Search results for "Mathematics::Dynamical Systems"
showing 10 items of 113 documents
Localization of hidden Chua's attractors
2011
Abstract The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria . In the present Letter for localization of hidden attractors of Chuaʼs circuit it is suggested to use a special analytical–numerical algorithm.
Abnormal escape rates from nonuniformly hyperbolic sets
1999
Consider a $C^{1+\epsilon}$ diffeomorphism $f$ having a uniformly hyperbolic compact invariant set $\Omega$, maximal invariant in some small neighbourhood of itself. The asymptotic exponential rate of escape from any small enough neighbourhood of $\Omega$ is given by the topological pressure of $-\log |J^u f|$ on $\Omega$ (Bowen–Ruelle in 1975). It has been conjectured (Eckmann–Ruelle in 1985) that this property, formulated in terms of escape from the support $\Omega$ of a (generalized Sinai–Ruelle–Bowen (SRB)) measure, using its entropy and positive Lyapunov exponents, holds more generally. We present a simple $C^\infty$ two-dimensional counterexample, constructed by a surgery using a Bowe…
Quantum Walk and Quantum Billiards. Towards a better understanding of Quantum Chaos
2019
Quantum billiards have been simulated so far in many ways, but in this work a new aproximation is considerated. This study is based on the quantum billiard already obtained by others authors via a tensor product of two 1-D quantum walks . Chaotic and non chaotic billiards are tested.
Attracteurs de Lorenz de variété instable de dimension arbitraire
1997
Abstract We construct the first examples of flows with robust multidimensional Lorenz-like attractors: the singularity contained in the attractor may have any number of expanding eigenvalues, and the attractor remains transitive in a whole neighbourhood of the initial flow. These attractors support a Sinai-Ruelle-Bowen SRB-measure and, contrary to the usual (low-dimensional) Lorenz models, they have infinite modulus of structural stability.
Existence of homoclinic orbits and heteroclinic cycle in a class of three-dimensional piecewise linear systems with three switching manifolds
2020
In this article, we construct a kind of three-dimensional piecewise linear (PWL) system with three switching manifolds and obtain four theorems with regard to the existence of a homoclinic orbit and a heteroclinic cycle in this class of PWL system. The first theorem studies the existence of a heteroclinic cycle connecting two saddle-foci. The existence of a homoclinic orbit connecting one saddle-focus is investigated in the second theorem, and the third theorem examines the existence of a homoclinic orbit connecting another saddle-focus. The last one proves the coexistence of the heteroclinic cycle and two homoclinic orbits for the same parameters. Numerical simulations are given as example…
Universality of level spacing distributions in classical chaos
2007
Abstract We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limacon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian dis…
Physical interpretation of laser phase dynamics
1990
The basic features characterizing the dynamical evolution of the phase of a detuned-laser field under an unstable regime are physically interpreted in terms of dispersive and dynamical effects. A general method for obtaining any attractor projection containing the phase information is established, which provides evidence for the heteroclinic character of the attractor in the presence of cavity detuning for any emission regime.
On the Multifractal Character of the Lorenz Attractor
1992
A detailed analysis of the Lorenz attractor in connection with generalized dimensions is presented in this work. Different methods have been employed to estimate these dimensions. Two of them are of standard type. A new method, based on the minimal spanning tree of the point distribution, is extensively tested in this work. It turns out that the Lorenz attractor is very appropriate for being analyzed through this technique, which produces a very clean estimate of the extrema scaling indices α min and α max . The different methods give qualitatively the same result: The Lorenz attractor has a multifractal character
Blenders near polynomial product maps of $\mathbb C^2$
2021
In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto(p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types : repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets : the bifurcation locus of $H_d(\mathbb P^2)$ and the set of endomorphisms having an attracting set of non-empty interior. In an independent part, we use perturbations of H\'enon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.
Generalised power series solutions of sub-analytic differential equations
2006
Abstract We show that if a solution y ( x ) of a sub-analytic differential equation admits an asymptotic expansion ∑ i = 1 ∞ c i x μ i , μ i ∈ R + , then the exponents μ i belong to a finitely generated semi-group of R + . We deduce a similar result for the components of non-oscillating trajectories of real analytic vector fields in dimension n. To cite this article: M. Matusinski, J.-P. Rolin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).