- Email: [email protected]
Homoclinic tubes and chaos in perturbed sine-Gordon equation
Chaos, Solitons and Fractals 20 (2004) 791–798 www.elsevier.com/locate/chaos
Homoclinic tubes and chaos in perturbed sine-Gordon equation Y. Charles Li
*
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Accepted 27 August 2003 Communicated by L. Reichl
Abstract Sine-Gordon equation under a quasi-periodic perturbation or a chaotic perturbation is studied. Existence of a homoclinic tube is proved. Established are chaos associated with the homoclinic tube, and ‘‘chaos cascade’’ referring to the embeddings of smaller scale chaos in larger scale chaos. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Propagations of nonlinear waves through homogeneous media are often modeled by well-known nonlinear wave equations, for example, sine-Gordon equation. Studies were also drawn to variable media [2–4]. Variations of the media can speed up, slow down (even stop), or break the wave propagations. Studies have been focused upon such variations of the media, which are localized defects. In the current article, we will study quasi-periodic or chaotic media. The equation to be studied can be called a quasi-periodically or a chaotically defective sine-Gordon equation. This equation represents a concrete example realizing the chaos theorem proved in [1,5]. The chaos theorem, proved through a shadowing lemma, claims the existence of Bernoulli shift dynamics of tori in a neighborhood of a homoclinic tube. Bernoulli shift dynamics has sensitive dependence upon initial data in the product topology. In fact, Bernoulli shift has become a hallmark of chaos. Existence of a homoclinic tube and Bernoulli shift dynamics will be established for the defective sine-Gordon equation. In the case of a chaotic medium, often the dynamics inside the homoclinic tube is chaotic too. Such chaos is called ‘‘chaos in the small’’. The global chaos associated with the homoclinic tube is called ‘‘chaos in the large’’. We see the embedding of smaller scale chaos in larger scale chaos. Such embedding can be continued with even smaller scale chaos. We call this chain of embeddings of smaller scale chaos in larger scale chaos, a ‘‘chaos cascade’’. The article is organized as follows: In Section 2, we present the formulation of the problem. Section 3 is on an integrable theory. Section 4 is on the existence of a homoclinic tube and chaos. Section 5 is on a discussion of ‘‘chaos in the large’’ and ‘‘chaos in the small’’.
2. Formulations of the problem Consider the sine-Gordon equation under a quasi-periodic perturbation,
*
Tel.: +1-573-8840622; fax: +1-573-8821869. E-mail address: [email protected] (Y.C. Li).
0960-0779/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.013
792
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
utt ¼ c2 uxx þ sin u þ ½Du þ f ðtÞðsin u uÞ;
ð2:1Þ
which is subject to periodic boundary condition and odd constraint uðt; x þ 2pÞ ¼ uðt; xÞ;
uðt; xÞ ¼ uðt; xÞ;
ð2:2Þ
where u is a real-valued function of two real variables t P 0 and x, c is a parameter, 12 < c < 1, is a small parameter, P 0, D is a dissipative operator Du ¼ aut þ butxx ;
a P 0;
bP0
and f ðtÞ is quasi-periodic with a basis of frequencies x1 ; . . . ; xN , f ðtÞ ¼
N X
an cos½hn ðtÞ;
hn ðtÞ ¼ xn t þ h0n ;
n¼1
where an and h0n are parameters. The above system is invariant under the transform u ! u. On the other hand, the odd constraint prohibits the transform u ! u þ 2p. Eq. (2.1) is equivalent to the following system 0 u u ; ð2:3Þ þ ¼L sin u þ f ðtÞðsin u uÞ v v t where L
u v ¼ : v c2 uxx av þ bvxx
When ¼ 0; L generates a C0 semi-group on H 1 L2 (the Sobolev spaces H 1 and H 0 ¼ L2 on ½0; 2p), and the domain of L is H 2 H 1 . When 6¼ 0, L still generates a C0 semi-group on H 1 L2 , but if b 6¼ 0, then the domain of L is H 2 H 2 . Since the nonlinear term in (2.3) is uniformly Lipschitz, (2.3) is globally well-posed in Cð½0; 1Þ; H 1 L2 Þ. That is, for any ðu0 ; v0 Þ 2 H 1 L2 , there exists a unique mild solution ðuðtÞ; vðtÞÞ 2 Cð½0; 1Þ; H 1 L2 Þ such that ðuð0Þ; vð0ÞÞ ¼ ðu0 ; v0 Þ. One can introduce the evolution operator F t as ðuðtÞ; vðtÞÞ ¼ F t ðu0 ; v0 Þ. If b ¼ 0, F t is defined for all t 2 R, and for any fixed t, F t is a C 1 diffeomorphism. For a classical reference, see [6].
3. Integrable theory When ¼ 0, Eq. (2.1) reduces to the well-known sine-Gordon equation utt ¼ c2 uxx þ sin u;
ð3:1Þ
which is integrable through the Lax pair wx ¼ Bw;
ð3:2Þ
wt ¼ Aw;
ð3:3Þ
where 1 B¼ c A¼
i ðcux 4
þ ut Þ
1 iu e 16k
þk
1 iu e k 4i ðcux þ ut Þ 16k ! i 1 iu ðcux þ ut Þ 16k e þk 4 : 1 iu e k 4i ðcux þ ut Þ 16k
! ;
The Lax pair (3.2) and (3.3) possesses a symmetry. w1 w2 solves the Lax pair (3.2) and (3.3) at ðk; uÞ, then solves the Lax pair (3.2) and (3.3) at Lemma 3.1. If w ¼ w2 w1 ðk; uÞ. There is a Darboux transformation for the Lax pair (3.2) and (3.3).
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
793
Theorem 3.2 (Darboux transformation I). Let i/2 U ¼ u þ 2i ln ; /1 k m/2 =/1 W¼ w; k m/1 =/2 where / ¼ wjk¼m for some m, then W solves the Lax pair (3.2) and (3.3) at ðk; U Þ. Often in order to guarantee the reality condition (i.e. U needs to be real-valued), one needs to iterate the Darboux transformation by virtue of Lemma 3.1. The result corresponds to the counterpart of the Darboux transformation for the cubic nonlinear Schr€ odinger equation [7]. Theorem 3.3 (Darboux transformation II). Let " # mj/1 j2 þ mj/2 j2 U ¼ u þ 2i ln ; mj/1 j2 þ mj/2 j2 W ¼ Gw; where G¼
G1 G3
G1 ¼ jmj2 G2 ¼ G3 ¼
G2 ; G4
mj/1 j2 þ mj/2 j2 mj/1 j2 þ mj/2 j2
kðm2 m2 Þ/1 /2 mj/1 j2 þ mj/2 j2 kðm2 m2 Þ/1 /2 mj/1 j2 þ mj/2 j2
G4 ¼ jmj2
k2 ;
;
;
mj/1 j2 þ mj/2 j2 mj/1 j2 þ mj/2 j2
k2
and / ¼ wjk¼m for some m, then W solves the Lax pair (3.2) and (3.3) at ðk; U Þ. Proof. Let / be an eigenfunction solving the Lax pair (3.2) and (3.3) at ðk; uÞ. With (/; m; u), the Darboux transformation given in Theorem 3.2 leads to e ¼ u þ 2i ln i/2 ; ð3:4Þ U /1 k e ¼ m/2 =/1 w: ð3:5Þ W k m/1 =/2 / By Lemma 3.1, /^ ¼ 2 solves the Lax pair (3.2) and (3.3) at ð m; uÞ. Hence, /1 m b ¼ m/2 =/1 U /^ m m/1 =/2 b ; e Þ. With ( U e ), the Darboux transformation given in Theorem 3.2 leads solves the Lax pair (3.2) and (3.3) at ð m; U m; U to the expressions given in the current theorem. h Focusing upon the spatial part (3.2) of the Lax pair, one can develop a complete Floquet theory. Let MðxÞ be the fundamental matrix of (3.2), Mð0Þ ¼ I (2 · 2 identity matrix), then the Floquet discriminant is given as D ¼ trace Mð2pÞ:
794
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
The Floquet spectrum is given by r ¼ fk 2 Cj 2 6 DðkÞ 6 2g: Periodic and anti-periodic points k (which correspond to periodic and anti-periodic eigenfunctions, respectively) are defined by Dðk Þ ¼ 2: A critical point kðcÞ is defined by dD ðcÞ ðk Þ ¼ 0: dk A multiple point kðmÞ is a periodic or anti-periodic point which is also a critical point. The algebraic multiplicity of kðmÞ is defined as the order of the zero of DðkÞ 2 at kðmÞ . When the order is 2, we call the multiple point a double point, and denote it by kðdÞ . The order can exceed 2. The geometric multiplicity of kðmÞ is defined as the dimension of the periodic or anti-periodic eigenspace at kðmÞ , and is either 1 or 2. An important sequence of invariants Fj of the sine-Gordon equation can be defined by ðcÞ
Fj ðu; ut Þ ¼ Dðkj ðu; ut Þ; u; ut Þ: ðcÞ
If fkj g is a simple critical point of D, then oFj oD ¼ ; w ¼ u; ut : ow ow k¼kðcÞ j
As a function of three variables, D ¼ Dðk; u; ut Þ has the partial derivatives given by Bloch functions w (i.e. e are periodic in x of period 2p, and K is a complex constant): e ðxÞ, where w w ðxÞ ¼ eKx w pffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 4 oD i þ iu þ ¼ w1 w1 eiu wþ 4kcox ðwþ þ 1 w2 þ w2 w1 Þ þ e 2 w2 ; ou 16kc W ðw ; w Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 4 þ oD i ¼ w1 w2 þ wþ þ 2 w1 ; out 4c W ðw ; w Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2p D2 4 oD 1 1 iu 1 iu þ þ 1 w 1 w e w þ e w ¼ 2 2 1 1 dx; ok c W ðwþ ; w Þ 0 16k2 16k2 þ where W ðwþ ; w Þ ¼ wþ 1 w2 w2 w1 is the Wronskian. Of course, the sine-Gordon equation can be written in the Hamiltonian form
ut ¼
oH ; ov
vt ¼
oH ; ou
where the Hamiltonian is given by Z 2p 1 2 H¼ ðv þ c2 u2x Þ þ cos u dx: 2 0 It turns out that Fj ’s provide the perfect Melnikov vectors rather than the Hamiltonian or other invariants [7]. u ¼ 0 is a fixed point of the sine-Gordon equation. Linearization of the sine-Gordon equation at u ¼ 0 leads to utt ¼ c2 uxx þ u: P 0 Xk t Let u ¼ 1 sin kx, u0k and Xk are constants, then k¼1 uk e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xk ¼ 1 c2 k 2 ; k ¼ 1; 2; . . . Since 1=2 < c < 1, only k ¼ 1 is an unstable mode, the rest modes are neutrally stable. The corresponding nonlinear unstable foliation can be represented through the Darboux transformation given in Theorem 3.3. When u ¼ 0, the Bloch functions of the Lax pair (3.2) and (3.3) are
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
795
λ (d)
λ0
(d)
λ1
0
1/
4
Fig. 1. Floquet spectrum of the Lax pair at u ¼ 0, (d) double point, () critical point.
w ¼ eiðjxþxtÞ
1 ; i
1 1 and x ¼ k 16k where j ¼ 1c k þ 16k . Thus, " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 1 jc ðjcÞ2 : k¼ 2 4 The Floquet discriminant is given by D ¼ 2 cosð2pjÞ: The spectral data are depicted in Fig. 1. Noticing that the Darboux transformation in Theorem 3.3pdepends ffiffiffiffiffiffiffiffiffiffiffiffiffi upon quadratic products of eigenfunctions, one realizes that / should be chosen at j ¼ 12 and m ¼ 14 ½c þ i 1 c2 . m is a ðdÞ complex double point, m ¼ k1 in Fig. 1. (It turns out that for other soliton equations, e.g. Davey–Stewartson II equation [8], m may not be a double point.) The wise choice for / is rffiffiffiffiffi rffiffiffiffiffi cþ þ c / ; / þ /¼ c cþ where 1
r
/ ¼ ei2x2t
1 ; i
r¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c2
and c are arbitrary complex constants. Let cþ ¼ eqþih ; s ¼ rt q; n ¼ x þ h; c then 0 / ¼ 2@
cosh 2s cos n2 i sinh 2s sin n2 cosh 2s
sin n2
i sinh 2s
cos n2
1 A:
796
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
The Darboux transformation in Theorem 3.3 leads to hr i p p sech s sin x ; # 2 ; ; U ¼ 4#; # ¼ arctan c 2 2
ð3:6Þ
corresponding to h ¼ p2 (which in turn corresponds to the U ! U symmetry). Notice that det G ¼ ðk2 m2 Þðk2 m2 Þ: L’Hospital’s rule implies that pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 4 DðmÞD00 ðmÞ ¼ lim : þ 2 k!m W ðW ; W Þ 2mðm m2 ÞW ð/þ ; / Þ Moreover,
0
U ¼ G/ ¼ e
ðq2þih2Þ
mðm m ÞW ð/ ; / Þ@ 2
2
þ
/2 mj/1 j2 þmj/2 j2 /1 mj/1 j2 þ mj/2 j2
1 A:
Finally, oF1 pr2 ¼ pffiffiffi sech s½c2 þ r2 sech2 s sin2 x1 ½ tanh s sin x i cosx; out u¼U 2c
ð3:7Þ
corresponding to h ¼ p2. The real part of (3.7) is the Melnikov vector. 4. Existence of a homoclinic tube and chaos The defective sine-Gordon equation (2.1) can be related to an autonomous system by introducing extra phase variables h ¼ ðh1 ; . . . ; hN Þ, utt ¼ c2 uxx þ sin u þ ½Du þ f ðh1 ; . . . ; hN Þðsin u uÞ;
ð4:1Þ
dhn ¼ xn ; dt
ð4:2Þ
ðn ¼ 1; . . . ; N Þ:
For any h0 , solving (4.2), Eq. (4.1) becomes (2.1). u ¼ 0 corresponds to a N -torus denoted by e S . Linearization at u ¼ 0 leads to utt ¼ c2 uxx þ u þ Du; dhn ¼ xn ; ðn ¼ 1; . . . ; N Þ: dt Thus u ¼ 0 corresponds to a normally hyperbolic N -torus with one unstable mode (since 1=2 < c < 1), when > 0. Proofs of the following invariant manifold theorem have become standard after the works [9,10]. Theorem 4.1. The N -torus Se has an (N þ 1)-dimensional C m (m P 3) center-unstable manifold W cu and a 1-codimensional C m center-stable manifold W cs in the phase space ðu; ut ; hÞ 2 H 1 L2 TN . W cu \ W cs ¼ e S . W cu is C 1 in for 2 ½0; 0 Þ 2 2 cs 1 and some 0 > 0. When b 6¼ 0, for ðu; ut Þ 2 H H , W is C in for 2 ½0; 0 Þ. When b ¼ 0, W cs is always C 1 in for 2 ½0; 0 Þ. Inside W cu and W cs respectively, there are a C m invariant family of 1-dimensional C m unstable fibers S g and a C m invariant family of C m stable fibers fFs ðhÞ : h 2 e S g, such that fFu ðhÞ : h 2 e [ [ W cu ¼ Fu ðhÞ; W cs ¼ Fs ðhÞ: h2e S h2e S pffiffiffiffiffiffiffiffiffiffiffiffiffi There are positive constants ju ¼ 12 1 c2 , js ¼ 14 ða þ bÞ, and C such that kF t ðq Þ F t ðhÞk 6 Ceju t kq hk; kF t ðqþ Þ F t ðhÞk 6 Cejs t kqþ hk; kF t ðhþ Þ F t ðh Þk 6 Ckhþ h k
8t 2 ð1; 0; 8t 2 ½0; þ1Þ; 8t 2 ð1; þ1Þ;
where F t is the evolution operator of (4.1) and (4.2).
8h 2 e S; 8h 2 e S;
8q 2 Fu ðhÞ; 8qþ 2 Fs ðhÞ;
8hþ ; h 2 e S;
ð4:3Þ ð4:4Þ
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
797
In terms of the original setting (2.1), Fu ðhÞ and Fs ðhÞ are the unstable and stable manifolds of the fixed point u ¼ 0, which are C m smooth in h0 . As shown in [9,10], to the leading order, the signed distance between Fu ðhÞ and Fs ðhÞ (which is a certain coordinate difference) is given by the Melnikov integral Z þ1 Z 2p oF1 M¼ ½Du þ f ðtÞðsin u uÞ dx dt; out 1 0 u¼U 1 where U is given in (3.6) and oF j is given in (3.7). The signed distance between Fu ðhÞ and Fs ðhÞ is C m (m P 3) in h0 . out u¼U The zero of the Melnikov integral and implicit function theorem imply the following theorem, for detailed arguments, see [9,10].
Theorem 4.2. If b 6¼ 0, there is a region for ða; bÞ in Rþ Rþ , or if b ¼ 0, there is a region for a in Rþ , such that W cu and W cs intersect into a N -dimensional C m (m P 3) homoclinic tube b S asymptotic to the N -torus e S. pffiffi Additional remarks for the proof of the theorem are that the size of Fs ðhÞ is of order Oð Þ since the nonlinear term in (2.1) is cubic. Therefore, the so-called second measurement in [9,10] is not needed. The rest of this section only deals with the case b ¼ 0. The Poincare period map F determined by setting h1 ¼ 2np has the homoclinic tube n ¼ ð S1 S0 S1 Þ which is asymptotic to the (N 1)-torus S obtained from e S by setting h1 ¼ 0. S0 is a C m (N 1)-torus as a result of the smoothness of the signed distance with respect to h0 , and Sj ¼ F j S0 , 8j 2 Z. The rest of Assumption (A1) in [1] can be verified by noticing that the decay rates in (4.3) and (4.4) are uniform with respect to h, and the fact that Fu ðhÞ and Fs ðhÞ are the unstable and stable manifolds of the fixed point u ¼ 0 of (2.1). Since S is a finite-dimensional torus, n [ S is compact, thus Assumption (A2) in [1] is also satisfied. Therefore, we have the following theorem. Theorem 4.3 (Chaos theorem). When b ¼ 0, there is a Cantor set N of tori which is invariant under the iterated Poincare map F 2Kþ1 for some K. The action of F 2Kþ1 on N is topologically conjugate to the action of the Bernoulli shift on two symbols 0 and 1.
5. Chaos in the large and chaos in the small The previous theory also applies to the following system utt ¼ c2 uxx þ sin u þ ½Du þ f ðt; h2 ; . . . ; hN Þðsin u uÞ;
ð5:1Þ
dhn ¼ xn þ l gn ðh2 ; . . . ; hN Þ; dt
ð5:2Þ
ðn ¼ 2; . . . ; N Þ;
where f is periodic in t; h2 ; . . . ; hN , and ðh2 ; . . . ; hN Þ 2 TN 1 , by introducing the extra variable h1 ¼ x1 t þ h01 , f takes the same form as before f ðtÞ ¼
N X
an cos½hn ðtÞ:
n¼1
The function g is a smooth function defined on TN 1 , and l > 1. To the leading order in , hn ¼ xn t þ h0n , (n ¼ 2; . . . ; N ). The rest of the setup is the same as before. l > 1 guarantees that the growth and decay rates on the N -torus e S in Theorem 4.1 is at a higher order (in ) than those off e S . Theorems 4.2 and 4.3 hold for the above system (5.1) and (5.2). The dynamics of ðh2 ; . . . ; hN Þ 2 TN 1 can be chaotic too. Such chaos is called ‘‘chaos in the small’’. The chaos associated with the homoclinic tube, as given in Theorem 4.3, is called ‘‘chaos in the large’’. A concrete example of ‘‘chaos in the small’’ can be introduced as follows: Let N ¼ 4, hn ¼ xn t þ h0n þ l #n , n ¼ 2; 3; 4, l > 1, and #n ’s are given by the ABC flow [11] which is verified numerically to be chaotic, #_ 2 ¼ A sin #4 þ C cos #3 ; #_ 3 ¼ B sin #2 þ A cos #4 ; #_ 4 ¼ C sin #3 þ B cos #2 ; where A, B, and C are real parameters.
798
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
Here we see the embedding of smaller scale chaos in larger scale chaos. By introducing more variables, such embedding can be continued with even smaller scale chaos. This leads to a chain of embeddings. We call this chain of embeddings of smaller scale chaos in larger scale chaos, a ‘‘chaos cascade’’. We hope that such ‘‘chaos cascade’’ will be proved important.
Acknowledgement I would like to thank Brenda Frazier for artist work.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
Li Y. Chaos and shadowing aound a homoclinic tube. Abstr Appl Anal, in press. Zhang F, Kivshar Y, Vazquez L. Resonant kink-impurity interactions in the sine-Gordon model. Phys Rev A 1992;45:6019. Goodman R, Slusher R, Weinstein M. Stopping light on a defect. J Opt Soc Am B 2001. Cao X, Malomed B. Soliton-defect collisions in the nonlinear Schr€ odinger equation. Phys Lett A 1995;206(3–4):177. Li Y. Chaos and shadowing lemma for autonomous systems of infinite dimensions. J Dyn Differen Equat, in press. Pazy A. Semigroups of linear operators and applications to partial differential equations. In: Applied Mathematical Sciences, vol. 44. Springer-Verlag; 1983. Li Y, McLaughlin D. Morse and Melnikov functions for NLS PDEs. Commun Math Phys 1994;162(1):175. Li Y. B€ acklund–Darboux transformations and Melnikov analysis for Davey–Stewartson II equations. J Nonlinear Sci 2000;10(1):103. Li Y, McLaughlin D, Shatah J, Wiggins S. Persistent homoclinic orbits for a perturbed nonlinear Schr€ odinger equation. Commun Pure Appl Math 1996;49(11):1175. Li Y. Persistent homoclinic orbits for nonlinear Schr€ odinger equation under singular perturbation. Int J Math Math Sci, in press. Dombre T et al. Chaotic streamlines in the ABC flows. J Fluid Mech 1986;167:353.
Homoclinic tubes and chaos in perturbed sine-Gordon equation Y. Charles Li
*
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Accepted 27 August 2003 Communicated by L. Reichl
Abstract Sine-Gordon equation under a quasi-periodic perturbation or a chaotic perturbation is studied. Existence of a homoclinic tube is proved. Established are chaos associated with the homoclinic tube, and ‘‘chaos cascade’’ referring to the embeddings of smaller scale chaos in larger scale chaos. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Propagations of nonlinear waves through homogeneous media are often modeled by well-known nonlinear wave equations, for example, sine-Gordon equation. Studies were also drawn to variable media [2–4]. Variations of the media can speed up, slow down (even stop), or break the wave propagations. Studies have been focused upon such variations of the media, which are localized defects. In the current article, we will study quasi-periodic or chaotic media. The equation to be studied can be called a quasi-periodically or a chaotically defective sine-Gordon equation. This equation represents a concrete example realizing the chaos theorem proved in [1,5]. The chaos theorem, proved through a shadowing lemma, claims the existence of Bernoulli shift dynamics of tori in a neighborhood of a homoclinic tube. Bernoulli shift dynamics has sensitive dependence upon initial data in the product topology. In fact, Bernoulli shift has become a hallmark of chaos. Existence of a homoclinic tube and Bernoulli shift dynamics will be established for the defective sine-Gordon equation. In the case of a chaotic medium, often the dynamics inside the homoclinic tube is chaotic too. Such chaos is called ‘‘chaos in the small’’. The global chaos associated with the homoclinic tube is called ‘‘chaos in the large’’. We see the embedding of smaller scale chaos in larger scale chaos. Such embedding can be continued with even smaller scale chaos. We call this chain of embeddings of smaller scale chaos in larger scale chaos, a ‘‘chaos cascade’’. The article is organized as follows: In Section 2, we present the formulation of the problem. Section 3 is on an integrable theory. Section 4 is on the existence of a homoclinic tube and chaos. Section 5 is on a discussion of ‘‘chaos in the large’’ and ‘‘chaos in the small’’.
2. Formulations of the problem Consider the sine-Gordon equation under a quasi-periodic perturbation,
*
Tel.: +1-573-8840622; fax: +1-573-8821869. E-mail address: [email protected] (Y.C. Li).
0960-0779/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.08.013
792
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
utt ¼ c2 uxx þ sin u þ ½Du þ f ðtÞðsin u uÞ;
ð2:1Þ
which is subject to periodic boundary condition and odd constraint uðt; x þ 2pÞ ¼ uðt; xÞ;
uðt; xÞ ¼ uðt; xÞ;
ð2:2Þ
where u is a real-valued function of two real variables t P 0 and x, c is a parameter, 12 < c < 1, is a small parameter, P 0, D is a dissipative operator Du ¼ aut þ butxx ;
a P 0;
bP0
and f ðtÞ is quasi-periodic with a basis of frequencies x1 ; . . . ; xN , f ðtÞ ¼
N X
an cos½hn ðtÞ;
hn ðtÞ ¼ xn t þ h0n ;
n¼1
where an and h0n are parameters. The above system is invariant under the transform u ! u. On the other hand, the odd constraint prohibits the transform u ! u þ 2p. Eq. (2.1) is equivalent to the following system 0 u u ; ð2:3Þ þ ¼L sin u þ f ðtÞðsin u uÞ v v t where L
u v ¼ : v c2 uxx av þ bvxx
When ¼ 0; L generates a C0 semi-group on H 1 L2 (the Sobolev spaces H 1 and H 0 ¼ L2 on ½0; 2p), and the domain of L is H 2 H 1 . When 6¼ 0, L still generates a C0 semi-group on H 1 L2 , but if b 6¼ 0, then the domain of L is H 2 H 2 . Since the nonlinear term in (2.3) is uniformly Lipschitz, (2.3) is globally well-posed in Cð½0; 1Þ; H 1 L2 Þ. That is, for any ðu0 ; v0 Þ 2 H 1 L2 , there exists a unique mild solution ðuðtÞ; vðtÞÞ 2 Cð½0; 1Þ; H 1 L2 Þ such that ðuð0Þ; vð0ÞÞ ¼ ðu0 ; v0 Þ. One can introduce the evolution operator F t as ðuðtÞ; vðtÞÞ ¼ F t ðu0 ; v0 Þ. If b ¼ 0, F t is defined for all t 2 R, and for any fixed t, F t is a C 1 diffeomorphism. For a classical reference, see [6].
3. Integrable theory When ¼ 0, Eq. (2.1) reduces to the well-known sine-Gordon equation utt ¼ c2 uxx þ sin u;
ð3:1Þ
which is integrable through the Lax pair wx ¼ Bw;
ð3:2Þ
wt ¼ Aw;
ð3:3Þ
where 1 B¼ c A¼
i ðcux 4
þ ut Þ
1 iu e 16k
þk
1 iu e k 4i ðcux þ ut Þ 16k ! i 1 iu ðcux þ ut Þ 16k e þk 4 : 1 iu e k 4i ðcux þ ut Þ 16k
! ;
The Lax pair (3.2) and (3.3) possesses a symmetry. w1 w2 solves the Lax pair (3.2) and (3.3) at ðk; uÞ, then solves the Lax pair (3.2) and (3.3) at Lemma 3.1. If w ¼ w2 w1 ðk; uÞ. There is a Darboux transformation for the Lax pair (3.2) and (3.3).
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Theorem 3.2 (Darboux transformation I). Let i/2 U ¼ u þ 2i ln ; /1 k m/2 =/1 W¼ w; k m/1 =/2 where / ¼ wjk¼m for some m, then W solves the Lax pair (3.2) and (3.3) at ðk; U Þ. Often in order to guarantee the reality condition (i.e. U needs to be real-valued), one needs to iterate the Darboux transformation by virtue of Lemma 3.1. The result corresponds to the counterpart of the Darboux transformation for the cubic nonlinear Schr€ odinger equation [7]. Theorem 3.3 (Darboux transformation II). Let " # mj/1 j2 þ mj/2 j2 U ¼ u þ 2i ln ; mj/1 j2 þ mj/2 j2 W ¼ Gw; where G¼
G1 G3
G1 ¼ jmj2 G2 ¼ G3 ¼
G2 ; G4
mj/1 j2 þ mj/2 j2 mj/1 j2 þ mj/2 j2
kðm2 m2 Þ/1 /2 mj/1 j2 þ mj/2 j2 kðm2 m2 Þ/1 /2 mj/1 j2 þ mj/2 j2
G4 ¼ jmj2
k2 ;
;
;
mj/1 j2 þ mj/2 j2 mj/1 j2 þ mj/2 j2
k2
and / ¼ wjk¼m for some m, then W solves the Lax pair (3.2) and (3.3) at ðk; U Þ. Proof. Let / be an eigenfunction solving the Lax pair (3.2) and (3.3) at ðk; uÞ. With (/; m; u), the Darboux transformation given in Theorem 3.2 leads to e ¼ u þ 2i ln i/2 ; ð3:4Þ U /1 k e ¼ m/2 =/1 w: ð3:5Þ W k m/1 =/2 / By Lemma 3.1, /^ ¼ 2 solves the Lax pair (3.2) and (3.3) at ð m; uÞ. Hence, /1 m b ¼ m/2 =/1 U /^ m m/1 =/2 b ; e Þ. With ( U e ), the Darboux transformation given in Theorem 3.2 leads solves the Lax pair (3.2) and (3.3) at ð m; U m; U to the expressions given in the current theorem. h Focusing upon the spatial part (3.2) of the Lax pair, one can develop a complete Floquet theory. Let MðxÞ be the fundamental matrix of (3.2), Mð0Þ ¼ I (2 · 2 identity matrix), then the Floquet discriminant is given as D ¼ trace Mð2pÞ:
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Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
The Floquet spectrum is given by r ¼ fk 2 Cj 2 6 DðkÞ 6 2g: Periodic and anti-periodic points k (which correspond to periodic and anti-periodic eigenfunctions, respectively) are defined by Dðk Þ ¼ 2: A critical point kðcÞ is defined by dD ðcÞ ðk Þ ¼ 0: dk A multiple point kðmÞ is a periodic or anti-periodic point which is also a critical point. The algebraic multiplicity of kðmÞ is defined as the order of the zero of DðkÞ 2 at kðmÞ . When the order is 2, we call the multiple point a double point, and denote it by kðdÞ . The order can exceed 2. The geometric multiplicity of kðmÞ is defined as the dimension of the periodic or anti-periodic eigenspace at kðmÞ , and is either 1 or 2. An important sequence of invariants Fj of the sine-Gordon equation can be defined by ðcÞ
Fj ðu; ut Þ ¼ Dðkj ðu; ut Þ; u; ut Þ: ðcÞ
If fkj g is a simple critical point of D, then oFj oD ¼ ; w ¼ u; ut : ow ow k¼kðcÞ j
As a function of three variables, D ¼ Dðk; u; ut Þ has the partial derivatives given by Bloch functions w (i.e. e are periodic in x of period 2p, and K is a complex constant): e ðxÞ, where w w ðxÞ ¼ eKx w pffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 4 oD i þ iu þ ¼ w1 w1 eiu wþ 4kcox ðwþ þ 1 w2 þ w2 w1 Þ þ e 2 w2 ; ou 16kc W ðw ; w Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 4 þ oD i ¼ w1 w2 þ wþ þ 2 w1 ; out 4c W ðw ; w Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2p D2 4 oD 1 1 iu 1 iu þ þ 1 w 1 w e w þ e w ¼ 2 2 1 1 dx; ok c W ðwþ ; w Þ 0 16k2 16k2 þ where W ðwþ ; w Þ ¼ wþ 1 w2 w2 w1 is the Wronskian. Of course, the sine-Gordon equation can be written in the Hamiltonian form
ut ¼
oH ; ov
vt ¼
oH ; ou
where the Hamiltonian is given by Z 2p 1 2 H¼ ðv þ c2 u2x Þ þ cos u dx: 2 0 It turns out that Fj ’s provide the perfect Melnikov vectors rather than the Hamiltonian or other invariants [7]. u ¼ 0 is a fixed point of the sine-Gordon equation. Linearization of the sine-Gordon equation at u ¼ 0 leads to utt ¼ c2 uxx þ u: P 0 Xk t Let u ¼ 1 sin kx, u0k and Xk are constants, then k¼1 uk e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xk ¼ 1 c2 k 2 ; k ¼ 1; 2; . . . Since 1=2 < c < 1, only k ¼ 1 is an unstable mode, the rest modes are neutrally stable. The corresponding nonlinear unstable foliation can be represented through the Darboux transformation given in Theorem 3.3. When u ¼ 0, the Bloch functions of the Lax pair (3.2) and (3.3) are
Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
795
λ (d)
λ0
(d)
λ1
0
1/
4
Fig. 1. Floquet spectrum of the Lax pair at u ¼ 0, (d) double point, () critical point.
w ¼ eiðjxþxtÞ
1 ; i
1 1 and x ¼ k 16k where j ¼ 1c k þ 16k . Thus, " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1 1 jc ðjcÞ2 : k¼ 2 4 The Floquet discriminant is given by D ¼ 2 cosð2pjÞ: The spectral data are depicted in Fig. 1. Noticing that the Darboux transformation in Theorem 3.3pdepends ffiffiffiffiffiffiffiffiffiffiffiffiffi upon quadratic products of eigenfunctions, one realizes that / should be chosen at j ¼ 12 and m ¼ 14 ½c þ i 1 c2 . m is a ðdÞ complex double point, m ¼ k1 in Fig. 1. (It turns out that for other soliton equations, e.g. Davey–Stewartson II equation [8], m may not be a double point.) The wise choice for / is rffiffiffiffiffi rffiffiffiffiffi cþ þ c / ; / þ /¼ c cþ where 1
r
/ ¼ ei2x2t
1 ; i
r¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 c2
and c are arbitrary complex constants. Let cþ ¼ eqþih ; s ¼ rt q; n ¼ x þ h; c then 0 / ¼ 2@
cosh 2s cos n2 i sinh 2s sin n2 cosh 2s
sin n2
i sinh 2s
cos n2
1 A:
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Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
The Darboux transformation in Theorem 3.3 leads to hr i p p sech s sin x ; # 2 ; ; U ¼ 4#; # ¼ arctan c 2 2
ð3:6Þ
corresponding to h ¼ p2 (which in turn corresponds to the U ! U symmetry). Notice that det G ¼ ðk2 m2 Þðk2 m2 Þ: L’Hospital’s rule implies that pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 4 DðmÞD00 ðmÞ ¼ lim : þ 2 k!m W ðW ; W Þ 2mðm m2 ÞW ð/þ ; / Þ Moreover,
0
U ¼ G/ ¼ e
ðq2þih2Þ
mðm m ÞW ð/ ; / Þ@ 2
2
þ
/2 mj/1 j2 þmj/2 j2 /1 mj/1 j2 þ mj/2 j2
1 A:
Finally, oF1 pr2 ¼ pffiffiffi sech s½c2 þ r2 sech2 s sin2 x1 ½ tanh s sin x i cosx; out u¼U 2c
ð3:7Þ
corresponding to h ¼ p2. The real part of (3.7) is the Melnikov vector. 4. Existence of a homoclinic tube and chaos The defective sine-Gordon equation (2.1) can be related to an autonomous system by introducing extra phase variables h ¼ ðh1 ; . . . ; hN Þ, utt ¼ c2 uxx þ sin u þ ½Du þ f ðh1 ; . . . ; hN Þðsin u uÞ;
ð4:1Þ
dhn ¼ xn ; dt
ð4:2Þ
ðn ¼ 1; . . . ; N Þ:
For any h0 , solving (4.2), Eq. (4.1) becomes (2.1). u ¼ 0 corresponds to a N -torus denoted by e S . Linearization at u ¼ 0 leads to utt ¼ c2 uxx þ u þ Du; dhn ¼ xn ; ðn ¼ 1; . . . ; N Þ: dt Thus u ¼ 0 corresponds to a normally hyperbolic N -torus with one unstable mode (since 1=2 < c < 1), when > 0. Proofs of the following invariant manifold theorem have become standard after the works [9,10]. Theorem 4.1. The N -torus Se has an (N þ 1)-dimensional C m (m P 3) center-unstable manifold W cu and a 1-codimensional C m center-stable manifold W cs in the phase space ðu; ut ; hÞ 2 H 1 L2 TN . W cu \ W cs ¼ e S . W cu is C 1 in for 2 ½0; 0 Þ 2 2 cs 1 and some 0 > 0. When b 6¼ 0, for ðu; ut Þ 2 H H , W is C in for 2 ½0; 0 Þ. When b ¼ 0, W cs is always C 1 in for 2 ½0; 0 Þ. Inside W cu and W cs respectively, there are a C m invariant family of 1-dimensional C m unstable fibers S g and a C m invariant family of C m stable fibers fFs ðhÞ : h 2 e S g, such that fFu ðhÞ : h 2 e [ [ W cu ¼ Fu ðhÞ; W cs ¼ Fs ðhÞ: h2e S h2e S pffiffiffiffiffiffiffiffiffiffiffiffiffi There are positive constants ju ¼ 12 1 c2 , js ¼ 14 ða þ bÞ, and C such that kF t ðq Þ F t ðhÞk 6 Ceju t kq hk; kF t ðqþ Þ F t ðhÞk 6 Cejs t kqþ hk; kF t ðhþ Þ F t ðh Þk 6 Ckhþ h k
8t 2 ð1; 0; 8t 2 ½0; þ1Þ; 8t 2 ð1; þ1Þ;
where F t is the evolution operator of (4.1) and (4.2).
8h 2 e S; 8h 2 e S;
8q 2 Fu ðhÞ; 8qþ 2 Fs ðhÞ;
8hþ ; h 2 e S;
ð4:3Þ ð4:4Þ
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797
In terms of the original setting (2.1), Fu ðhÞ and Fs ðhÞ are the unstable and stable manifolds of the fixed point u ¼ 0, which are C m smooth in h0 . As shown in [9,10], to the leading order, the signed distance between Fu ðhÞ and Fs ðhÞ (which is a certain coordinate difference) is given by the Melnikov integral Z þ1 Z 2p oF1 M¼ ½Du þ f ðtÞðsin u uÞ dx dt; out 1 0 u¼U 1 where U is given in (3.6) and oF j is given in (3.7). The signed distance between Fu ðhÞ and Fs ðhÞ is C m (m P 3) in h0 . out u¼U The zero of the Melnikov integral and implicit function theorem imply the following theorem, for detailed arguments, see [9,10].
Theorem 4.2. If b 6¼ 0, there is a region for ða; bÞ in Rþ Rþ , or if b ¼ 0, there is a region for a in Rþ , such that W cu and W cs intersect into a N -dimensional C m (m P 3) homoclinic tube b S asymptotic to the N -torus e S. pffiffi Additional remarks for the proof of the theorem are that the size of Fs ðhÞ is of order Oð Þ since the nonlinear term in (2.1) is cubic. Therefore, the so-called second measurement in [9,10] is not needed. The rest of this section only deals with the case b ¼ 0. The Poincare period map F determined by setting h1 ¼ 2np has the homoclinic tube n ¼ ð S1 S0 S1 Þ which is asymptotic to the (N 1)-torus S obtained from e S by setting h1 ¼ 0. S0 is a C m (N 1)-torus as a result of the smoothness of the signed distance with respect to h0 , and Sj ¼ F j S0 , 8j 2 Z. The rest of Assumption (A1) in [1] can be verified by noticing that the decay rates in (4.3) and (4.4) are uniform with respect to h, and the fact that Fu ðhÞ and Fs ðhÞ are the unstable and stable manifolds of the fixed point u ¼ 0 of (2.1). Since S is a finite-dimensional torus, n [ S is compact, thus Assumption (A2) in [1] is also satisfied. Therefore, we have the following theorem. Theorem 4.3 (Chaos theorem). When b ¼ 0, there is a Cantor set N of tori which is invariant under the iterated Poincare map F 2Kþ1 for some K. The action of F 2Kþ1 on N is topologically conjugate to the action of the Bernoulli shift on two symbols 0 and 1.
5. Chaos in the large and chaos in the small The previous theory also applies to the following system utt ¼ c2 uxx þ sin u þ ½Du þ f ðt; h2 ; . . . ; hN Þðsin u uÞ;
ð5:1Þ
dhn ¼ xn þ l gn ðh2 ; . . . ; hN Þ; dt
ð5:2Þ
ðn ¼ 2; . . . ; N Þ;
where f is periodic in t; h2 ; . . . ; hN , and ðh2 ; . . . ; hN Þ 2 TN 1 , by introducing the extra variable h1 ¼ x1 t þ h01 , f takes the same form as before f ðtÞ ¼
N X
an cos½hn ðtÞ:
n¼1
The function g is a smooth function defined on TN 1 , and l > 1. To the leading order in , hn ¼ xn t þ h0n , (n ¼ 2; . . . ; N ). The rest of the setup is the same as before. l > 1 guarantees that the growth and decay rates on the N -torus e S in Theorem 4.1 is at a higher order (in ) than those off e S . Theorems 4.2 and 4.3 hold for the above system (5.1) and (5.2). The dynamics of ðh2 ; . . . ; hN Þ 2 TN 1 can be chaotic too. Such chaos is called ‘‘chaos in the small’’. The chaos associated with the homoclinic tube, as given in Theorem 4.3, is called ‘‘chaos in the large’’. A concrete example of ‘‘chaos in the small’’ can be introduced as follows: Let N ¼ 4, hn ¼ xn t þ h0n þ l #n , n ¼ 2; 3; 4, l > 1, and #n ’s are given by the ABC flow [11] which is verified numerically to be chaotic, #_ 2 ¼ A sin #4 þ C cos #3 ; #_ 3 ¼ B sin #2 þ A cos #4 ; #_ 4 ¼ C sin #3 þ B cos #2 ; where A, B, and C are real parameters.
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Y.C. Li / Chaos, Solitons and Fractals 20 (2004) 791–798
Here we see the embedding of smaller scale chaos in larger scale chaos. By introducing more variables, such embedding can be continued with even smaller scale chaos. This leads to a chain of embeddings. We call this chain of embeddings of smaller scale chaos in larger scale chaos, a ‘‘chaos cascade’’. We hope that such ‘‘chaos cascade’’ will be proved important.
Acknowledgement I would like to thank Brenda Frazier for artist work.
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