Klaudiusz Wójcik

ON SOME NONAUTONOMOUS CHAOTIC SYSTEM ON THE PLANE

We prove the existence of the chaotic behavior in dynamical systems generated by some class of time periodic nonautonomous equations on the plane. We use topological methods based on the Lefschetz Fixed Point Theorem and the Wazewski Retract Theorem.

Remark on complicated dynamics of some planar system

Abstract In this paper we study the complicated dynamics generated by the planar periodic system z = z 1+e iκt |z| 2 , for 0 κ ⩽0.495. We prove the existence of infinitely many (geometrically distinct) periodic solutions with some special properties.

Periodic segment implies infinitely many periodic solutions

In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.

Isolating segments for Carathéodory systems and existence of periodic solutions

The method of isolating segments is introduced in the context of Caratheodory systems. We define isolating segments and extend the results of Srzednicki (1994) to Caratheodory systems.

Periodic segments and Nielsen numbers

We prove that the Poincare map φ(0,T ) has at least N(h, cl(W0 \W− 0 )) fixed points (whose trajectories are contained inside the segment W ) where the homeomorphism h is given by the segment W .

On the discrete Conley index in the invariant subspace

Abstract We present theorems concerning the relations between the discrete homotopy Conley index in the affine invariant subspace and the index calculated in the entire space or in the half space. Our basic tool is the notion of a representable index pair, i.e., an index pair composed of hypercubes. As an application we prove an index theorem for homeomorphisms f : R n − 1 × [0, +∞) → R n − 1 × [0, +∞) of compact attraction.

Conley index and permanence in dynamical systems

The motivation for our problem comes from permanence theory, which plays an important role in mathematical ecology. Roughly speaking, a flow f on R× [0,∞) is said to be permanent (or uniformly persistent) whenever R×{0} is a repeller (see [7]). Other closely related terminology includes cooperativity, persistence and ecological stability. For a discussion of how these terms are related, see [1], [9]. The criterion of permanence for biological systems is a condition ensuring the long-term survival of all species. Sufficient conditions for permanence have been given for a wide variety of models. For more details and extensive bibliographies concerning the problem, we refer the reader to [2], [8]. In this paper we show that if S ⊂ R × {0} is an isolated invariant set with nonzero homological Conley index, then there exists an x in R × (0,∞) such that ω(x) is contained in S. This may be understood as a strong violation of permanence. We first give a brief account of the Conley index theory.

On Fixed Point Index Formula and its Applications in Dynamics

Abstract The aim of this note is to present a generalization of the topological method for detecting chaotic dynamics in a periodic local processes generated by non-autonomous ordinary differential equations, based on the notion of periodic segments.

On the cohomology of an isolating block and its invariant part

We give a sufficient condition for the existence of an isolating block

Mayer-Vietoris property of the fixed point index

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