Chapter 22. Mathematical Chaos Stable and Unstable Manifolds Saddle Fixed Point

Transcription

1 Chapter 22 Mathematcal Chaos The sets generated as the long tme attractors of physcal dynamcal systems descrbed by ordnary dfferental equatons or dscrete evoluton equatons n the case of maps such as the Hénon map appear to be very complcated, often wth unpleasant propertes such as the lack of robustness (the parameters leadng to chaotc and nonchaotc soluton may be ntertwned on an arbtrarly small scale). Few proofs are avalable for the conjectured propertes of these sets. To make mathematcal progress t s necessary to make restrctons on the type of dynamcal systems consdered, that go far beyond standard smoothness assumptons. Often the assumptons needed to construct proofs are so strct that t s known that all common physcal attractors volate the assumptons. (However, numercal tests of the results proven for the restrcted systems often show that they apply, at least as an excellent approxmaton, to the physcal systems.) In ths chapter the goal s to present some of the flavor of ths mathematcal approach, snce many of our deas on chaotc systems have arsen n ths context. A key assumpton n much of the mathematcal development s the noton of hyperbolcty. To defne ths concept t we frst ntroduce the general dea of stable and unstable manfolds. Useful references are: Eckmann and Ruelle [1], sectons IIIE, F; Ott [2], sectons 4.3 and 9.5; Algood et al. [3], secton 2.6 and chapter 10; and Guckenhemer and Holmes [4] secton Stable and Unstable Manfolds Saddle Fxed Pont 1

2 CHAPTER 22. MATHEMATICAL CHAOS 2 The dea of stable and unstable manfolds s most easly ntroduced n the context of a saddle fxed pont of a two dmensonal map. They are the natural extensons of the lnear egenvectors of the stablty analyss of the fxed pont nto the nonlnear regme. Consder a fxed pont x f of a dfferentable two dmensonal map F wth a dfferentable nverse map. If x f has one unstable egenvalue s wth s > 1 and correspondng egenvector E s, and one stable egenvalue u wth u < 1 and correspondng egenvector E u, t s called a saddle fxed pont. The stable manfold of x f denoted W s (x f ) s the set of ponts y such that F n (y) F n (x f ) 0as n ; the unstable manfold of x f denoted W u (x f ) s the set of ponts y such that F n (y) F n (x f ) 0asn. Both are one dmensonal manfolds contanng x f. A manfold s bascally a nce set (e.g. wthout fractal propertes): a one dmensonal manfold s defned as a set that s locally a curve and can be produced locally by bendng a lne. The letters D (n a sans serf font!) and O are 1-manfolds, the letters A and X are not, snce there are ponts where no small neghborhood looks lke a lne (Algood et al. [3]). Also W s (x f ) s tangent to E s and W p (x f ) s tangent to E u at x f. The extenson to perodc saddles s gven by notng that these are fxed ponts of F q for some q, and there s a straghtforward generalzaton to hgher dmensonal maps. The dea of stable and unstable manfolds can be defned locally at an arbtrary pont x n the phase space: the stable manfold W s of x s the set of ponts y such that F n (y) F n (x) 0asn, and the unstable manfold W u of x s the set of ponts y such that F n (y) F n (x) 0asn. Useful results can be proven for stable and unstable manfolds. For example f x s n an attractng set then W u (x) s contaned n. Also the number of postve Lyapunov exponents of the set s a lower bound for the capacty dmenson of Hyperbolc Invarant Sets A hyperbolc nvarant set n a sense s the generalzaton of a saddle fxed pont and can be defned n terms of the propertes of the lnearzed map about the pont x on the set,.e. the tangent space T x at the pont x. Note that the defnton apples to both attractng and nonattractng sets. An nvarant set under the map F s sad to be hyperbolc f there s a drect sum decomposton of T x nto stable and unstable spaces T x = Ex s Eu x for all x n such that: () the splttng nto Ex s, Eu x vares contnuously wth x;

3 CHAPTER 22. MATHEMATICAL CHAOS 3 () the splttng s nvarant n the sense that DF (x)ex s,u = E s,u F(x),.e. the evolvng the stable and unstable spaces at x wth the tangent space map gves the same result as the stable and unstable spaces at the evolved pont F(x); () there are numbers K>0 and 0 <ρ<1 such that for all n>0 DF n (x)v Kρ n v for v n Ex s (22.1) DF n (x)v Kρ n v for v n Ex u. (22.2) (In these expressons DF (x) s the Jacobean matrx of F at x.) The latter condton says that the exponental decay rate of vectors n the stable subspace and the exponental growth rate n the unstable subspace are bounded away from zero. Agan the lnear spaces Ex s, Eu x may be extended nto the nonlnear regme far from x to gve stable and unstable manfolds W s (x), W u (x) at each pont x on the attractor that are tangent to Ex s, Eu x at x (e.g. Guckenhemer and Holmes [4], Theorem 5.2.8). Two ponts on the stable (unstable) manfold approach (separate from) each other exponentally. Most of the mathematcal understandng of chaotc attractors s restrcted to hyperbolc attractors. Further restrctons are often needed on smoothness and other propertes. For example an Axom A attractor s an attractor of a dfferentable map that s hyperbolc and mxng. The property of mxng s that for any two sets S a and S b n the phase space lm n µ [ S a F n (S b ) ] = µ(s a )µ(s b ) (22.3) where µ s the natural measure of the attractor. The property of mxng s that ntal condtons get spread over the attractor accordng to the measure. Axom A attractors are partcularly nce, and many results have been proven for these attractors, for example the exstence of a natural nvarant measure that s smooth along the expandng drectons. Axom A attractors are also structurally stable, whch means that even the delcate chaotc structure survves a small perturbaton of the map. Most physcal attractors are non-hyperbolc because there are ponts on the attractor where the stable and unstable manfolds are tangent to one another (see fgure 22.1 for the Hénon map). Structural stablty does not seem to be a property of many physcal attractors. The bakers map s hyperbolc, although the map s not dfferentable so t s not Axom A. In the next chapter the horseshoe map, that

4 CHAPTER 22. MATHEMATICAL CHAOS 4 Fgure 22.1: Plot of the Hénon attractor (red) and the stable manfold (blue) of the fxed pont that les wthn the attractor. The stable component Ex s of the tangent space (.e. the contractng drecton) les along ths curve at the ponts x where t ntersects the attractor. Snce the expandng drecton Ex u les along the attractor, we see that Ex s and Eu x are parallel at ponts where the blue curve s tangent to the attractor. At these ponts they do not span the tangent space T x ndcatng a breakdown of hyperbolcty. The calculaton of the stable manfold s dscussed n reference [5], and the fgure was constructed usng the program suppled there.

5 CHAPTER 22. MATHEMATICAL CHAOS 5 s both dfferentable and hyperbolc, s ntroduced. Ths map shows a chaotc set, but ths set s not an attractor. The Ansov map or cat map x n+1 = x n + y n mod 1 (22.4) y n+1 = x n + 2y n mod 1 s a dfferentable, area preservng hyperbolc map (the egenvalues of the Jacobean (3 ± 5)/2 and the egenvectors are ndependent of poston). The orbt from a typcal ntal condton flls the whole unt square wth unform measure. The map s therefore Axom A. The Sna map x n+1 = x n + y n + δ cos(2πy n ) mod 1 (22.5) y n+1 = x n + 2y n mod 1 can be consdered a perturbaton of ths map, and so by structural stablty for small enough δ wll also be hyperbolc and the attractor wll be of capacty dmenson 2. The measure of the Sna map s however no longer unform and n fact shows nterestng structure, and dagnostcs nvolvng the measure (e.g. the nformaton dmenson) wll vary wth δ. The propertes of the Sna map are llustrated n the demonstraton Illustraton To llustrate the flavor of the use of hyperbolcty to prove propertes of chaotc attractors consder the followng [6]. For a map F wth an Axom A (.e. hyperbolc and mxng) attractor the natural measure of the attractor contaned n some closed set S s µ(s) = lm n L 1 (22.6) where the sum s over the unstable fxed ponts of F n whch le n S, and L s the product of the unstable egenvalues of the lnearzaton of F n at the th fxed pont. In partcular f the set S s the entre attractor so that µ(s) = 1wehave 1 = lm n L 1. (22.7)

6 CHAPTER 22. MATHEMATICAL CHAOS 6 cell C k e' c' h' e c F n a' x n b' h a f d x 0 b g f' d' g' fxed pont Fgure 22.2: Pont x 0 s mapped nto x n under F n. Then f ab a b and cd c d the parallelogram ef gh e f g h, and by contnuty there must be a sngle saddle fxed pont of F n n the ntersecton regon. In all cases the horzontal (vertcal) lnes n the fgure are segments of the stable (unstable) manfolds. The fgure s drawn wth orthogonal lnes for smplcty, but ths s not essental to the argument Proof We wll llustrate the proof for two dmensonal maps. Partton the phase space nto small cells C where each cell has as ts boundares the stable and unstable manfolds. Small enough cells may be consdered parallelograms. Consder the teraton of a large number of ntal condtons dstrbuted over a partcular cell C k accordng to the natural measure of the attractor. Iterate a large number of tmes n then a fracton gven by µ(c k ) wll return to the cell (the mxng property). Suppose x 0 s an ntal condton that returns to x n n C k. If ab s the stable manfold segment passng through x 0 and a b ts mage after n teratons, and c d s the unstable manfold segment passng through x n and cd ts premage under n teratons, then the parallelogram ef gh s mapped to the parallelogram e f g h after n terates, where the boundares of ef gh and e f g h are segments of stable and unstable manfolds (see fgure 22.2). Ths means there must be a sngle saddle

7 CHAPTER 22. MATHEMATICAL CHAOS 7 fxed pont of F n n the ntersecton regon, and conversely any saddle fxed pont of F n can be surrounded by smlar parallelograms. If the unstable egenvector of the fxed pont s λ u > 1, then the heght of ef gh s a fracton λ 1 u of the heght of C k. The measure of an Axom A attractor s smooth along the unstable drectons, and snce C k s small so that the dstrbuton of the measure across the heght can be consdered unform, the fracton of the measure of C k n the strp ef gh s then just λ 1 u. Thus the fracton of ntal condtons that return to C k under F n, whch by the mxng property s µ(c k ) for large n, s gven by summng the λ 1 u over the saddle fxed ponts of F n wthn C k. Summng over the C k n S then proves the result 22.6 snce for a two dmensonal map L s just equal to the unstable egenvalue λ. Further results may be derved relatng the propertes of chaotc attractors to the unstable perodc orbts. A few of these results, proven for hyperbolc attractors, are quoted here for a two dmensonal map. 1. The Lyapunov exponents are 1 λ 1,2 = lm n n 1 λ () u ln λ () u,s (22.8) wth λ () u, λ () s the unstable and stable egenvalues at the th perod n pont. 2. The capacty dmenson D C s gven by lm n ( λ () s ) DC 1 = 1. (22.9) 3. The egenvalues of the Perron-Frobenus operator δ(y f(x))for the evoluton of the measure can be expanded n the unstable fxed ponts [7]. Auerbach et al. [8] have tested some of these results on the (nonhyperbolc) Hénon map, whch has 1, 3, 1, 7, 1, 15, 29, 63, 55, 103 order 1 to 10 cycles respectvely. For example the calculate values for D C based on (22.9) and get values of 1.26, 1.29, 1.30, 1.26, 1.27 usng cycles of order 6 to 10. There s a large lterature n both math and physcs on ths area. February 29, 2000

8 Bblography [1] J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985) [2] E. Ott, Chaos n Dynamcal Systems (Cambrdge, 1993) [3] K.T.Algood, T.D. Sauer, and J.A.Yorke, Chaos: an Introducton to Dynamcal Systems (Sprnger, 1997) [4] J. Guckenhemer and P. Holmes, Nonlnear Oscllatons, Dynamcal Systems, and Bfurcatons of Vector Felds (Sprnger, 1983) [5] H.E. Nusse and J.A. Yorke, Dynamcs: Numercal Exploratons (2nd Edton, Sprnger, New York) chapter 9. [6] C. Grebog, E. Ott, and J.A. Yorke, Phys. Rev. A37, 1711 (1988) [7] H.G. Schuster, Determnstc Chaos (3rd Edton, VCH 1995), Appendx H. [8] D. Auerbach, P. Cvtanovć, J.-P. Eckmann, G. Gunaratne and I. Procacca, Phys. Rev. Lett. 58, 2387 (1987) 8