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A class of continua that are not attractors of any IFS

Marcin Kulczycki, Magdalena Nowak (2012)

Open Mathematics

This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.

Compact Global Chaotic Attractors of Discrete Control Systems

David Cheban (2014)

Nonautonomous Dynamical 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...

Connection graphs

Piotr Bartłomiejczyk (2006)

Fundamenta Mathematicae

We introduce connection graphs for both continuous and discrete dynamical systems. We prove the existence of connection graphs for Morse decompositions of isolated invariant sets.

Dependence of hidden attractors on non-linearity and Hamilton energy in a class of chaotic system

Ge Zhang, Chunni Wang, Ahmed Alsaedi, Jun Ma, Guodong Ren (2018)


Non-linearity is essential for occurrence of chaos in dynamical system. The size of phase space and formation of attractors are much dependent on the setting of nonlinear function and parameters. In this paper, a three-variable dynamical system is controlled by different nonlinear function thus a class of chaotic system is presented, the Hamilton function is calculated to find the statistical dynamical property of the improved dynamical systems composed of hidden attractors. The standard dynamical...

Determining the domain of attraction of hybrid non–linear systems using maximal Lyapunov functions

Szabolcs Rozgonyi, Katalin M. Hangos, Gábor Szederkényi (2010)


In this article a method is presented to find systematically the domain of attraction (DOA) of hybrid non-linear systems. It has already been shown that there exists a sequence of special kind of Lyapunov functions V n in a rational functional form approximating a maximal Lyapunov function V M that can be used to find an estimation for the DOA. Based on this idea, an improved method has been developed and implemented in a Mathematica-package to find such Lyapunov functions V n for a class of hybrid (piecewise...

Dynamical systems and shapes.

J.J. Sánchez-Gabites (2008)


This survey is an introduction to some of the methods, techniques and concepts from algebraic topology and related areas (homotopy theory, shape theory) which can be fruitfully applied to study problems concerning continuous dynamical systems. To this end two instances which exemplify the interaction between topology and dynamics are considered, namely, Conley’s index theory and the study of some properties of certain attractors.

Existence, blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions

Le Thi Phuong Ngoc, Nguyen Thanh Long (2016)

Applications of Mathematics

In this paper we consider a nonlinear Love equation associated with Dirichlet conditions. First, under suitable conditions, the existence of a unique local weak solution is proved. Next, a blow up result for solutions with negative initial energy is also established. Finally, a sufficient condition guaranteeing the global existence and exponential decay of weak solutions is given. The proofs are based on the linearization method, the Galerkin method associated with a priori estimates, weak convergence,...

Existence, uniqueness and global asymptotic stability for a class of complex-valued neutral-type neural networks with time delays

Manchun Tan, Desheng Xu (2018)


This paper explores the problem of delay-independent and delay-dependent stability for a class of complex-valued neutral-type neural networks with time delays. Aiming at the neutral-type neural networks, an appropriate function is constructed to derive the existence of equilibrium point. On the basis of homeomorphism theory, Lyapunov functional method and linear matrix inequality techniques, several LMI-based sufficient conditions on the existence, uniqueness and global asymptotic stability of equilibrium...

Flows near compact invariant sets. Part I

Pedro Teixeira (2013)

Fundamenta Mathematicae

It is proved that near a compact, invariant, proper subset of a C⁰ flow on a locally compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. Theorem 1 shows that the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour in the vicinity of compact invariant sets than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings...

Global φ-attractor for a modified 3D Bénard system on channel-like domains

O.V. Kapustyan, A.V. Pankov (2014)

Nonautonomous Dynamical Systems

In this paper we prove the existence of a global φ-attractor in the weak topology of the natural phase space for the family of multi-valued processes generated by solutions of a nonautonomous modified 3D Bénard system in unbounded domains for which Poincaré inequality takes place.

Lyapunov quasi-stable trajectories

Changming Ding (2013)

Fundamenta Mathematicae

We introduce the notions of Lyapunov quasi-stability and Zhukovskiĭ quasi-stability of a trajectory in an impulsive semidynamical system defined in a metric space, which are counterparts of corresponding stabilities in the theory of dynamical systems. We initiate the study of fundamental properties of those quasi-stable trajectories, in particular, the structures of their positive limit sets. In fact, we prove that if a trajectory is asymptotically Lyapunov quasi-stable, then its limit set consists...

Multivalued Lyapunov functions for homeomorphisms of the 2-torus

Patrice Le Calvez (2006)

Fundamenta Mathematicae

Let F be a homeomorphism of 𝕋² = ℝ²/ℤ² isotopic to the identity and f a lift to the universal covering space ℝ². We suppose that κ ∈ H¹(𝕋²,ℝ) is a cohomology class which is positive on the rotation set of f. We prove the existence of a smooth Lyapunov function of f whose derivative lifts a non-vanishing smooth closed form on 𝕋² whose cohomology class is κ.

On gradient-like random dynamical systems

Aya Hmissi, Farida Hmissi, Mohamed Hmissi (2012)

ESAIM: Proceedings

This paper deals with some characterizations of gradient-like continuous random dynamical systems (RDS). More precisely, we establish an equivalence with the existence of random continuous section or with the existence of continuous and strict Liapunov function. However and contrary to the deterministic case, parallelizable RDS appear as a particular case of gradient-like RDS.The obtained results are generalizations of well-known analogous theorems in the framework of deterministic dynamical systems....

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