Topological Chaos for Differential Inclusions with Multivalued Impulses on Tori

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Jan Andres
Keyword(s):  
Periodic Solutions ◽  
Topological Entropy ◽  
Similar Condition ◽  
Differential Inclusions ◽  
Lower Estimate ◽  
Nielsen Numbers ◽  
Positive Topological Entropy ◽  
Topological Chaos ◽  
Translation Operators ◽  
Admissible Maps

A multivalued version of the Ivanov inequality for the lower estimate of topological entropy of admissible maps is applied to differential inclusions with multivalued impulses on tori via the associated Poincaré translation operators along their trajectories. The topological chaos in the sense of a positive topological entropy is established in terms of the asymptotic Nielsen numbers of the impulsive maps being greater than 1. This condition implies at the same time the existence of subharmonic periodic solutions with infinitely many variety of periods. Under a similar condition, the coexistence of subharmonic periodic solutions of all natural orders is also carried out.

Chaos for Differential Equations with Multivalued Impulses

2021 ◽  
Vol 31 (07) ◽  
pp. 2150113
Author(s):  
Jan Andres
Keyword(s):  
Differential Equations ◽  
Topological Entropy ◽  
Deterministic Chaos ◽  
Impulsive Differential Equations ◽  
Multivalued Maps ◽  
Nielsen Numbers ◽  
Positive Topological Entropy ◽  
Translation Operators ◽  
Principal Tool ◽  
Admissible Maps

The deterministic chaos in the sense of a positive topological entropy is investigated for differential equations with multivalued impulses. Two definitions of topological entropy are examined for three classes of multivalued maps: [Formula: see text]-valued maps, [Formula: see text]-maps and admissible maps in the sense of Górniewicz. The principal tool for its lower estimates and, in particular, its positivity are the Ivanov-type inequalities in terms of the asymptotic Nielsen numbers. The obtained results are then applied to impulsive differential equations via the associated Poincaré translation operators along their trajectories. The main theorems for chaotic differential equations with multivalued impulses are formulated separately on compact subsets of Euclidean spaces and on tori. Several illustrative examples are supplied.

scholarly journals Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1602
Author(s):  
Jan Andres ◽  
Jerzy Jezierski
Keyword(s):  
Differential Equations ◽  
Topological Entropy ◽  
Lower Estimate ◽  
Deterministic Chaos ◽  
Impulsive Differential Equations ◽  
Nielsen Numbers ◽  
Positive Topological Entropy ◽  
Admissible Pairs ◽  
Impulsive Dynamics ◽  
Differential Equations And Inclusions

The main aim of this article is two-fold: (i) to generalize into a multivalued setting the classical Ivanov theorem about the lower estimate of a topological entropy in terms of the asymptotic Nielsen numbers, and (ii) to apply the related inequality for admissible pairs to impulsive differential equations and inclusions on tori. In case of a positive topological entropy, the obtained result can be regarded as a nontrivial contribution to deterministic chaos for multivalued impulsive dynamics.

scholarly journals Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1701
Author(s):  
Jan Andres
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Vector Fields ◽  
Hausdorff Metric ◽  
Invariant Sets ◽  
Devaney’S Chaos ◽  
Nielsen Numbers ◽  
Multiple Periodic Solutions ◽  
Translation Operators ◽  
Compact Subsets

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

scholarly journals Periodic Solutions for Semilinear Fourth-Order Differential Inclusions via Nonsmooth Critical Point Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bian-Xia Yang ◽  
Hong-Rui Sun
Keyword(s):  
Critical Point ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Critical Point Theory ◽  
Differential Inclusions ◽  
Point Theory ◽  
Nonsmooth Critical Point Theory ◽  
Critical Point Theorem ◽  
Related Literature ◽  
Nonsmooth Critical Point

Three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem. Some results of previous related literature are extended.

A vector p-Laplacian type approach to multiple periodic solutions for the p-relativistic operator

2017 ◽  
Vol 19 (03) ◽  
pp. 1650029 ◽  
Author(s):  
Petru Jebelean ◽  
Jean Mawhin ◽  
Călin Şerban
Keyword(s):  
Differential Operator ◽  
Periodic Solutions ◽  
Differential Inclusions ◽  
Singular Operator ◽  
Multiple Periodic Solutions ◽  
Vector Formula

We prove the existence of at least [Formula: see text] geometrically distinct [Formula: see text]-periodic solutions for a differential inclusions system of the form [Formula: see text] Here, [Formula: see text] is a monotone homeomorphism, [Formula: see text] is periodic with respect to each component of the second variable and [Formula: see text] stands for the generalized Clarke gradient of [Formula: see text] at [Formula: see text]. The monotonicity assumptions on [Formula: see text] highlight the vector [Formula: see text]-Laplacian as being the prototype differential operator. The main interesting feature of this approach is that it also provides a useful framework to treat the case of the [Formula: see text]-relativistic singular operator.

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