Topological entropy of composition and impulsive differential equations satisfying a uniqueness condition

Andres, Jan; Ludvík, Pavel

doi:10.1016/j.chaos.2022.111800

Cited by 3 publications

(5 citation statements)

References 22 publications

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“…Let us note that in our papers [3,4,7] still formally another extension of Definition 1 was employed, matching with the Nielsen fixed point theory on tori R n /Z n . On the other hand, here we would like to follow rather the ideas from [8], where single-valued arguments were, however, exclusively applied on compact subsets in Euclidean spaces R n .…”

Section: Remark 4 the Inequality In Lemma 3 Can Be Generalized To The...mentioning

confidence: 99%

“…Because of a uniqueness condition, there is, for any k ∈ N, an evident one-to-one correspondence between the kω-periodic solutions x(•) of (4), x(t) ≡ x(t + kω) and x(t) ≡ x(t + jω) for j < k, and k-periodic points x 0 of T ω such that x(0) = x 0 , i.e. x 0 = T k ω (x 0 ) and x 0 = T j ω (x 0 ) for j < k. Now, consider the impulsive differential equation (8) x = F (t, x), t = t j := jω , for ω > 0 , x(t…”

Section: Some Further Preliminariesmentioning

confidence: 99%

“…Therefore, we can naturally introduce the following definition of topological entropy for (4), (7) and (8). Definition 5.…”

Section: Some Further Preliminariesmentioning

confidence: 99%

“…Similarly, equation (8), where F (t, x) := A(t)x and I is a real (n × n)-matrix, has a topological entropy…”

Section: Some Further Preliminariesmentioning

confidence: 99%

“…For nonlinear impulsive systems (see e.g. [2,5,8]), and for multivalued or discontinuous impulsive systems (see e.g. [3,4,7,13,14]).…”

Section: Introduction and Some Preliminariesmentioning

confidence: 99%

See 4 more Smart Citations

Topological entropy and differential equations

Andres¹,

Ludvík²

2023

Arch. Math. (Brno)

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On the background of a brief survey panorama of results on the topic in the title, one new theorem is presented concerning a positive topological entropy (i.e. topological chaos) for the impulsive differential equations on the Cartesian product of compact intervals, which is positively invariant under the composition of the associated Poincaré translation operator with a multivalued upper semicontinuous impulsive mapping.

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Section: Remark 4 the Inequality In Lemma 3 Can Be Generalized To The...mentioning

confidence: 99%

Section: Some Further Preliminariesmentioning

confidence: 99%

“…Therefore, we can naturally introduce the following definition of topological entropy for (4), (7) and (8). Definition 5.…”

Section: Some Further Preliminariesmentioning

confidence: 99%

“…Similarly, equation (8), where F (t, x) := A(t)x and I is a real (n × n)-matrix, has a topological entropy…”

Section: Some Further Preliminariesmentioning

confidence: 99%

“…For nonlinear impulsive systems (see e.g. [2,5,8]), and for multivalued or discontinuous impulsive systems (see e.g. [3,4,7,13,14]).…”

Section: Introduction and Some Preliminariesmentioning

confidence: 99%

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Topological entropy and differential equations

Andres¹,

Ludvík²

2023

Arch. Math. (Brno)

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Parametric topological entropy and differential equations with time–dependent impulses II: Multivalued case

Andres

2023

Journal of Differential Equations

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Parametric Topological Entropy for Multivalued Maps and Differential Inclusions with Nonautonomous Impulses

Andres

Ludvík

2023

Int. J. Bifurcation Chaos

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The main purpose of this paper is to investigate a parametric topological entropy for impulsive differential inclusions on tori. In this way, besides other matters, we would like to extend our recent results concerning impulsive differential equations as well as those on “nonparametric” topological entropy to impulsive differential inclusions. Parametric topological entropy, which is usually called a topological entropy for nonautonomous dynamical systems, is considered here via the compositions of associated multivalued Poincaré translation operators with the single-valued time-dependent impulsive maps. On compact polyhedra and, in particular on tori, parametric topological entropy for families of admissible multivalued maps can be estimated from below by means Ivanov-type inequality in terms of the asymptotic Nielsen and Lefschetz numbers which are, unlike the topological entropy, homotopy invariants. In the scalar case, an effective criterion for a positive parametric topological entropy can be given by topological degree arguments for equi-continuous impulsive maps. In a single-valued nonparametric case, a positive topological entropy usually signifies topological chaos. Some simple illustrative examples are provided.

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Topological entropy of composition and impulsive differential equations satisfying a uniqueness condition

Cited by 3 publications

References 22 publications

Topological entropy and differential equations

Topological entropy and differential equations

Parametric topological entropy and differential equations with time–dependent impulses II: Multivalued case

Parametric Topological Entropy for Multivalued Maps and Differential Inclusions with Nonautonomous Impulses

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