The efficient approximation of coherent pairs in non-autonomous dynamical systems

Let be a nonautonomous dynamical system. In this paper, we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new definition of asymptotic periodicity. This definition is not only very natural but also resistant to changes of the beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for pointwise convergent nonautonomous systems topological transitivity together with a dense set of asymptotically periodic points imply sensitivity. We also show that even for uniformly convergent systems, the nonautonomous analogue of Sharkovsky's theorem is not valid for most definitions of periodic points.

Weak stability of non-autonomous discrete dynamical systems

Multi-transitivity in non-autonomous discrete systems

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

Sensitivity of non-autonomous discrete dynamical systems

We introduce the Ellis semigroup of a nonautonomous discrete dynamical system (X, f1,∞) when X is a metric compact space. The underlying set of this semigroup is the pointwise closure of in the space XX. By using the convergence of a sequence of points with respect to an ultrafilter it is possible to give a precise description of the semigroup and its operation. This notion extends the classical Ellis semigroup of a discrete dynamical system. We show several properties that connect this semigroup and the topological properties of the nonautonomous discrete dynamical system.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

In this study, the author investigates controllability of discrete‐time bilinear systems by using invariant sets. An invariant set defined in this study is a set on which the system is invariant, namely for any initial state that belongs to the set, the system cannot be steered away from this set. If a system is controllable, then it has no invariant set in its state space other than the state space itself. Invariant sets are thus closely related to controllability. By studying the invariant sets of discrete‐time bilinear systems, the author first presents a necessary and sufficient condition for controllability of the systems in dimension two, which covers a classical result under the same condition. Then the author considers the high‐dimensional systems and improve the recent results on controllability by relaxing a condition concerning invariant sets, where more general controllability criteria are derived. Finally, examples are given to illustrate the obtained results.

THE PERIODICITY OF CHAOTIC IMPACT OSCILLATORS IN HAUSDORFF PHASE SPACES

We study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

We review the recent notion of a nonautonomous dynamical system (NDS), which has been introduced as an abstraction of both random dynamical systems and continuous skew product flows. Our focus is on fundamental analogies and discrepancies brought about by these two principal classes of NDS. We discuss base dynamics mainly through almost periodicity and almost automorphy, and we emphasize the importance of these concepts for NDS which are generated by differential and difference equations. Nonautonomous dynamics is presented by means of selected examples. We also mention several natural yet unresolved questions.

A set oriented definition of finite-time Lyapunov exponents and coherent sets

Let (Y, d) be a nontrivial metric space and (Y, g1,∞) be a nonautonomous discrete dynamical system given by sequences $(g_{l})_{l = 1}^{\infty }$ of continuous maps gl : Y → Y and let $\mathcal {F}$, $\mathcal {F}_{1}$ and $\mathcal {F}_{2}$ be given shift-invariant Furstenberg families. In this paper, we study stronger forms of transitivity and sensitivity for nonautonomous discrete dynamical systems by using Furstenberg family. In particular, we discuss the $\mathcal {F}$-transitivity, $\mathcal {F}$-mixing, $\mathcal {F}$-sensitivity, $\mathcal {F}$-collective sensitivity, $\mathcal {F}$-synchronous sensitivity, $(\mathcal {F}_{1},\mathcal {F}_{2})$-sensitivity and $\mathcal {F}$-multi-sensitivity for the system (Y, g1,∞) and show that under the conditions that gj is semi-open and satisfies gj ∘ g = g ∘ gj for each j ∈ {1, 2, ⋯ } and that $$\sum\limits_{j = 1}^{\infty}D(g_{j},g) $$exists (i.e., $\sum \limits _{j = 1}^{\infty }D(g_{j},g)<+\infty $), the following hold: The above results extend the existing ones.

In this paper, we first study the existence and uniqueness of random uniform attractors to a class of non-autonomous stochastic lattice dynamical systems driven by multiplicative noise in a weighted space. Then, we establish the upper semicontinuity of random uniform attractors of the corresponding limiting system as the intensity of noise approaches ɛ0 &amp;gt; 0 or as the finite-dimensional approximation approaches the infinite-dimensional case. The study of random uniform attractors in weighted spaces brings about essential difficulties. Some special estimates will overcome the difficulties. Uniform-tail estimates are needed to prove the asymptotic compactness of solution operators, which is a major step towards proving the existence of attractors. The upper semicontinuity of random uniform attractors is proved when the union of all perturbed random uniform attractors is tempered.

In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results of Rempe-Gillen and Urbanski on non-autonomous iterated function systems to the setting of non-autonomous graph directed Markov systems and then show that the Hausdorff dimension of the fractal limit set is equal to the zero of the associated pressure function provided the size of the alphabets at each time step do not grow too quickly. In trying to remove these growth restrictions, we present several other systems for which Bowen's formula holds, most notably ascending systems. We then use these various constructions to investigate the Hausdorff dimension of various subsets of the Julia set for different large classes of transcendental meromorphic functions of finite order which have been perturbed non-autonomously. In particular we find lower and upper bounds for the dimension of the subset of the Julia set whose points escape to infinity, and in many cases we find the exact dimension. While the upper bound was known previously in the autonomous case, the lower bound was not known in this setting, and all of these results are new in the non-autonomous setting. We also use transfer operator techniques to prove an almost sure invariance principle for random dynamical systems for which the thermodynamical formalism has been well established. In particular, we see that if a system exhibits a fiberwise spectral gap property and the base dynamical system is sufficiently well behaved, i.e. it exhibits an exponential decay of correlations, then the almost sure invariance principle holds. We then apply these results to uniformly expanding random systems like those studied by Mayer, Skorulski, and Urbanski and Denker and Gordin.