applies to single degree of freedom oscillators forced periodically with an excitation having 0 ( 1) amplitude and 0 ( t ) frequency. Thus our method should give information concerning chaotic dynamics in an entirely new class of problems. Coupling these results with a theorem 0375-9601/88/S (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
of Arnold [ 51 we conclude that in some situations the action of orbits near separatrices may not wander far from its initial value yet the orbit evolves in a chaotic manner. We illustrate the theory with an example of a pendulum whose base is oscillated slowly and periodically in time. We remark that simultaneously Escande [ 61 has considered the possibility of chaos in the periodic case and has obtained similar results although with different methods. Also, Cary et al. [ 7 ] have studied similar problems arising in the context of plasma physics. We consider an analytic hamiltonian function, H=H(x, y; z),z=wt, periodic in z with period T which defines the following hamiltonian vector field:
dqx,y;z), )I=-g(x)y;z)) dY I-=tCO
(lb
and we will refer to the following parametrized ily of systems as the unperturbed system
fam-
i=
qx, aY
i=o.
y; z),
li= -
+;(x,y;z)
)
(110
We have the following assumption on the structure of the phase space of the unperturbed system. Assumption. Vz E (0, T] ( 1 ). has a hyperbolic fixed B.V.
339
Volume
128, number
PHYSICS
6.7
point y(z) which is connected to itself by a homoclinic orbit qt: (t). Moreover we assume that y(z) and q; ( t ) are differentiable (at least C’ ) VZE (0, T] See fig. 1 for an illustration of the homoclinic structure of the unperturbed system ( l)(). Thus the two-dimensional stable and unstable manifolds of y(z), denoted W’( u(i) ) and W’(v(z) ), respectively, coincide along l-s
u qi;(t). z-(0./l
We now turn our attention to the perturbed system ( 1 )c By invariant manifold theory (see ref. [ 81 or ref. [ 9 ] ) for t sufficiently small v(z) becomes a hyperbolic periodic orbit, denoted yt( z) = Y(Z) + 0 ( t ), of period 2~1~10 having two-dimensional stable and unstable manifolds denoted W’( ye(z) ) and W” (ye(z) ), respectively. which need not intersect as in the unperturbed system. In ref. [ 31 it is shown that (up to a normalization constant) the O(t) term in the Taylor expansion of the distance between W’(yt(z)) and W”(y,(z)) is given by
LETTERS
then for t sufficiently small W ‘(y,(z)) and W” ( yt( z) ) intersect transversely. Proqf See ref. [ 3 1. We refer to M(Z) as the adiabatic Melnikov,function. Recall from the Smale-Birkhoff homoclinic theorem (see ref. [ lo] ) that the transversal intersection of W’( ye( 3) ) and W” ( y,( z) ) implies the existence of Smale horseshoes and their attendant chaotic dynamics. Thus (2 ) allows one to explicitly determine the existence of chaos for systems such as ( 1)‘. We now want to show how our techniques are complementary to results of Arnold [ 51 concerning the infinite time adiabatic invariance of the action in ( 1),. In an open set U c IR’ bounded away from r suppose that the unperturbed system (1 )(, contains a one parameter family of closed orbits for each I’E (0, T] (note: this situation is typical of planar hamiltonian systems). Now using the hamiltonian structure of ( 1 ),], a z-dependent canonical transformation to action-angle variables [ 111 can be made in U. Then in action-angle variables ( 1 )(, is written as i=o.
(2) and the following theorem is proved Theorem I. Suppose there exists z= FE (0, T] such that
(2) d+#O, ,’
I I .April 1988
A
&sz(I,z),
of the phase space of
( 1 Jo.
(3)
Thus the structure of the unperturbed phase space in U consists of a family of invariant 2-tori with 0 and -7 being the angular variables on the tori. Arnold defines an adiabatic invariant of the system ( l)E to be a function J(x, y; Z) which changes O(t) on a time scale 0( l/t). It is a standard result that the action is an adiabatic invariant for ( 1 )L [ 111. however Arnold proves that the action not only experiences small changes on finite time intervals. but also that it only changes O(E) on infinite time intervals. More precisely we have the following theorem: Theorem 2 [ 51. Suppose (3) satisfies the hypotheses of the KAM theorem (see ref. [ 12 ] ) . Consider the action. I. of the orbits of ( 1 ), in U. Then for any 6> 0 there exists an e,,(6) >O such that for O
Fig. 1. Geometry
i=o.
<6
forall
--co
The theorem is proved using the techniques of RAM theory [ 121 and is based on the fact that in U the invariant 2-tori of the unperturbed system (3) having “sufficiently irrational” rotation numbers are
Volume 128, number
PHYSICS
6,7
LETTERS
11April 1988
A
preserved for the perturbed system ( 1),, thus these codimension one invariant surfaces form obstacles preventing orbits starting between neighboring nonresonant tori from escaping. The estimates needed to prove the theorem break down as I is approached, thus no conclusions can be made concerning orbits of (1 )c in arbitrarily small neighborhoods of r. However, this is precisely the region where our version of Melnikov’s theory may be applied. The adiabatic Melnikov function gives us information concerning the behavior of I- in the perturbed system and indicates if and how I breaks UP.
We illustrate the theory with an example. Consider a simple planar pendulum whose base is subjected to a slow periodic vertical acceleration. The equation of motion for this system is given by (see ref. [ 13 ] ). 8=21, &-(l-cusinz)sin0,
i=eW,
(4)
where CX>0 represents the amplitude of the vertical oscillation. The unperturbed system is given by B=v,
ti=-(l-cwsinz)sin&
and is hamiltonian
i=O,
with hamiltonian
(5) function
given
by H(B,v;z)=$v2-(l-crsinz)cos0.
(6)
The unperturbed system has a hyperbolic fixed point at (&G)=(rt, 0)=(-n, 0) foreachzc(0, 2x] provided CY< 1. These fixed points are connected to themselves by pairs of homoclinic trajectories given
Fig. 2. Phase space of (5). fixed z.
4ao
cos z (9)
=JiGZz’
So we see that at z=z=krt/2, k= + 1, _+2, ... (9) is zero with dM(zQ a!, w)/dz#O for o#O, O
by (0,’ (l)> 6
(t))
=(+-2sin-‘[tanh(Jmt)], ?cZ,/msech(dmt)). From
(2 ), the adiabatic
(7) Melnikov
function
is given
by M’(z;
Y>w) rn
=
Using
[cuotcoszu$(t)sint?$(t)]dt. s -CL(7), we compute
(8) and obtain
(8)
Fig. 3. PoincarC map of (4).
341
Volume
128. number
6.7
PHYSICS
tion of the 19-&z phase space on which a Poincart map is constructed. The Poincare map is constructed by defining images of points on the cross-section to be their points of first return to the cross-section (fig. 3 ). This is only defined for the perturbed system and occurs after time 27r/t0. Now in the unperturbed system for fixed z, the frequencies of the periodic orbits both inside and outside the homoclinic orbits decrease to zero monotonically as the homoclinic orbits are approached. Thus by Arnold’s theorem, for t sufficiently small, we can find a KAM torus .qinside and close to the homoclinic orbits and KAM tori .“i; and .&outside and close to the homoclinic orbits which serve to create an invariant set in a neighborhood of the homoclinic orbits, i.e., points starting in the region bounded by ,z .c and 4forever remain in this region. However, this region also contains Smale horseshoes associated with the homoclinic orbits. Thus, although the action of orbits starting in this region remains close to its initial value, the orbits nevertheless evolve chaotically due to the presence of the Smale horseshoes. We close by remarking that our methods have recently been generalized to multidegree of freedom systems undergoing multiply periodic excitations with weak damping, see ref. [ 3 1.
342
LETTERS
A
I I April 1988
References [ I] J.L. Tennyson, J.R. Gary. and D.F. Escande. Phys. Rev. Lett. 56 (1986) 2117. [2] J.H. Hannay, J. Phys. .4 19 (1986) L1067. [3] S. Wiggins. SIAM J. Appl. Math.. April 1988. to be published: Global bifurcations and chaos-analytical methods, Springer Applied Mathematical Sciences Series (Springer. Berlin). to be published. [4] V.K. Melnikov. Trans. Moscow Math. Sot. 12 ( 1963) I. [5] V.I. Arnold, Soy. Math. Dokl. 3 (1962) 136. [6] D.F. Escande. Proc. Int. Workshop on Plasma theory and nonlinear and turbulent processes in physics, Kiev, 1987. [7] J.R. Gary. D.L. Bruhwiler, P. Rusu and R.T. Skodje, Bull. Am. Phys. Sot. 32 (1987) 1861. [ 8] M.W. Hirsch. CC. Pugh and M. Shub, Springer lecture notes m mathematics, Vol. 583. Invariant manifolds (Springer, Berlin, 1977). [9 N. Fenichel. Indiana Univ. Math. J. 2 I ( I97 I ) 193. and P.J. Holmes. Nonlinear oscillattons. ]I0 J. Guckenheimer dynamical systems and bifurcations of vector fields (Sprtnger. Berlin. 1983). V.I. Arnold. Mathematical methods of classical mechantcs ]ll (Sprtnger, Berlin. 1977). ]I2 V.I. -\rnold, Russ. Math. Surv. I8 ( 1963) 9. [l3 A.H. Nayfeh and D.T. Mook. Nonlinear oscillations (Wiley. New York, 1979).