A limit set trichotomy for order-preserving systems on time scales.
A uniform dichotomy for generic cocycles over a minimal base
We consider continuous -cocycles over a minimal homeomorphism of a compact set of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.
Analytic enclosure of the fundamental matrix solution
This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing...
Asymptotic stability of switching systems.
Chaos in nonautonomous dynamical systems.
Compact Global Chaotic Attractors of Discrete Control Systems
The paper is dedicated to the study of the problem of existence of compact global chaotic attractors of discrete control systems and to the description of its structure. We consider so called switched systems with discrete time xn+1 = fν(n)(xn), where ν : ℤ+ ⃗ {1,2,...,m}. If m ≥ 2 we give sufficient conditions (the family M := {f1,f2,...,fm} of functions is contracting in the extended sense) for the existence of a compact global chaotic attractor. We study this problem in the framework of non-autonomous...
Complicated dynamics in nonautonomous ODEs.
Equivalence of ill-posed dynamical systems
The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic...
From transverse heteroclinic cycles to transverse homoclinic orbits
Group actions on monotone skew-product semiflows with applications
We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group -action has been considered on a strongly monotone skew-product semiflow. Here we relax the strong monotonicity and compactness requirements, and establish a theory concerning symmetry or monotonicity properties of uniformly stable 1-cover minimal sets. We then apply this theory to show rotational symmetry of certain stable entire solutions for a class of...
Growth conditions for the stability of a class of time-varying perturbed singular systems
In this paper, we investigate the problem of stability of linear time-varying singular systems, which are transferable into a standard canonical form. Sufficient conditions on exponential stability and practical exponential stability of solutions of linear perturbed singular systems are obtained based on generalized Gronwall inequalities and Lyapunov techniques. Moreover, we study the problem of stability and stabilization for some classes of singular systems. Finally, we present a numerical example...
Heteroclinic points of multi-dimensional dynamical systems.
Higher dimensional Melnikov mappings
Metric Entropy of Nonautonomous Dynamical Systems
We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (Xn, μn) of probability spaces and a sequence of measurable maps fn : Xn → Xn+1 with fnμn = μn+1. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion...
Nonautonomous attractors of skew-product flows with digitized driving systems.
On a criterion of Yakubovich type for the absolute stability of nonautonomous control processes.
On Funnels of Multis and Their Behavior
On geometric detection of periodic solutions and chaotic dynamics.
On the estimation of averages over infinite intervals with an application to average persistence in population models.
Pullback incremental attraction
A pullback incremental attraction, a nonautonomous version of incremental stability, is introduced for nonautonomous systems that may have unbounded limiting solutions. Its characterisation by a Lyapunov function is indicated.