Existence of chaos in evolution equations

MATHEMATICAL AND COMPUTER MODELLING PERGAMON

Mathematical and Computer Modelling 36 (2002) 1211-1219 www.elsevier.com/locate/mcm

E x i s t e n c e of C h a o s in E v o l u t i o n E q u a t i o n s Y A N G U A N G C . L1 School of Mathematics, Institute for Advanced Study Princeton, NJ 08540, U.S.A.

(Received August 2001; revised and accepted August 2002) A b s t r a c t - - F o r a general evolution equation with a Silnikov homoclinic orbit, Smale horseshoes are constructed with the tools of [1] and in the same way as in [1]. The linear part of the evolution equation has a finite number of unstable modes. For evolution equations with infinitely many linearly unstable modes, the problem is still open. (~ 2002 Elsevier Science Ltd. All rights reserved. 1. I N T R O D U C T I O N In recent years, t h e e x i s t e n c e of chaos in p a r t i a l differential e q u a t i o n s has b e e n e s t a b l i s h e d [1-3]. T h e s e s t u d i e s set up a scheme for a t t a c k i n g p r o b l e m s on chaos in P D E s . T h e t y p e of chaos s t u d i e d in t h e s e works is t h e so-called h o m o c l i n i c chaos g e n e r a t e d in a n e i g h b o r h o o d of a h o m o c l i n i c o r b i t . T w o t y p e s of h o m o c l i n i c o r b i t s have b e e n s t u d i e d . O n e t y p e is t h e so-called t r a n s v e r s a l h o m o c l i n i c o r b i t [4]. T h e o t h e r t y p e is t h e so-called Silnikov h o m o c l i n i c o r b i t , which is n o n t r a n s v e r s a l [1-3]. For l o w e r - d i m e n s i o n a l a n d general f i n i t e - d i m e n s i o n a l s y s t e m s , Silnikov s t u d i e d t h e s y m b o l i c d y n a m i c s s t r u c t u r e s in t h e n e i g h b o r h o o d s of such h o m o c l i n i c o r b i t s [5-8]. In [9] a n d [1], we have d e v e l o p e d a different c o n s t r u c t i o n of S m a l e h o r s e s h o e s in t h e n e i g h b o r h o o d of a Silnikov h o m o c l i n i c o r b i t . T h e a d v a n t a g e s of our c o n s t r u c t i o n s have b e e n fully a d d r e s s e d in [1,9]. In t h i s note, we generalize t h e c o n s t r u c t i o n in [1] t o m o r e g e n e r a l e v o l u t i o n e q u a t i o n s w i t h finitely m a n y l i n e a r l y u n s t a b l e modes. For e v o l u t i o n e q u a t i o n s w i t h infinitely m a n y linearly u n s t a b l e m o d e s , t h e p r o b l e m is still open. 2.

THE

SET-UP

Consider the evolution equation =

+

(1)

w h e r e L is a linear o p e r a t o r which is c o n s t a n t in t i m e , a n d :N is t h e n o n l i n e a r t e r m . Following are t h e a s s u m p t i o n s for t h e setup. ( A 1 ) u = 0 is a saddle and the linear operator L has only point spectrum a s follows:

o(L) = { - a + i j 3 , % A ~ , j E S + C Z + } , This work is supported by a Guggenheim Fellowship and an AMS Centennial Fellowship. Current address: Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. Emaih cli©math .missouri. edu

0895-7177/02/$ - see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. PII: S0895-7177(02)00270-4

Typeset by Afl~-TEX

1212

Y.C. LI

where c~ > 0, fl > 0, 3' > 0, Re{A~} <> 0. T h e n u m b e r of elements in S + is finite, d e n o t e d by N , ct < ~, and

c~
inf { - R e { A ; } } ,

~,
j6S-

inf { R e { A + } } .

j6S+

(A2) T h e evolution equation (1) is globally well p o s e d in a Hilbert space 9-C, that is, there exists a unique solution to the Cauchy problem of (I), u(t, uo) E C ° [ ( - o c , oc), 9/], u(0, u0) = u0. Moreover, we assume the regularity condition on initial data that for any t C ( - o c , oo), the evolution operator F ( u o ) t = u(t, uo) is C n in Uo for s o m e n > 2. (A3) W i t h respect to the saddle u = O, the evolution operator F t ( u ) a d m i t s a C 2 s m o o t h linearization, i.e., there exists a C 2 diffeomorphism J~ : ~K --* J-C, such that in terms of the n e w variable (z = J~ u, the evolution equation (1) is transformed into the linear form ot~ = h e

in a neighborhood of u = O. T h e conjugated evolution operator [.t = j t F t j t - 1 is still C o in t i m e t and C 2 in (L. In fact, we assume that the evolution equation (1) takes the normal form Jc = -c~x - fly + Gz ( x, y, z, v± ) , = Zx - ~y + G

(x,y,z,v±),

= ~z + Gz ( x , y , z , v ± ) , Otv ± = L e v ± + G,± ( x , y , z , v ± ) ,

where G = 0 in a neighborhood f~ of O, v + = (v+, . . . , v+ ) ', N is the number o f elements in S +, x, y, z, and v+ are tea/variables, and e L+t

~ c+e ;~+t,

as t --* --oo,

e L-t

< c-e -x-t,

as t ~ +oo.

For references on such linearization results, see, for example, [10], etc. (A4) There exists a Silnikov homoclinic orbit h(t) a s y m p t o t i c to O. A s t --* +(xD, h is tangent to the (x, y)-plane at O, and as t ---* - o o , h is tangent to the z-axis ( w i t h o u t loss of generality, positive z-axis) at O. T h e stable and unstable manifolds o f 0 are C 2 smooth, and dim {~rvW ~ A ~YvW s } = 1, where v c h(t), g ' v W u is the tangent space of the unstable manifold of 0 at the point v on the homoclinic orbit h(t ), similarly for ~YvW s.

REMARK. Proving the existence of Silnikov homoclinic orbits in partial differential equations is a rather nontrivial question. So far, this has been done for perturbed nonlinear Schr6dinger equations [3,11], perturbed vector nonlinear Schr6dinger equations [12], and perturbed discrete nonlinear Schr6dinger equations [13]. The above and later assumptions have been either verified or discussed for these equations in [1,9,14]. The perturbed Davey-Stewartson II equation has been studied along this direction [15,16]. Unfortunately, existence of Silnikov homoclinic orbits has not been proved due to some technical difficulty [16]. I would like to comment on equations that have the potential of being cast into the above setup: 1. perturbations of the modified KdV equation Ut q- 6U2Ux q- Uzxx = O,

Evolution Equations

1213

2. perturbations of the derivative nonlinear Schr6dinger equation [17-20], a > 0,

x + 21ul2u'

and 3. perturbations of the derivative nonlinear Schr6dinger equation iut = Uzx -- iau2uz + 2]u12u + 2a2[u[ 4u.

All three of the above equations are integrable systems.

3. T H E C O N S T R U C T I O N

OF SMALE HORSESHOES

3.1. D e f i n i t i o n s

DEFINITION. The Poincard section ~ o is defined by the constraint y=O,

r / e x p { - ~ -~} < x < r h

o < = < ,7,

II v+ II < ,7,

where r1 is a sufficiently small constant so that ~-~0 is included in the neighborhood f~ of 0 where the dynamics is given by the linear system.

DEFINITION. The auxiliary section ~ + is defined by the constraints

[ 2~ }

v=o,

~ exp 1.---7

-,7 < z < ,7,


II v± II <

"~.

The homoclinic orbit h intersects the (z = O):boundary Of ~ o at w (+) with coordinates denoted by

x=x.,

y=0,

z=0,

v+ = 0 ,

v- = v , .

There exists T > 0 such that the point w (-) = F - T ( w (+)) on h (where F t is the evolution operator) has the z coordinate equal to r]. Denote the coordinates of w (-) by =0,

x=y=v-

z=rl,

v + = v +.

DEFINITION. The Poincard section ~-~t is defined as ~1 =

F-To

N~.

DEFINITION. The map P~ from ~ o to ~ 1 is defined as

P~ :Vo~ y:, 0

vw ~ Uo,

'Z, 1

Po~(w)= u"(~) c Z , 1

where t, = t . ( w ) > 0 is the smallest time t such that F t ( w ) c ~1" The map from E 1 to ~ o (= E o UO E o ) is defined as

~ ° : < ~ Z1 ~ Z

o,

po(~) = F~(,,,) ~ ~ o

v,,, ~ Z , 1

The Poincard map P from ~ o to itself is defined as

":':v~X:' O

,X:, O

,,:,:pOopS.

1214

Y.C. LI

3.2. F i x e d P o i n t s o f t h e P o i n c a r ~ M a p P On the Poincard section ~ 0 , we center the origin of the coordinate frame at w (+), and denote the new coordinates by (x(°),z(°),v(+,°)). On the Poincar~ section Y~I, we center the origin of the coordinate frame at w(-), and denote the new coordinates by (x (1), y(1), z(1), v(+,z)), which satisfy the constraint equation f(V)

X(1),y(1),Z (1) +rl, v (+'1) +V,+ , V(-'1)) ----O,

(2)

where, for any w E J£, :(~.)

--

(S <~)(w),s(~)(~.), s(z)(w),s <~÷)(~0),f(v )(w)).

Denote the vector v + in component form, v+ =~(V+l, v+2,. .. , v+) '. Then we have the following lemma. LEMMA 1. 0 w ~ (( - ) ) 0 z

and °/(") Or+ ~:w(-)~: (j = 1 , . . . , N ) cannot be zero simultaneously.

PROOF. Assume that they are zero simultaneously, then 7 - . V f ( Y ) ( w (-)) = 0 , where ~- is the tangent vector of h at w(-) and " V " denotes gradient. Then this implies that h is tangent to ~ 0 at w (+). This contradiction proves the lemma. | Let ( be one of the coordinates {z, v+ (j = 1 , . . . ' N)}, such that °/(') :w(-)~: # O, and denote by 0~ ~ v(+,l) v(+1) the vector (z (1), j (j = 1 , . . . , N))' \ {~}, i.e., with components consisting of {z (1), vj(+,l) (j = 1 , . . . , N ) } without ~. LEMMA 2. In a neighborhood of w ( - ) , the Poincarg section ~ 1 function

can be represented as a C 2

~ =~ (X(1) y(1),V(-'l),v(l)) . PROOF. Applying the implicit function theorem to (2).

|

The map P01 has the representation x(1) = e - ~ t * ( x ( ° ) + x . ) c o s / 3 t . , y(1) = e-~t. ( x ( 0 ) + x . ) s i n / 3 t . , z (1) + r/ = z(°)e "~t" , v (+'1) + v.+ = eL+t.v(+,O) :

eL

÷

The map P1° can be approximated by its linearization at w(-), / X(1) x (°) ] y(1) z (°) / = A | z (1) v (+'°) v (-,°) ] \v(-,1)

|v(+,l)

,

Evolution Equations

where

A =

1215

Of(~) Ox Of( z ) Ox

Of(~) Oy Of( z ) Oy

Of(~) Oz Of( z ) Oz

Of(~) Or+ Of( z ) Or+

Of( x ) OrOf( z ) Or-

Of(~+) Ox Of(v-) Ox

Of(~+) Oy Of(v-) Oy

Of(~÷) Oz Of(v-) Oz

Of(~+) Ov+ Of(v-) Or+

Of@ +) OvOf(~-) Or-

(w(-))

The constraint equation (2) can be approximated by its linearization at w (-), y(1) B

Z (1)

----~O,

v(+,l) v(-,1) where

( O f (y) Of (v) Of(v) Of(v) Of (y)) B = \ -~x Oy Oz Or+ Or- (,~(_)) W i t h the above preparations, we can write the equations of the fixed points of P as follows in terms of the Silnikov coordinates {t., x (°), v(-'°), z(1), v(+,l) }:

(~_o, (~(o) + ~.)cos ~,.

x (0)

(z(') + ~) e-~" e_L+t. (v(+,l) + v+ )

I e-at* (x(0) + x . ) sin Jt. = A /

z(1)

v(-'°)

\

J

+ :2,

v(+,l)

eL-. (v(-,O) +v:)

(3)

where :2 = O(e -Èt* ) as t. --~ +co for some ~ > 0. B y rescaling the coordinates as follows:

{t.,:~(O)

= x(O)eat.,~)(-,O)

_-- V(-,O)eat., ~(1) = z(1)eat.,~(+,l)

= v(+,l)eat. }

we can rewrite equation (3) in the form

(~(0)) /x.cos~t,, / x . sin j3t.

(~ =A / ~(') ~(-,0) (¢o '1'

+:2,.

(4)

where :21 = O(e - " i t ' ) as t, ~ +co for some Ul > O, and the constraint equation (2) takes the form [ x. cos f~t. '~ | x, sin f~t,

~ ~(;'~)

B |

~(~)

J

+ :25 = 0,

(5)

where :22 = O(e -"2t*) as t. -~ +co for some v2 > 0. Solving the leading order term of (4) for (~(1), ~)(+,1)), we have C

~(+,1)] = - D \ x . s i n Z t .

'

(6)

1216

Y.C.

LI

,

D =

where

C =

Of(V+) Oz

Of(v+) Or+

Of@+ ) Ox

(~-~)

Of(V+) Oy

(~-~)

LEMMA 3. Matrix C is invertible. PROOF. Assume t h a t C is noninvertible; then there exists a nonzero vector (~(+.~)), such t h a t

C t 0(+,1 )

- - O,

thus,

w=A(~(~)

/0(; '1>

GT~(+~W~O-~(+)W ~,

where 0"w(+)W s is the t a n g e n t space of W s at w (+), similarly for 9"w(+)W ~. Since A is invertible, w # 0. We also know t h a t ~Y~(+>h • Ir~(+)W s n :T~(+IW ~. Since [Tw(+) h is transversal to ~ o and w lies in ~-~-o, we have dim {9"w(+~W~ N g'w(+) W s} = 2, which contradicts A s s u m p t i o n (A4). This completes the proof. Solving (6), we have

('") 0 (+'1)

= _C_ID

(

x. cos/3t. x. sinflt.

)

(7)

"

T h e n solving the constraint equation (5), to the leading order, we have x . COS/3t. x , sin/3t. B

\

J

=0,

_ C - 1 D ( x* c°s /3t* ) x . sin/3t. 0

which can be rewritten as

(s)

A1 cos 13t. + A2 sin fit. = 0. We assume the following condition. (A5) If A1 and A2 do not vanish simultaneously, then (8) has a sequence of solutions

t~~) = ~ / ~

- ~],

t c z +,

(9)

where =

arctan

~2

"

~(o)

Then substituting (7) and (9) into the leading order terms of (4), we can solve for ( ~( .o~ ) to the leading order. Finally, applying the implicit function theorem, we have the fixed-point theorem.

Evolution Equations

1217

THEOREM 1. FIXED-POINT THEOREM. Under A s s u m p t i o n (A5), there exists an integer to > O, such that there exists a sequence of solutions to equations (4) and (5) labeled by ~ (~ >_ 6o): t. = T (~), ~(1)

~(o) = x(e)

z(e),

=

©(+,1)

0(-,o) = v(e)

,

(-,o),

~ (e)

= u(+,l ) ,

where, as t ---* +oo,

T (e) = 1 [(rr - ~] + o(1). For a complete proof of this theorem, see [1]. 3.3. S m a l e H o r s e s h o e s DEFINITION. For sufficiently large n u m b e r £, we define the slab Seil] Y~o as follows:

_< 'r/exp

- ~

,

so that it contains two fixed points of P denoted by p+ and Pe.

We choose a basis on the tangent space T~, ) ~ 1 represented in the coordinates

(X(1) y(1) y(--']) ~,V(+1)) as follows: Ex(1 ) =

' l , O , O , V ~ ( w (-)) ,-,(1,0,0,0),0),

/~y(1) =

"0, 1, O, ~ 7 ( ( w ( - ) ) o (0, 1, O, 0), O) ,

Ei~( ,1) = E v (+1) =

o,o,o

where It represents a basis for the corresponding components. Denote by {e(o) ,e(o),e,~+,o~, e c_ ,,) } the unit vectors along (x (°) , z (°), v (+'°), v(-'°))-directions in ~ 0 " In this coordinate frame, Se has the product representation as shown in Figure 1.

ez(O)

ev(-,0)

3

® 1

2

ev(+,°)

4 ,,

ex(o)

Figure 1. The product representation of Sg.

1218

Y . C . LI

P-I

®

~'v<2

®

Figure 2. A geometric representation of V P ° ( w ( - ) ) o VP~ (w (+)) o St. Under the linear map A (the linearization of pO at w(-)), the coordinate frame {Ex¢l~,Ey~l,, Ev(-,1) , Ev¢+l)} is mapped into a coordinate frame {~x(1), ~y(l>, £v(-.1~, £v¢+U} on ~ 0 with origin at w (+). In this coordinate frame, VP°(w (-)) o VP01(w (+)) o Se has the representation as shown in Figure 2 on Y~0" We introduce a system of curvilinear coordinates ( ~ , ~s) on the (gxIu, ~y(~ )plane such that { ~ = 0},

{ ~ = c~ (a constant)},

{~ = 0},

{~ = cs (a constant)}

correspond to the boundaries 3, 4, 1, 2 of the annulus on the (£~(~/, gy(,~ )-plane. Let g~- = '~C (~ be the tangent vector to the ~ coordinate at p+. We make the following assumption. (A6) Span {e(o) , e (_ o), ~ - , Ev(+~>} = ~0Under Assumptions (A1)-(A6), we can verify the Conley-Moser conditions [21] in the same way as in [1], which lead to the construction of Smale horseshoes. Let W be a set which consists of elements of the doubly infinite sequence form a = (... a - 2 a - l a o , ala2 • • .) ,

where ak c {0, 1}, k E Z. We introduce a topology in W by taking as neighborhood basis of a*=(

. . . . . • ..a_2a_la

o, a l a 2 . . .

)

,

the set Wj = {acW

I ak =a~,(Ikl < j ) } ,

for j = 1, 2 , . . . . This makes W a topological space. The shift automorphism X is defined on W by x:W' ,W, VacW, x(a) = b , wherebk = a k + l . The shift automorphism )4 exhibits sensitive dependence on initial conditions, which is the hallmark of chaos. THEOREM 2.

SMALE HORSESHOE THEOREM. Under A s s u m p t i o n s ( A 1 ) - ( A 6 ) for the evolution equation (1), for all sufficiently large integers g, there exists a sequence of compact Cantor subsets Ae of Se, Ae consists of points and is invariant under P. P restricted to Ae is topologicM1y conjugate to the shift a u t o m o r p h i s m X on two symbols 0 and 1. T h a t is, there exists a homeomorphism ¢e : W ,

, At,

Evolution Equations

1219

such that the following diagram commutes: W

W

¢~

¢,

>

At

,

Ae.

PROOF. With preliminaries given above, the proof follows in the same way as in [1].

4. C O N C L U S I O N

AND

DISCUSSION

In this note, we have generalized the construction of Smale horseshoes in [1] to a general evolution equation, which indicates that our techniques [1,9] on constructing Smale horseshoes in a neighborhood of a Silnikov homoclinic orbit has a much wider application. On the other hand, so far we can only handle evolution equations with finitely many linearly unstable modes. For evolution equations with infinitely many linearly unstable modes, we cannot invert certain linear operators in establishing the existence of fixed points of the Poincar@ map. Nevertheless, this note furnishes an initiation for studying Silnikov homoclinic orbits for general evolution equations, thereby proving the existence of chaos for general evolution equations.

REFERENCES 1. Y. Li, Smale horseshoes and symbolic dynamics in perturbed nonlinear Schr6dinger equations, Journal of Nonlinear Sciences 9 (4), 363 (1999). 2. Y. Li and D.W. McLaughlin, Morse and Melnikov functions for NLS pdes, Comm. Math. Phys. 162, 175 (1994). 3. Y. Li et al., Persistent homoclinic orbits for a perturbed nonlinear Schr6dinger equation, Comm. Pure Appl. Math. X L I X , 1175 (1996). 4. Y. Li, Chaos and shadowing lemma for autonomous systems of infinite dimensions (submitted) Available at

http ://www. math. missouri, edu/~cli (2002). 5. L.P. Silnikov, A case of the existence of a countable number of periodic motions, Soviet Math. Doklady 6, 163 (1965). 6. L.P. Silnikov, The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus, Soviet Math. Doklady 6, 163 (1965). 7. L.P. Silnikov, On a Poincar@-Birkoff problem, Math. USSR Sb. 3 (1967). 8. L.P. Silnikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sb. 10 (1970). 9. Y. Li and S. Wiggins, Homoclinic orbits and chaos in discretized perturbed NLS system, Part II. Symbolic dynamics, Journal of Nonlinear Sciences 7, 315 (1997). 10. N.V. Nikolenko, The method of Poincar@ normal forms in problems of integrability of equations of evolution type, Russian Math. Surveys 41 (5), 63 (1986). 11. Y. Li, Persistent homoclinic orbits for nonlinear Schr6dinger equation under singular perturbation, (submitted) (2002). 12. Y. Li, Singularly perturbed vector and scalar nonlinear Schr6dinger equations with persistent homoclinic orbits, Studies in Applied Mathematics 109, 19-38 (2002). 13. Y. Li and D.W. McLaughlin, Homoclinic orbits and chaos in discretized perturbed NLS system, Part I. Homoclinic orbits, Journal of Nonlinear Sciences 7, 211 (1997). 14. Y. Li, Existence of chaos for a singularly perturbed NLS equation, Acta. Math (submitted). 15. Y. Li, B~icklund-Darboux transformations and Melnikov analysis for Davey-Stewartson II equations, Journal of Nonlinear Sciences 10 (1), 103 (2000). 16. Y. Li, Melnikov analysis for singularly perturbed DSII equation, Results in Math. (submitted). 17. V.S. Shchesnovich and E.V. Doktorov, Perturbation theory for the modified nonlinear Schr5dinger solitons, Phys. D 129 (1/2), 115 (1999). 18. V.S. Gerdjikov, E.V. Doktorov and J. Yang, Adiabatic interaction of N ultrashort solitons: Universality of the complex Toda chain model, Phys. Rev. E 64 (5, Part 2) (2001). 19. K. Mio e t al., Modified nonlinear Schr6dinger equation for Alfven waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1), 265 (1976). 20. M. Wadati et al., A generalization of inverse scattering method, J. Phys. Soc. Japan 46 (6), 1965 (1979). 21. J. Moser, Stable and Random Motions in Dynamical Systems, Annals of Mathematics Studies, Volume 77, Princeton University Press, (1973).