Fractal Dimension of Cantori

Li, Wentian; Bak, Per

doi:10.1103/physrevlett.57.655

Cited by 29 publications

(6 citation statements)

References 8 publications

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“…In certain cases, the classical phase space becomes increasingly intricate, showing both selfsimilar and fractal properties [47]. We notice that such a structure is reminiscent of the way that the KAM tori break up into "fractal" orbits of zero dimension [48] (cantori) as the system becomes chaotic. Classically, cantori represent strong obstacles to phase space transport.…”

Section: Critical Statistics and Quantum Chaosmentioning

confidence: 95%

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

García-García

Verbaarschot²

2003

Phys. Rev. E

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We investigate the spectral properties of a generalized GOE (Gaussian Orthogonal Ensemble) capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland Hamiltonian (C-S). Taking advantage of the correspondence between C-S particles and eigenvalues, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar/fractal Poincaré section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the ∆ 3 -statistic is accurately described by our analytical results.

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Section: Critical Statistics and Quantum Chaosmentioning

confidence: 95%

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

García-García

Verbaarschot²

2003

Phys. Rev. E

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“…This implies that the Hausdorff dimension of the cantorus is zero (Li and Bak, 1986;MacKay, 1987). This is remarkable, since it implies that when an invariant circle breaks, its length falls immediately to zero; furthermore, its dimension discontinuously changes from 1 to zero (providing it becomes hyperbolic).…”

Section: E Cantorimentioning

confidence: 97%

Symplectic maps, variational principles, and transport

1992

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Program in Applied Mathematics, Vni versi ty of Colorado, Boo!der, Colorado 80309 Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and Quid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamicsfrom simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the Aux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.

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“…The Single Intersection property is observed for all k, and can also be proven by perturbative techniques (Simo, personal communication) for small k (though not uniform in the rotation number). The hyperbolicity of Aubry Mather sets (which is an essential ingredient in this work) is observed numerically as soon as the invariant curve has broken up (Li and Bak 1986). We expect that the structure of the invariant manifolds that we outlined in Theorem 2.4 is very regular for all k: it is not necessarily uniformly Lipschitz, but it will probably consist of finitely many graphs, which may be sufficient to prove many of the results of this and previous works.…”

Section: Discussionmentioning

confidence: 88%

Intersection properties of invariant manifolds in certain twist maps

Veerman

Tangerman

1991

Commun.Math. Phys.

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Abstract. We consider the space N of C 2 twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constant k (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps in N with nonlinearity k large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem).In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.

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Fractal Dimension of Cantori

Cited by 29 publications

References 8 publications

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Symplectic maps, variational principles, and transport

Intersection properties of invariant manifolds in certain twist maps

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