A survey on the modeling of hybrid behaviors: How to account for impulsive jumps properly

Feketa, Petro; Klinshov, Vladimir; Lücken, Leonhard

doi:10.1016/j.cnsns.2021.105955

Cited by 36 publications

(4 citation statements)

References 129 publications

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“…There are many limitations to our work. It is known that asymptotic stability of solutions to impulsive systems can be treated in both weak (convergence towards the solution depends only on the elapsed time) and strong (convergence depends on the elapsed on the elapsed time and the number of impulses) flavors [29,30]. We deal with the classical weak stability in this paper.…”

Section: Discussionmentioning

confidence: 99%

Exponential Stability of Hopfield Neural Network Model with Non-Instantaneous Impulsive Effects

Fečkan

Wang

2023

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Section: Discussionmentioning

confidence: 99%

Exponential Stability of Hopfield Neural Network Model with Non-Instantaneous Impulsive Effects

Fečkan

Wang

2023

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“…where α ∈ X, ϕ : X → X are given. Note that formulation (4) is natural in many mechanical problems where there is an instantaneous change in velocity [13,47], i.e., impulsive action (4) causes discontinuity not only in y but also in y t .…”

Section: Problem Statementmentioning

confidence: 99%

“…There are several mathematical frameworks available in the literature for the proper modeling and analysis of these processes, namely impulsive differential equations, discontinuous dynamical systems [6][7][8], hybrid dynamical systems [9][10][11], and differential equations with Dirac delta functions [12]. We refer the interested reader to [13], which provides a concise overview and comparison of these frameworks. More recent studies of discontinuous dynamics include results for delay [14][15][16], fractional order [17][18][19], stochastic equations [20][21][22], fuzzy differential systems [23,24], and applications of impulses for stabilization and control purposes [25][26][27].…”

Section: Introductionmentioning

confidence: 99%

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

Feketa,

Fedorenko,

Bezushchak

et al. 2023

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In this paper, we investigate the qualitative behavior of an evolutionary problem consisting of a hyperbolic dissipative equation whose trajectories undergo instantaneous impulsive discontinuities at the moments when the energy functional reaches a certain threshold value. The novelty of the current study is that we consider the case in which the entire infinite-dimensional phase vector undergoes an impulsive disturbance. This substantially broadens the existing results, which admit discontinuities for only a finite subset of phase coordinates. Under fairly general conditions on the system parameters, we prove that such a problem generates an impulsive dynamical system in the natural phase space, and its trajectories have nonempty compact ω-limit sets.

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“…The impulsive differential equations have been widely studied and have seen significant progress in recent years by many authors [10][11][12][13]. It has found widespread application in a variety of fields, including biological technology, medicine dynamics, physics, economy, population dynamics, and epidemiology [14]. It is widely accepted that both natural and human-induced phenomena can have impulsive effects on population dynamics and epidemiology.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Behaviors in a Stage-Structured Model with a Birth Pulse

Liu,

Guo,

Liu

2023

Mathematics

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This paper presents an exploitation model with a stage structure to analyze the dynamics of a fish population, where periodic birth pulse and pulse fishing occur at different fixed time. By utilizing the stroboscopic map, we can obtain an accurate cycle of the system and investigate the stability thresholds. Through the application of the center manifold theorem and bifurcation theory, our research has shown that the given model exhibits transcritical and flip bifurcation near its interior equilibrium point. The bifurcation diagrams, maximum Lyapunov exponents and phase portraits are presented to further substantiate the complexity. Finally, we present high-resolution stability diagrams that demonstrate the global structure of mode-locking oscillations. We also describe how these oscillations are interconnected and how their complexity unfolds as control parameters vary. The two parametric planes illustrate that the structure of Arnold’s tongues is based on the Stern–Brocot tree. This implies that the periodic occurrence of birth pulse and pulse fishing contributes to the development of more complex dynamical behaviors within the fish population.

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A survey on the modeling of hybrid behaviors: How to account for impulsive jumps properly

Cited by 36 publications

References 129 publications

Exponential Stability of Hopfield Neural Network Model with Non-Instantaneous Impulsive Effects

Exponential Stability of Hopfield Neural Network Model with Non-Instantaneous Impulsive Effects

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

Dynamical Behaviors in a Stage-Structured Model with a Birth Pulse

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