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.     

Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship,Characterization and Robustness

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.     

Single Parameter Dissipativity and Attractors in DiscreteTirne Asynchronous Systems ∗

Discrete time nonautonomous dynamical systems generated by nonautonomous difference equations are formulated as discrete time skew—product systems consisting of cocycle state mappings that are driven by discrete time autonomous dynamical systems. Forwards and pullback attractors are two possible generalizations of autonomous attractors to such systems. Their existence follows from appropriate forwards or pullback dissipativity conditions. For discrete time nonautonomous dynamical systems generated by asynchronous systems with frequency updating components such a dissipativity condition is usually known for a single starting parameter value of the driving system. Additional conditions that then ensure the existence of a forwards or pullback attractor for such an asynchronous system are investigated here     

Morse Decompositions for Nonautonomous General Dynamical Systems

The Morse decomposition theory for nonautonomous general dynamical systems (set-valued dynamical systems) and differential inclusions is established. The stability of Morse decompositions of pullback attractors is also addressed.     

Attractors for discrete periodic dynamical systems

A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the union of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an autonomous map, properties that the nonautonomous attractor inherits from the autonomous attractor are discussed. Examples from population biology are presented.     

Bounds for attractors and the existence of homoclinic orbits in the lorenz system

Frequency estimates are derived for the Lyapunov dimension of attractors of non-linear dynamical systems. A theorem on the localization of global attractors is proved for the Lorenz system. This theorem is applied to obtain upper bounds for the Lyapunov dimension of attractors and to prove the existence of homoclinic orbits in the Lorenz system.     

Impulsive non‐autonomous dynamical systems and impulsive cocycle attractors

In this work, we define the notions of ‘impulsive non‐autonomous dynamical systems’ and ‘impulsive cocycle attractors’. Such notions generalize (we will see that not in the most direct way) the notions of autonomous dynamical systems and impulsive global attractors in the current published literature. We also establish conditions to ensure the existence of an impulsive cocycle attractor for a given impulsive non‐autonomous dynamical system, which are analogous to the continuous case. Moreover, we prove the existence of such attractor for a non‐autonomous 2D Navier–Stokes equation with impulses, using energy estimates. Copyright © 2016 John Wiley & Sons, Ltd.     

On Attractors of Multivalued Semi-Flows and Differential Inclusions

In this paper we study the existence of global attractors for multivalued dynamical systems. These theorems are then applied to dynamical systems generated by differential inclusions for which the solution is not unique for a given initial state. Finally, some boundary-value problems are considered.     

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 for partly dissipative lattice dynamic systems in weighted spaces

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in weighted spaces. We first establish the dynamic systems on infinite lattice, and then prove the existence of the global attractor in weighted spaces by the asymptotic compactness of the solutions. It is shown that the global attractors contain traveling waves. The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.     

Limit theorems and Markov approximations for chaotic dynamical systems

Summary We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.     

On grazing and strange attractors fragmentation in non-smooth dynamical systems

This paper presents some new ideas to understand the strange attractor fragmentation caused by grazing in non-smooth dynamic systems. The sufficient and necessary conditions for grazing bifurcations in non-smooth dynamic systems are presented. The initial sets of grazing mapping are introduced and the corresponding initial grazing manifolds are discussed. The grazing-induced fragmentation of strange attractors of chaotic motions in non-smooth dynamical systems is presented. The mathematical theory for such a fragmentation of strange attractors should be further developed.     

Upper semicontinuity of pullback attractors for multi-valued random cocycle

In this paper we study the upper semicontinuity of random attractors for multi-valued random cocycle when small random perturbations approach zero or small perturbation for random cocycle is considered. Furthermore, we consider the upper semicontinuity of random attractors for multi-valued random cocycle under the condition which the metric dynamical systems is ergodic.     

A note on attractors with finite fractal dimension

Conditions for the finite fractal dimension of precompact invariantsets are formulated and exponential attractors for discreteand continuous dynamical systems are constructed.     

The number of attractors in dynamical systems associated with cycles

The number of attractors in dynamical systems of binary vectors associated with cycles is calculated.     

Striped attractor generation and synchronization analysis for coupled Rössler systems

A model of networked chaotic Rössler systems with periodic couplings is discussed. New phenomena, including individual attractors in striped rectangular shapes and partial synchronization (or clustering), are shown for these locally coupled systems. Coupling-induced attractors with multiple stripes can be easily controlled by coupling parameters. Moreover, various interconnection topologies are also taken into consideration in the synchronization analysis, and dynamical behaviors of the coupled systems are illustrated by numerical results.     

Random Point Attractors Versus Random Set Attractors

The notion of an attractor for a random dynamical system withrespect to a general collection of deterministic sets is introduced.This comprises, in particular, global point attractors and globalset attractors. After deriving a necessary and sufficient conditionfor existence of the corresponding attractors it is proved thata global set attractor always contains all unstable sets ofall of its subsets. Then it is shown that in general randompoint attractors, in contrast to deterministic point attractors,do not support all invariant measures of the system. However,for white noise systems it holds that the minimal point attractorsupports all invariant Markov measures of the system.     

Dynamical behavior for stochastic lattice systems

Random attractors describe the long term behavior of the random dynamical systems. This paper is devoted to a general first order stochastic lattice dynamical systems (SLDS) with some dissipative nonlinearity. We prove the asymptotic compactness of the random dynamical system and obtain the random attractor, which is a compact random invariant set with tempered bound.     

Algebraic approximation of attractors of dynamical systems on manifolds

We consider an algebraic approximation of attractors of dynamical systems defined on a Euclidean space, a flat cylinder, and a projective space. We present the Foias-Temam method for the approximation of attractors of systems with continuous time and apply it to the investigation of Lorenz and Rössler systems. A modification of this method for systems with discrete time is also described. We consider elements of the generalization of the method to the case of an arbitrary Riemannian analytic manifold.     

Maximal Unstable Dissipative Interval to Preserve Multi-scroll Attractors via Multi-saturated Functions

In this paper, we present families of piecewise linear systems which are controlled by a continuous piecewise monoparametric control function for the generation of monoparametric families of multi-scroll attractors. Thus, the maximum range of values that the parameter set can take in order to preserve the useful dynamics for generating of multi-scroll attractors is found and it will be called maximal robust dynamics interval. This class of dynamical systems is the result of combining two or more unstable “one-spiral” trajectories. We give necessary and sufficient conditions in order to preserve multi-scroll attractors in terms of a parameter, i.e., a family of multi-scroll attractors is generated by means of a family of switching systems with multiple monoparametric companion matrices. Lastly, we provide an example to show how the developed theory works.