The abundance of strange attractor in the families of nearby singular diffeomorphism

In this paper, we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space, such that for each point of the set the corresponding diffeomorphism possesses strange attractor. For some families of one-dimensional mapping satisfying certain transversality condition, we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor. Furthermore, we study the dynamics of this type of strange attractor. Project Supported by Fund of National Science of China     

Coupling leads to chaos

The main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two Brusselators linearly coupled by diffusion. We will show that such a family contains a generic unfolding of a 4-dimensional nilpotent singularity of codimension 4. On the other hand, we will prove that in any generic unfolding Xμ of an n-dimensional nilpotent singularity of codimension n there are bifurcation curves of (n−1)-dimensional nilpotent singularities of codimension n−1 which are in turn generically unfolded by Xμ. Arguments conclude recalling that any generic unfolding of the 3-dimensional nilpotent singularity of codimension 3 exhibits strange attractors.     

Diffusive Lorenz dynamics: Coherent structures and spatiotemporal chaos

In this paper, we are interested in collective behaviors of many interacting Lorenz strange attractors. With an intermediate diffusion coupling between the attractors, a new remarkable synchronization of well organized structures merges as a result of two competing mechanisms: temporal chaos and spatial diffusive stabilization. A window of the coupling parameter for coherent structures is found numerically. Different from all existing scenarios of routes to chaos (period doubling, intermittency and strange attractors), an algorithmetic increase of wavenumbers before an abrupt change to chaos (compared to the periodic doubling geometrical) is unexpectedly discovered. Meta-stable states axe also observed in simulations.     

A periodically forced,piecewise linear system,Part II: The fragmentation mechanism of strange attractors and grazing

In the first part of this work, the local singularity of non-smooth dynamical systems was discussed and the criteria for the grazing bifurcation were presented mathematically. In this part, the fragmentation mechanism of strange attractors in non-smooth dynamical systems is investigated. The periodic motion transition is completed through grazing. The concepts for the initial and final grazing, switching manifolds are introduced for six basic mappings. The fragmentation of strange attractors in non-smooth dynamical systems is described mathematically. The fragmentation mechanism of the strange attractor for such a non-smooth dynamical system is qualitatively discussed. Such a fragmentation of the strange attractor is illustrated numerically. The criteria and topological structures for the fragmentation of the strange attractor need to be further developed as in hyperbolic strange attractors. The fragmentation of the strange attractors extensively exists in non-smooth dynamical systems, which will help us better understand chaotic motions in non-smooth dynamical systems.     

Strange attractors in saddle-node cycles: prevalence and globality

 We consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle-node cycles. We show that they always exhibit Hénon-like strange attractors for a set of parameter values with positive Lebesgue density at the bifurcation value. In open classes of such families the strange attractors are of global type: their basins contain an a priori defined neighbourhood of the cycle. Furthermore, the bifurcation parameter may also be a point of positive density of hyperbolic dynamics. Oblatum VIII-1993 & 23-II-1995     

Time averages for unpredictable orbits of deterministic systems

In many cases, the orbits of deterministic systems displaying highly irregular oscillations yield smoothly converging time averages. It may happen, however, that these time averages do not converge and themselves display wild oscillations. This is analyzed for heteroclinic attractors and hyperbolic strange attractors.     

Projective limits of Banach-Lie groups

In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.     

Cubic tangencies and hyperbolic diffeomorphisms

We show that the loss of hyperbolicity of an Anosov diffeomorphism of the torusT ² can be produced by a cubic tangency at a heteroclinic point. Such a first bifurcation is generic for 3-parameters families of diffeomorphisms. Our construction may also be applied to any basic set of a surface diffeomorphism. Moreover, if the pointq of cubic tangency corresponds to a lateral point of then the bifurcation is generic for two parameters. In this case the pointq may be a homoclinic intersection.Dedicated to the memory of R. MañéPartially supported by CNPq (Brazil) and CNRS (France).Supported by CNRS (France), Rectorat Université de Bourgogne (France) and CNPq (Brazil).     

Existence of generic homoclinic tangencies for henon mappings

It is an open question whether there is a strange attractor for the Henon mapping. We show that for certain maps close to the Henon map there are strange attractors.     

Strange attractors in higher dimensions

We consider generic one-parameter families of diffeomorphisms on a manifold of arbitrary dimension, unfolding a homoclinic tangency associated to a sectionally dissipative saddle point (the product of any pair of eigenvalues has norm less than 1). We prove that such families exhibit strange attractors in a persistent way: for a positive Lebesgue measure set of parameter values. In the two-dimensional case this had been obtained in a joint work with L. Mora, based on and extending the results of Benedicks-Carleson on the quadratic family in the plane.     

Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.     

Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors

A chaotic motion of gyrostats in resistant environment is considered with the help of well known dynamical systems with strange attractors: Lorenz, Rössler, Newton–Leipnik and Sprott systems. Links between mathematical models of gyrostats and dynamical systems with strange attractors are established. Power spectrum of fast Fourier transformation, gyrostat longitudinal axis vector hodograph and Lyapunov exponents are find. These numerical techniques show chaotic behavior of motion corresponding to strange attractor in angular velocities phase space. Cases for perturbed gyrostat motion with variable periodical inertia moments and with periodical internal rotor relative angular moment are considered; for some cases Poincaré sections areobtained.     

Attractors of Discrete Controlled Systems in Metric Spaces

A new scenario is described for the creation of a strange attractor in discrete dynamical systems acting in metric spaces. We investigate attractors for ensembles of dynamical systems and attractors for controlled systems with programmed piecewise-constant controls taking finitely many values.     

A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems

In this paper, we consider characteristic polynomials of n-dimensional systems that determine a segment of polynomials. One parameter is used to characterize this segment of polynomials in order to determine the maximal interval of dissipativity and unstability. Then we apply this result to the generation of a family of attractors based on a class of unstable dissipative systems (UDS) of type affine linear systems. This class of systems is comprised of switched linear systems yielding strange attractors. A family of these chaotic switched systems is determined by the maximal interval of perturbation of the matrix that governs the dynamics for still having scroll attractors.     

A representation of one-dimensional attractors ofA-diffeomorphisms by hyperbolic homeomorphisms

We give a representation for the restrictions ofA-diffeomorphisms of closed orientable surfaces of genus > 1 from a homotopy class containing a pseudo-Anosov diffeomorphism to all one-dimensional attractors that do not contain special pairs of boundary periodic points. The representation is given by the restriction of a hyperbolic homeomorphism to an invariant zero-dimensional set formed by the intersection of two transversal geodesic laminations. It is shown how this result can be generalized to the representation of the restrictions ofA-diffeomorphisms defined on a closed surface of any genus to arbitrary one-dimensional attractors. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 76–87, July, 1997. Translated by V. E. Nazaikinskii     

Richness of chaotic dynamics in nonholonomic models of a celtic stone

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.     

STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS FOR DISSIPATIVE ZAKHAROV EQUATIONS

In this paper, the pseudospectral method to solve the dissipative Zakharov equations is used. Its convergence is proved by priori estimates. The existence of the global attractors and the estimates of dimension are presented. A class of steady state solutions is also disscussed. The numerical results show that if the steady state solutions satisfy some special conditions, they become unstable and limit cycles and strange attractors will occur for very small perturbations . The largest Lyapunov exponent and analysis of the linearized system are applied to explain these phenomena.     

The density of the transversal homoclinic points in the Henon-like strange attractors

We consider the Henon-like strange attractors Λ in a family which is a nonsingular perturbation of a d-modal family. The existence of the Henon-like strange attractors in this family was proved by Diaz et al. [Inventions Math. 125 (1996) 37]. We prove that the transversal homoclinic points are dense in Λ, and that hyperbolic periodic points are dense in Λ. Moreover, the hyperbolic periodic points that are heteroclinically related to the primary periodic point (transversal intersection of stable and unstable manifolds) are dense in Λ.     

Bifurcation analysis and linear control of the Newton–Leipnik system

In this paper, we study a sort of chaotic system—Newton–Leipnik system which possesses two strange attractors. The static and dynamic bifurcations of the system are studied. The chaos controlling is performed by a simpler linear controller, and numerical simulation of the control is supplied. At the same time, Lyapunov exponents of the system show that the result of the chaos controlling is right.