1,720,968 research outputs found
Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps
No full textWe consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension
Rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems
No full textWe study the problem of the rigorous computation of the stationary measure and of the rate of convergence to equilibrium of an iterated function system described by a stochastic mixture of two or more dynamical systems that are either all uniformly expanding on the interval, either all contracting. In the expanding case, the associated transfer operators satisfy a Lasota-Yorke inequality, we show how to compute a rigorous approximations of the stationary measure in the L 1 norm and an estimate for the rate of convergence. The rigorous computation requires a computer-aided proof of the contraction of the transfer operators for the maps, and we show that this property propagates to the transfer operators of the IFS. In the contracting case we perform a rigorous approximation of the stationary measure in the Wasserstein-Kantorovich distance and rate of convergence, using the same functional analytic approach. We show that a finite computation can produce a realistic computation of all contraction rates for the whole parameter space. We conclude with a description of the implementation and numerical experiments
An elementary way to rigorously estimate convergence to equilibrium and escape rates
No full textWe show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rates for systems satisfying a Lasota Yorke inequality. The bounds are deduced from the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiments on some nontrivial example
Decay of Correlations, Quantitative Recurrence and Logarithm Law for Contracting Lorenz Attractors
No full textIn this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation
Existence of multiple noise-induced transitions in Lasota–Mackey maps
Get PDFWe prove the existence of multiple noise-induced transitions in the Lasota-Mackey map, which is a class of one-dimensional random dynamical system with additive noise. The result is achieved with the help of rigorous computer assisted estimates. We first approximate the stationary distribution of the random dynamical system and then compute certified error intervals for the Lyapunov exponent. We find that the sign of the Lyapunov exponent changes at least three times when increasing the noise amplitude. We also show numerical evidence that the standard non-rigorous numerical approximation by finite-time Lyapunov exponent is valid with our model for a sufficiently large number of iterations. Our method is expected to work for a broad class of nonlinear stochastic phenomena
A rigorous computational approach to linear response
Get PDFWe present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the ##IMG## [http://ej.iop.org/images/0951-7715/31/3/1073/nonaa9a88ieqn001.gif] {} -norm, the response of expanding circle maps under stochastic and deterministic perturbations. Moreover, we present an example where we compute, up to a pre-specified error in the L 1 -norm, the response of the intermittent family at the boundary; i.e. when the unperturbed system is the doubling map
Rigorous Approximation of Diffusion Coefficients for Expanding Maps
Get PDFWe use Ulam’s method to provide rigorous approximation of diffusion coefficients for uniformly expanding maps. An algorithm is provided and its implementation is illustrated using Lanford’s map
ON THE CONTINUITY OF LYAPUNOV EXPONENTS FOR SYSTEMS WITH ADDITIVE NOISE (Research on the Theory of Random Dynamical Systems and Fractal Geometry)
Get PDFIn this short essay I show a small result on continuity of Lyapunov exponents for systems with additive noise
A general approach to Lehmann-Suwa-Khanedani index theorems: partial holomorphic connections and extensions of foliations
No full textThis thesis stresses the strong link between the existence of partial holomorphic connections on the normal bundle of a foliation seen as a quotient of the ambient tangent bundle and the extendability of a foliation to an infinitesimal neighborhood of a submanifold. We find some obstructions to extendability and thanks to the theory developed we obtain some new Khanedani-Lehmann-Suwa type index theorems, for foliations and holomorphic self maps
How does noise induce order?
Get PDFIn this paper we present a general result with an easily checkable condition
that ensures a transition from chaotic regime to regular regime in random
dynamical systems with additive noise. We show how this result applies to a
prototypical family of nonuniformly expanding one dimensional dynamical
systems, showing the main mathematical phenomenon behind Noise Induced Order.
Accepted at Journal of Statistical PhysicsComment: 32 pages, 3 figure
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