25 research outputs found

    A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

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    Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system

    Effective estimates of ergodic quantities illustrated on the Bolyai-R\'enyi map

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    We present a practical and effective method for rigorously estimating quantities associated to top eigenvalues of transfer operators to very high precision. The method combines explicit error bounds of the Lagrange-Chebyshev approximation with an established min-max method. We illustrate its applicability by significantly improving rigorous estimates on various ergodic quantities associated to the Bolyai-R\'enyi map.Comment: 21 pages, 3 figure

    Anomalous dynamics in symmetric triangular irrational billiards

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    We identify a symmetry induced mechanism which dominates the long time behaviour in symmetric triangular billiards. We rigorously prove the existence of invariant sets in symmetric irrational billiards on which the dynamics is governed by an interval exchange transformation. Counterintuitively, this property of symmetric irrational billiards is analogous to the case of general rational billiards, and it highlights the non-trivial impact of symmetries in non-hyperbolic dynamical systems. Our findings provide an explanation for the logarithmic subdiffusive relaxation processes observed in certain triangular billiards. In addition we are able to settle a long standing conjecture about the existence of non-periodic and not everywhere dense trajectories in triangular billiards

    Dynamic mode decomposition for analytic maps

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    Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps.Comment: 14 pages. 3 figure

    Non-periodic not everywhere dense trajectories in triangular billiards

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    Building on tools that have been successfully used in the study of rational billiards, such as induced maps and interval exchange transformations, we provide a construction of a one-parameter family of isosceles triangles exhibiting non-periodic trajectories that are not everywhere dense. This provides, by elementary means, a definitive negative answer to a long-standing open question on the density of non-periodic trajectories in triangular billiards.Comment: 15 pages, 2 figure

    Transfer operator approach to ray-tracing in circular domains

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    The computation of wave-energy distributions in the mid-to-high frequency regime can be reduced to ray-tracing calculations. Solving the ray-tracing problem in terms of an operator equation for the energy density leads to an inhomogeneous equation which involves a Perron–Frobenius operator defined on a suitable Sobolev space. Even for fairly simple geometries, let alone realistic scenarios such as typical boundary value problems in room acoustics or for mechanical vibrations, numerical approximations are necessary. Here we study the convergence of approximation schemes by rigorous methods. For circular billiards we prove that convergence of finite-rank approximations using a Fourier basis follows a power law where the power depends on the smoothness of the source distribution driving the system. The relevance of our studies for more general geometries is illustrated by numerical examples

    Complete spectral data for analytic Anosov maps of the torus

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    Using analytic properties of Blaschke factors we construct a family of analytic hyperbolic diffeomorphisms of the torus for which the spectral properties of the associated transfer operator acting on a suitable Hilbert space can be computed explicitly. As a result, we obtain explicit expressions for the decay of correlations of analytic observables without resorting to any kind of perturbation argument.Comment: 19 pages, 4 figure

    Replicate Periodic Windows in the Parameter Space of Driven Oscillators

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    In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have periodicity and pattern similar to stable and unstable periodic orbits already existent for the unperturbed oscillator. These features indicate that the reported replicate periodic windows are associated with chaos control of the considered oscillators
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