Sensitivity of Fuzzy Nonautonomous Dynamical Systems

This paper is devoted to a study of relations between two forms of sensitivity of nonautonomous dynamical system and its induced fuzzy systems. More specially, we study strong sensitivity and mean sensitivity in an original nonautonomous system and its connections with the same ones in its induced system, including set-valued system and fuzzified system.

Dynamics of Weakly Mixing Nonautonomous Systems

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

Dynamics of a rolling robot

Equations describing the rolling of a spherical ball on a horizontal surface are obtained, the motion being activated by an internal rotor driven by a battery mechanism. The rotor is modeled as a point mass mounted inside a spherical shell and caused to move in a prescribed circular orbit relative to the shell. The system is described in terms of four independent dimensionless parameters. The equations governing the angular momentum of the ball relative to the point of contact with the plane constitute a six-dimensional, nonholonomic, nonautonomous dynamical system with cubic nonlinearity. This system is decoupled from a subsidiary system that describes the trajectories of the center of the ball. Numerical integration of these equations for prescribed values of the parameters and initial conditions reveals a tendency toward chaotic behavior as the radius of the circular orbit of the point mass increases (other parameters being held constant). It is further shown that there is a range of values of the initial angular velocity of the shell for which chaotic trajectories are realized while contact between the shell and the plane is maintained. The predicted behavior has been observed in our experiments.

The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weakD-pullback exponential attractor, which is a family of setsM(t)∣t≤T, for anyT∈R, satisfying the following: (i)M(t)is compact, (ii)M(t)is positively invariant, that is,U(t,τ)M(τ)⊂M(t), and (iii) there existk,l>0such thatdist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is,M(t)pullback exponential attractsB(τ). Then we give a method to obtain the existence of weakD-pullback exponential attractors for a process. As an application, we obtain the existence of weakD-pullback exponential attractor for reaction diffusion equation inH01with exponential growth of the external force.

Directional entropy of ℤ+k-actions

In this paper, topological and measure-theoretic directional entropies are investigated for [Formula: see text]-actions. Let [Formula: see text] be a [Formula: see text]-action on a compact metric space. For each ray [Formula: see text] in [Formula: see text] we introduce a notion of positive expansivity for [Formula: see text] along [Formula: see text]. We apply the technique of “coding” which was given by Boyle and Lind in [1] to show that these directional entropies are both continuous at positively expansive directions. We relate the directional entropies of a [Formula: see text]-action at a ray [Formula: see text] to the entropies of a nonautonomous dynamical system which induced by the compositions of a sequence of maps along [Formula: see text]. And hence the variational principle relating topological and measure-theoretic directional entropies is given at positively expansive directions. Applying some known results relating entropies and other invariants (such as preimage entropies, degrees and Lyapunov exponents), we obtain the formulas of directional entropies for some classic examples, such as the [Formula: see text]-subshift actions on [Formula: see text], [Formula: see text]-actions on finite graphs and certain smooth [Formula: see text]-actions on Riemannian manifolds.

Remarks on Topological Entropy of Nonautonomous Dynamical Systems

In this paper, we give several classical definitions of topological entropy (on a noncompact and noninvariant subset) for nonautonomous dynamical system. Furthermore, their relationships are established.

SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval$[0,1)$, viewed as$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.