Intersection properties of invariant manifolds in certain twist maps

Veerman, J. J. P.; Tangerman, F. M.

doi:10.1007/bf02352495

Cited by 15 publications

(6 citation statements)

References 14 publications

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“…Cantori are typically hyperbolic, though I do not know of any theorem that guarantees this in general [when k is large enough all cantori of the standard map are hyperbolic; see Goroff (1985) and Veerman and Tangerman (1991)]. The hyperbolicity is measured by a Lyapunov multiplier, which is obtained from the linearized mapping along a segment of length n of the orbit:…”

Section: E Cantorimentioning

confidence: 99%

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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Section: E Cantorimentioning

confidence: 99%

Symplectic maps, variational principles, and transport

1992

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“…A necessary condition that a Q4-the0ry evolves chaotically in fictive time is that the bare coupling parameters (the bare mass and the bare quartic coupling) approach infinity. This l i t stands in certain analogy to the anti-integrable l i i i t of Frenkel-Kontorova-lie models [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Chaotic quantization of field theories

Beck

1995

Nonlinearity

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We introduce a class of coupled map lattices that simulate quantum field theories in an appropriate scaling limit. The Parisi-Wu approach of stochastic quantization is extended to more general spatio-temporal stochastic processes generated by wupled chaotic dynamical systems. We show that such complicated processes can invinsically arise from a quantized @-theory with formally infinite bare wupling parameters. This field theory leads to a lattice of diffusively coupled cubic maps, for example, coupled third-order Tschebyscheff polynomials.W e dmlate effective potentials for this model. The effective potentials turn out to exhibit interesting uansition scenarios when the spatial wupling strength is changed: there are several 'critical points' where single-wells change into double-wells. We point out a possible application of ow approach in cosmology: The equation of motion of the inflaton field, when quantized by the Parisi-Wu method, reduces to just the type of wupled map lattices studied here, provided the bare m m and bare quartic coupling approach infinity.

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“…A key remark is that all the results that we present here are for values of ε small: the anti-integrable limit scenario. In the litterature there are several results on this direction, see [3,6,25,31]. In all these papers they deal with the case that the potential V does not vanish at the anti-integrable limit.…”

Section: Formulation Of the Resultsmentioning

confidence: 99%

Existence of non-smooth bifurcations of uniformly hyperbolic invariant manifolds in skew product systems

Figueras

Lilja

2018

Nonlinearity

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In this paper we study the anti-integrable limit scenario of skew-product systems. We consider a generalization of such systems based on the Frenkel-Kontorova model, and prove the existence of orbits with any fibered rotation number in systems of both one and two degrees of freedom. In particular, our results also apply to two dimensional maps with degenerate potentials (vanishing second derivative), so extending the results of existence of Cantori for more general twist maps.We also prove that under certain mild regularity conditions on the potential the structure of the orbits is of Cantor type. From our results we deduce the existence of the non-smooth folding bifurcation (conjectured by Figueras-Haro, Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps, Chaos:25 (2015)).Lastly we present a pair of results which are useful in determining if a potential satisfies the regularity conditions required for the Cantor sets of orbits to exist and are also of independent interest.

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Intersection properties of invariant manifolds in certain twist maps

Cited by 15 publications

References 14 publications

Symplectic maps, variational principles, and transport

Symplectic maps, variational principles, and transport

Chaotic quantization of field theories

Existence of non-smooth bifurcations of uniformly hyperbolic invariant manifolds in skew product systems

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