Isolating Segments, Fixed Point Index, and Symbolic Dynamics: III. Applications

Wójcik, Klaudiusz; Zgliczyński, Piotr

doi:10.1006/jdeq.2001.4099

Cited by 11 publications

(3 citation statements)

References 8 publications

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“…Let ϕ be a local process on X, T > 0 and W, U be two subsets of R × X. We consider the following conditions (see [20], [23]):…”

Section: Dynamical Systems and Ważewski Setsmentioning

confidence: 99%

“…In this section we investigate the equation (1.1) which is a pertubation of the well known (see [23]) ż = 1 + e iκt |z| 2 z 2 (4.1)…”

Section: Applicationsmentioning

confidence: 99%

“…For N = 0 there is a trivial periodic solution ψ ≡ 0 of the equation (1.1). It was proved in [23] that there exist a compact subset I of the Poincaré section C and semiconjugacy g : I −→ g( I) ⊂ Σ 4 of σ| g( I) and ϕ (0,T ) such that Π ⊂ g( I) where Π is a sofic shift which presentation is given in Figure 1.…”

Section: Applicationsmentioning

confidence: 99%

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A study of chaos for processes under small perturbations

Oprocha¹,

Wilczyński²

2010

Publ. Math. Debrecen

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The method of isolating segments was developed to prove topological chaos (positive topological entropy) for Poincaré maps of time-periodic processes. The authors recently proved that it may also be used to verify distributional chaos, when the obtained semi-conjugacy covers a periodic point exactly one to one (so the solution giving raise to this preimage is also periodic). When we make a small perturbation of dynamics then usually the same isolating segments may be used (and as a result the same semi-conjugacy but possibly on a different set is obtained), however the periodic solution may be destroyed (then we have infinite set in the preimage), or if we are lucky, it may bifurcate to a finite number of periodic solutions.In this article we cover the case when two periodic solutions are continued from the previous one. We prove that in this case distributional chaos survives. Homoclinic and heteroclinic connections between these two solutions are also discussed.

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“…Let ϕ be a local process on X, T > 0 and W, U be two subsets of R × X. We consider the following conditions (see [20], [23]):…”

Section: Dynamical Systems and Ważewski Setsmentioning

confidence: 99%

“…In this section we investigate the equation (1.1) which is a pertubation of the well known (see [23]) ż = 1 + e iκt |z| 2 z 2 (4.1)…”

Section: Applicationsmentioning

confidence: 99%

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A study of chaos for processes under small perturbations

Oprocha¹,

Wilczyński²

2010

Publ. Math. Debrecen

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Fixed Point Results Based on the Ważewski Method

Srzednicki¹,

Wójcik²,

Zgliczyśki³

Handbook of Topological Fixed Point Theory

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Entropy for Symbolic Dynamics with Overlapping Alphabets

2016

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Abstract. We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces, give several techniques for computing lower bounds for it, and show that it is equal to a limit of entropies of (standard) full shifts. When a shift space with overlaps arises as a model for a discrete dynamical system with a finite set of overlapping neighborhoods, the entropy gives a lower bound for the topological entropy of the dynamical system.

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Isolating Segments, Fixed Point Index, and Symbolic Dynamics: III. Applications

Abstract: We prove the existence of complicated dynamics in some class of periodic systems on the plane. The proof is based on the continuation of the considered system to the model one. # 2002 Elsevier Science (USA)

Cited by 11 publications

References 8 publications

A study of chaos for processes under small perturbations

A study of chaos for processes under small perturbations

Fixed Point Results Based on the Ważewski Method

Entropy for Symbolic Dynamics with Overlapping Alphabets

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