Attractors Of Dynamical Systems Research Articles

Spatio‐temporal patterns with hyperchaotic dynamics in diffusively coupled biochemical oscillators

We present three examples how complex spatio‐temporal patterns can be linked to hyperchaotic attractors in dynamical systems consisting of nonlinear biochemical oscillators coupled linearly with diffusion terms. The systems involved are: (a) a two‐variable oscillator with two consecutive autocatalytic reactions derived from the Lotka–Volterra scheme; (b) a minimal two‐variable oscillator with one first‐order autocatalytic reaction; (c) a three‐variable oscillator with first‐order feedback lacking autocatalysis. The dynamics of a finite number of coupled biochemical oscillators may account for complex patterns in compartmentalized living systems like cells or tissue, and may be tested experimentally in coupled microreactors.

On the Connectedness of Attractors for Dynamical Systems

For a dynamical system on a connected metric spaceX, the global attractor (when it exists) is connected provided that either the semigroup is time-continuous orXis locally connected. Moreover, there exists an example of a dynamical system on a connected metric space which admits a disconnected global attractor.

The random attractor of the stochastic Lorenz system

We study the existence of attractors for dynamical systems under the influence of random parameters. Such a random attractor is a measurable multi-function with compact images fulfilling particular invariance and attracting properties. In particular, this uniquely determined attractor attracts measurable multi-functions. Under certain conditions we estimate the Hausdorff dimension of the attractor. These results will be applied to the Lorenz system under the influence of a random external parameter perturbation.

Driving trajectories to a desirable attractor by using small control

Driving trajectories to a desirable attractor for dynamical systems with multiple coexisting attractors has been a challenging problem in the field of chaos control. We develop an algorithm to steer most trajectories to the desirable attractor by using only small feedback control. The idea is to build a hierarchy of paths to the desirable attractor and then stabilize trajectories around one of the paths in the hierarchy. A substantial improvement in the probability for a random trajectory to asymptote to the desirable attractor has been achieved when there are fractal basin boundaries in the phase space.

Dynamics of behavior: Theory and applications for autonomous robot architectures

Abstract Limitations both for the further development as well as for the actual technical application of autonomous robots arise from the lack of a unifying theoretical language. We propose three concepts for such a language: (1) Behaviors are represented by variables, specific constant values of which correspond to task demands; (2) Behaviors are generated as attractors of dynamical systems; (3) Neural field dynamics lift these dynamic principles to the representation of information. We show how these concepts can be used to design autonomous robots. Because behaviors are generated from attractor states of dynamical systems, design of a robot architecture addresses control-theoretic stability. Moreover, flexibility of the robot arises from bifurcations in the behavioral dynamics. Therefore techniques from the qualitative theory of dynamical systems can be used to design and tune autonomous robot architectures. We demonstrate these ideas in two implementations. In one case, visual sensory information is integrated to achieve target acquisition and obstacle avoidance in an autonomous vehicle minimizing the known problem of spurious states. In a second implementation of the same behavior, a neural dynamic field endows the system with a form of obstacle memory. A critical discussion of the approach highlights strengths and weaknesses and compares to other efforts in this direction.

Attractors of dynamical systems with control: topology of purposeful formation

The definitions of homogeneous and mosaic attractors of codimension one are given. A topological method for their purposeful formation by using the feedback control laws of controlled dynamical systems is suggested.

Attractor properties of dynamical systems: neural network models

We propose tools to probe the nature of attractors in dynamical systems. These include the activity distribution, the evolutions of the state damage, activity damage and temporal correlation damage. When they are used to study the retrieval attractors in dilute asymmetric neural networks, a transition from a partially frozen phase to an unfrozen phase is found for networks trained with sufficiently noisy data near storage saturation, and this confirms that the retrieval attractors are more chaotic in this case. We are also able to demonstrate that the retrieval attractors in dilute asymmetric neural networks are not clouds of attractors, but consist of a single chaotic attractor for each stored pattern. Furthermore, they facilitate the device of effective freezing procedures, which significantly improve the quality of retrieval in neural networks.

Stabilization of quasiperiodic orbits for line-coupled diode resonator systems.

Chaotic transitions in line-coupled diode resonator systems were experimentally found to follow the Curry-Yorke model. [The Structure of Attractors in Dynamical Systems (Springer, Berlin, 1977), p. 48]. The standard diode model from the computer program spice was used to simulate these systems; the complete Lyapunov spectra from simulated systems are in good agreement with the spectra computed from the experimental time series. By applying the proportional-feedback technique to these systems, we can stabilize periodic orbits of increasing periods as well as unstable quasiperiodic orbits densely covering a torus.

Some problems of attractors of dynamical systems

In this paper, we discuss the problem of homeomorphism of attractors of dynamical systems, the classification problem and the problem of the topological structure of differentiable manifolds which allow dynamical systems with strange attractors.

Detecting the symmetry of attractors

This paper addresses the issue of how to determine numerically the symmetry of an attractor for dynamical systems. (The symmetries of attractors in phase space are related to patterns in the time-average of the solution.) Our approach to this question proceeds in two parts. First, we prove a general theorem, based on group-theoretic and differential topological ideas, which states that generically the symmetry of a (thickened) attractor can be computed from the symmetries of a point in an auxilliary space. This theorem proceeds by integrating an equivariant mapping over the thickened attractor. Once this is done, the numerical computation of symmetries reduces to showing that a certain nonnegative number is zero. Numerically, demonstrating that this number is zero can be difficult. Thus the second part of the algorithm is to consider how this number varies with parameters and noting that sudden jumps towards zero can be associated with increases in symmetry. The paper is divided into two parts. In the first we prove the general theorem and in the second we illustrate how the numerical techniques work on several examples including discrete dynamical systems with tetrahedral symmetry in R 3 and systems of three coupled cells. In high dimensions the integral mentioned previously is difficult to compute. For such examples, we assume that an ergodic theorem is valid and that symmetries can be computed using a time-average. We compare both of these methods on the low-dimensional examples as well as detect points of symmetry creation for a reaction-diffusion equation on an interval. This technique can also be used in principle to compute the symmetries of an attractor in an experiment from a time-series.

Effects of strong noise on attractors of dynamical systems

Stationary solutions of the Fokker-Planck equation are found by expansions of the probability distribution with respect to the reciprocal noise strength. It is shown that this expansion is convergent. Explicit representations for the probability distribution are obtained by numerical simulations for the Lorenz model and for a model of generators with inertial nonlinearity (GIN). The obtained distributions show an increasing amount of fine structure with decreasing noise which more and more reflects the fractal attractor structure. Results of measurements of the power spectrum of the GIN and of the distribution in the phase space are presented in dependence on the noise strengths.

Estimation of dimension for spatially distributed data and related limit theorems

In this paper we investigate the dimensional structure of probability distributions on Euclidean space and characterize a class of regular distributions. We obtain a consistent estimator of dimension based on a nearest neighbor statistic and in addition obtain asymptotic confidence intervals for dimension in the case of regular distributions. Although many examples of point estimation of dimension have recently appeared in the literature on chaotic attractors in dynamical systems, questions of consistency and interval estimation have not previously been addressed systematically.

Universal strange attractors on wrinkled tori

Strange attractors in dynamical systems that go to chaos via quasiperiodicity are considered. It is shown that there exists an infinite number of points in parameter space where the topology of the strange attractors is universal. At such points the periodic points belonging to unstable periodic orbits can be organised on ternary trees which are pruned by local rules. The grammar is universal, and thus the topological entropy is universal at each of these points in parameter space. The complete understanding of the topology is used to calculate systematically the metric properties of the attractors. The spectrum of scaling indices f( alpha ) is computed. It is found that there is no metric universality, although some aspects of the metric properties are universal. Experiments to test some of the predictions of this theory are suggested.

Catastrophe model for micromagnetics.

It is shown that transition at the Curie point and uniaxial crystalline anisotropy exhibit catastrophes which have the same morphology as well-known elementary catastrophes like the cusp and the swallowtail. It is possible to understand micromagnetics as conflict between attractors of dynamical systems. Catastrophe theory allows great geometrical insight into the mathematical background of micromagnetics. It appears as a geometrical study of metastability that can have general significance.

Spectra of scaling indices for fractal measures: Theory and experiment

We propose a new formalism to characterize fractal measures, such as strange attractors of dynamical systems, which uses a function f(α). Here α is the scaling index of the measure about a point on the fractal and f(α) is the dimension of the set of points on the fractal with the same value of α. The spectrum describes the global scaling structure of the measure. The spectrum includes as special points the Hausdorff dimension of the fractal and the scaling indices of the most rarefied and the most concentrated regions of the measure. We apply this formalism to two well-known transitions to chaos: the route via period-doubling and the route via quasiperiodicity. In both cases we find a universal spectrum which includes previously discovered scaling numbers. Both spectra have been measured experimentally on a forced Rayleigh-Bénard system.

Computation of the largest Lyapunov exponent by the generalized cell mapping

The method of cell mappings has been developed as an efficient tool for the global study of dynamical systems. One of them, the generalized cell mapping (GCM), describes the behavior of a system in a probabilistic sense, and is essentially a Markov chain analysis of dynamical systems. Since the largest Lyapunov exponent is widely used to characterize attractors of dynamical systems, we propose an algorithm for that quantity by the GCM. This allows us to examine the persistent groups of the GCM in terms of their Lyapunov exponent, thereby connecting them with their counterparts in point mapping systems.

Estimating the Dimensions of Weather and Climate Attractors

The dimensions of attractors are estimated from phase space trajectories of observed weather and climate variables (local surface pressure and relative sunshine duration, zonal wave amplitude; a δ18O-record). They provide primary information for descriptions of properties of the attractors of dynamical systems and give a lower limit to the number of the essential variables necessary to model the dynamics. These estimates are based on distance distributions of pairs of points on the single variable trajectory evolving in phase spaces which embed the attractor. One observes a low fractal dimensionality between three and four for the weather attractor, if interannual variability and seasonal changes are eliminated. The physical interpretation is based on the three dominating scales of cyclones, cyclone families and index-cycle; the irregularity of the flow and strong dependence on initial conditions amount for the fractal value. The climate variable also reveals a low dimensionality (between four and five) of the climate attractor. This is supported by an independent estimate based on eigenfunction expansion of the embedded phase space trajectory. These types of analyses suggest how to extend the standard data evaluation and model verification techniques to an analysis of the phase space behavior of observed and simulated dynamical systems.

Statistics of strange attractors by generalized cell mapping

It is proposed in this paper to use the generalized cell mapping to locate strange attractors of dynamical systems and to determine their statistical properties. The cell-to-cell mapping method is based upon the idea of replacing the state space continuum by a large collection of state space cells and of expressing the evolution of the dynamical system in terms of a cell-to-cell mapping. This leads to a Markov chain which in turn allows us to compute all the statistical properties as well as the invariant distribution. After a general discussion, the method is applied in this paper to strange attractors of a variety of systems governed either by point mappings or by differential equations. The results indicate that it is a viable, effective and attractive method. Some comments on this method in comparison with the method of direct iteration will also be made.

On global weak attractors in dynamical systems

On global weak attractors in dynamical systems