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.

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.

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.

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.

Coexistence of Random Subharmonic Solutions of Random Impulsive Differential Equations and Inclusions on a Circle

2020 ◽  
Vol 30 (10) ◽  
pp. 2050152
Author(s):  
Jan Andres
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Differential Inclusions ◽  
Impulsive Differential Equations ◽  
Subharmonic Solutions ◽  
Random Differential Equations ◽  
Random Impulses ◽  
Differential Equations And Inclusions

The coexistence of random periodic solutions with various periods (i.e. subharmonics) is proved to random differential equations on a circle with random impulses of all integer orders. One of the theorems is also extended to random differential inclusions on a circle with multivalued deterministic impulses. These results can be roughly characterized as a further application of the randomized Sharkovsky type theorems to random impulsive differential equations and inclusions on a circle.

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