Impulsive Differential Equations

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An introduction to chaotic dynamical systems

A new 3-D chaotic dynamical system with a peanut-shaped closed curve of equilibrium points is introduced in this work. Since the new chaotic system has infinite number of rest points, the new chaotic model exhibits hidden attractors. A detailed dynamic analysis of the new chaotic model using bifurcation diagrams and entropy analysis is described. The new nonlinear plant shows multi-stability and coexisting convergent attractors. A circuit model using MultiSim of the new 3-D chaotic model is designed for engineering applications. The new multi-stable chaotic system is simulated on a field-programmable gate array (FPGA) by applying two numerical methods, showing results in good agreement with numerical simulations. Consequently, we utilize the properties of our chaotic system in designing a new cipher colour image mechanism. Experimental results demonstrate the efficiency of the presented encryption mechanism, whose outcomes suggest promising applications for our chaotic system in various cryptographic applications.

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A 3-D Multi-Stable System With a Peanut-Shaped Equilibrium Curve: Circuit Design, FPGA Realization, and an Application to Image Encryption

Abstract : The emerging discipline known as \"chaos theory\" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for \"chaos\" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.

Introduction of chaotic dynamical systems

This paper studies the existence and uniqueness of exponentially stable almost periodic solutions for abstract impulsive differential equations in Banach space. The investigations are carried out by means of the fractional powers of operators. We construct an example to illustrate the feasibility of our results.

Almost periodic solutions for abstract impulsive differential equations

Shunting inhibitory cellular neural networks with strongly unpredictable oscillations

The existence and uniqueness of unpredictable solutions in the dynamics of nonhomogeneous linear systems of differential and discrete equations are investigated. The hyperbolic cases are under discussion. The presence of unpredictable solutions confirms the existence of Poincaré chaos. Simulations illustrating the chaos are provided.

Unpredictable solutions of linear differential and discrete equations

M. Akhmet has been supported by a grant (118F161) from TUB¨ ˙ITAK,\nthe Scientific and Technological Research Council of Turkey.\nM. Tleubergenova and A. Zhamanshin have been supported in parts by the MES RK\ngrant (AP05132573) ”Cellular neural networks with continuous/discrete time and singular perturbations” (2018-2020). Republic of Kazakhstan.

Quasilinear differential equations with strongly unpredictable solutions

Control and optimal response problems for quasilinear impulsive integrodifferential equations

We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results.

https://www.mdpi.com/2227-7390/8/10/1798/pdf

Madina Tleubergenova

Papers

Shunting inhibitory cellular neural networks with strongly unpredictable oscillations

Unpredictable solutions of linear differential and discrete equations

Quasilinear differential equations with strongly unpredictable solutions

Control and optimal response problems for quasilinear impulsive integrodifferential equations

Unpredictable Solutions of Linear Impulsive Systems