Attractors Of Dynamical Systems Research Articles

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.

Approximation of box dimension of attractors using the subdivision algorithm

In this paper we use the subdivision algorithm to approximate the box dimension of attractors of dynamical systems. Although in theory the subdivision algorithm provides a covering of the attractor with boxes of arbitrarily small diameter, in practice we have to overcome two obstructions: (1) ensure that the covering is (almost) minimal and (2) enhance the speed of convergence to the box dimension. We solve both problems and apply our results to the Hénon, Lorenz, Rössler and Chua attractors. The method suggested in this paper uses information from several subdivision steps and converges to the box dimension much faster than the expression in the definition of the box dimension which uses only one covering of the attractor with boxes of a prescribed diameter.

Numerical estimates of local dimension by waiting time and quantitative recurrence

Waiting time indicators are defined by measuring the time needed for a point x to approach a given point y. Our aim is to test the use of waiting time indicators, and quantitative recurrence ones to numerically estimate the local dimension of attractors in dynamical systems, as suggested by some rigorous results. We test this approach on a collection of examples, whose dimensions are already known rigorously, or with a large amount of precision. The main outcome is that the use of waiting time, compared to recurrence increases largely the speed of convergence of the estimators.

Phase resetting effects for robust cycles between chaotic sets.

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a Rössler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.

Learning about Reality from Observation

Takens, Ruelle, Eckmann, Sano, and Sawada launched an investigation of images of attractors of dynamical systems. Let A be a compact invariant set for a map f on $\mathbb{R}^{n}$ and let $\phi : \mathbb{R}^{n} \to \mathbb{R}^{m}$ be a "typical" smooth map, where n > m. When can we say that A and $\phi (A)$ are similar, based only on knowledge of the images in $\mathbb{R}^{m}$ of trajectories in A? For example, under what conditions on $\phi (A)$ (and the induced dynamics thereon) are A and $\phi (A)$ homeomorphic? Are their Lyapunov exponents the same? Or, more precisely, which of their Lyapunov exponents are the same? This paper addresses these questions with respect to both the general class of smooth mappings $\phi$ and the subclass of delay coordinate mappings. In answering these questions, a fundamental problem arises about an arbitrary compact set A in $\mathbb{R}^{n}$. For $x \in A$, what is the smallest integer d such that there is a C1 manifold of dimension d that contains all points of A that lie in some neighborhood of x? We define a tangent space TxA in a natural way and show that the answer is d = dim (Tx A). As a consequence, we obtain a Platonic version of the Whitney embedding theorem.

Lyapunov Functions in Dimension Estimates of Attractors of Dynamical Systems

We discuss relations between the Lyapunov dimension and the topological, Hausdorff, and fractal dimensions of attractors. For the Henon and Lorenz systems, explicit formulas characterizing the Lyapunov dimension of their attractors are obtained. Bibliography: 23 titles.

Attractors for dynamical systems in topological spaces

In this paper we study the long time behaviour of dynamical systemsacting in topological spaces. We give suitable definitions of attractivity and provesome properties of limit sets, as well as theorems on existence of attractors in theclass of regular spaces.

An adaptive subdivision technique for the approximation of attractors and invariant measures: proof of convergence

We prove convergence of a recently introduced adaptive multilevel algorithm for the efficient computation of invariant measures and attractors of dynamical systems. The proof works in the context of (sufficiently regular) stochastic processes and essentially shows that the discretization of phase space leads to a small random perturbation of the original process. A generalized version of a lemma of Khasminskii then gives the desired result.

An adaptive subdivision technique for the approximation of attractors and invariant measures: proof of convergence

We prove convergence of a recently introduced adaptive multilevel algorithm for the efficient computation of invariant measures and attractors of dynamical systems. The proof works in the context of (sufficiently regular) stochastic processes and essentially shows that the discretization of phase space leads to a small random perturbation of the original process. A generalized version of a lemma of Khasminskii then gives the desired result.

Analyticity of global attractors and determining nodes for a class of damped non-linear wave equations

The Gevrey regularity of global attractors of dynamical systems generated by a certain class of coupled dissipative systems of damped non-linear wave equations with periodic boundary conditions is established. This result means that the elements of the attractor are real-analytic functions in the spatial variables. As an application the existence of two determining nodes for the corresponding problem in one spatial dimension is proved.

Topological properties of attractors for dynamical systems

For a dynamical system { S t } on a metric space X, we examine the question whether the topological properties of X are inherited by the global attractor A (if it exists). When { S t } is jointly continuous, we prove that the Čech–Alexander–Spanier cohomology groups of A are isomorphic to the corresponding cohomology groups of X. The same conclusion is obtained in the case where { S t } is a group and A has a bounded neighborhood which is a deformation retract of X.

On Locally Compact Attractors of Dynamical Systems

In this paper we consider the existence of locally compact (maybe unbounded) attractors for abstract dynamical systems. We study the case in which the trajectories are bounded but the semigroup is not dissipative; i.e., there is not a bounded set which attracts each point of the phase space. Then we apply the obtained results to differential inclusions and reaction–diffusions equations.

Certain new robust properties of invariant sets and attractors of dynamical systems

UDC 517.938 w Introduction In this paper, new robust properties of (partially hyperbolic) invariant sets and attractors of diffeomorphisms are obtained, including the coexistence of dense sets of periodic points with different indices and the existence of a dense orbit with zero Lyapunov exponent. Tlm precise statements of these properties are given below in Theorem A. Partially hyperbolic invariant sets with the property of robust transitivity were intensively studied [3-7]. A review of this field with a vast bibliography can be found in [8].

Hierarchical structure of attractors of dynamical systems

Some basic properties of the small random perturbed dynamical system of Freidlin-WentzeU type are elicited. A hierarchy structure of attractors is constructed and some further properties of this structure are confirmed.

More information regarding dynamical systems via convergence to correlation dimension

It is a well-known fact that a fractal number calculated from time-series data cannot fully describe the complex structure of a strange attractor in dynamical systems. Hence, it is necessary to find a procedure through which fractal dimension provides more information about the underlying dynamical system. In order to serve such a purpose, in this study correlation integrals have been calculated at every row or off-diagonal from the distance matrix. Subsequently, the slope of each integrals has been plotted versus the number of the row or off-diagonal considered in the distance matrix. This gives an asymptotic convergence to correlation dimension and has close relation with lacunarity and types of behavior, i.e., periodic or aperiodic.

On the influence of noise on the largest Lyapunov exponent of attractors of stochastic dynamic systems

In this paper, we prove that the magnitude of the largest Lyapunov exponent of attractors appartaining to stochastic dynamic systems is increasing under the influence of noise. Thus we offer an answer to the conjecture posed by Argyris et al. in [1]. We investigate also the influence of an additive output noise to the correlation dimension and on the largest Lyapunov exponent of the attractors of dynamic systems either in a discrete or in a continuous in time formulation.

On linearisable noisy systems

In this paper we elaborate a study of the influence of the alternative mathematical schemes of noise introduced by a recent paper of Argyris et al. in this journal and discuss their influence on the correlation dimension and on the largest Lyapunov exponent of attractors of dynamic systems arising in either a discrete or a continuous in time formulation. The concept of a linearizable noisy system is also introduced.

Upper semicontinuity of attractors for small random perturbations of dynamical systems

The relationship between random attractors and global attractors for dynamical systems is studied. If a partial differential equation is perturbed by an E-small random term and certain hypotheses are satisfied, the upper semicontinuity of the random attractors is obtalned as c goes to zero. The results are applied to the Navier-Stokes equations and a problem of reaction-diffusion type, both perturbed by an additive white noise.

Conditional and Relative Multifractal Spectra

In the study of the involved geometry of singular distributions, the use of fractal and multifractal analysis has shown results of outstanding significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to the ordinary metric or volume. Most prominent examples include self-similar measures and attractors of dynamical systems. In certain cases, the multifractal spectrum is known explicitly, providing a characterization in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel methods, the conditional and the relativemultifractal spectrum, which allow for a direct comparison of two distributions. These notions measure the extent to which the singularities of two distributions 'correlate'. Being based on multifractal concepts, however, they go beyond calculating correlations. As a particularly useful tool, we develop the multifractal formalism and establish some basic properties of the new notions. With the simple example of Binomial multifractals, we demonstrate how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and sparse parts of the distributions.

Attractors of dynamical systems

Attractors of dynamical systems