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.

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.

Randomized Sharkovsky-type theorems and their application to random impulsive differential equations and inclusions on tori

2019 ◽  
Vol 19 (05) ◽  
pp. 1950036 ◽  
Author(s):  
Jan Andres
Keyword(s):  
Differential Equations ◽  
Deterministic Chaos ◽  
Impulsive Differential Equations ◽  
Factor Space ◽  
State Variables ◽  
Harmonic Solution ◽  
Lefschetz Numbers ◽  
Differential Equations And Inclusions ◽  
Dimension One ◽  
The Given

Our randomized versions of the Sharkovsky-type cycle coexistence theorems on tori and, in particular, on the circle are applied to random impulsive differential equations and inclusions. The obtained effective coexistence criteria for random subharmonics with various periods are formulated in terms of the Lefschetz numbers (in dimension one, in terms of degrees) of the impulsive maps and their iterates w.r.t. the (deterministic) state variables. Otherwise, the forcing properties of certain periods of the given random subharmonics are employed, provided there exists a random harmonic solution. In the single-valued case, the exhibition of deterministic chaos in the sense of Devaney is detected for random impulsive differential equations on the factor space [Formula: see text]. Several simple illustrative examples are supplied.

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).

Sharp Block–Sharkovsky Type Theorem for Multivalued Maps on the Circle and Its Application to Differential Equations and Inclusions

2019 ◽  
Vol 29 (09) ◽  
pp. 1950127 ◽  
Author(s):  
Jan Andres ◽  
Karel Pastor
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
General Result ◽  
Type Theorem ◽  
Multivalued Maps ◽  
Interval Maps ◽  
Translation Operators ◽  
Differential Equations And Inclusions ◽  
Positive Integers ◽  
Full Analogy

This is a final part of the series of our papers devoted to a multivalued version of the (Sharkovsky type) Block cycle coexistence theorem. It improves our last general result in the sense that its part related to the usual ordering of positive integers becomes a full analogy of the standard single-valued case, while the alternative part related to the Sharkovsky ordering of positive integers is an analogy of the multivalued case for interval maps, provided there exists a fixed point. That is why we call the obtained theorem here as “sharp”. This theorem is still applied via the associated Poincaré translation operators to differential equations and inclusions on the circle. All the deterministic results are also randomized in an advantageous way.

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