Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities Introduction

Date

2017-01-01

Author

Akhmet, Marat
Kashkynbayev, Ardak

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https://hdl.handle.net/11511/99056

Journal

BIFURCATION IN AUTONOMOUS AND NONAUTONOMOUS DIFFERENTIAL EQUATIONS WITH DISCONTINUITIES

DOI

https://doi.org/10.1007/978-981-10-3180-9_1

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Department of Mathematics, Article

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Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities Akhmet, Marat (Springer, London/Berlin , 2017-01-01) This book is devoted to bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types. That is, those with jumps present either in the right-hand-side or in trajectories or in the arguments of solutions of equations. The results obtained in this book can be applied to various fields such as neural networks, brain dynamics, mechanical systems, weather phenomena, population dynamics, etc. Without any doubt, bifurcation theory should be further develope...
Bifurcation of discontinuous limit cycles of the Van der Pol equation Akhmet, Marat (2014-01-01) In this paper, we apply the methods of B-equivalence and psi-substitution to prove the existence of discontinuous limit cycle for the Van der Pol equation with impacts on surfaces. The result is extended through the center manifold theory for coupled oscillators. The main novelty of the result is that the surfaces, where the jumps occur, are not flat. Examples and simulations are provided to demonstrate the theoretical results as well as application opportunities. (C) 2013 IMACS. Published by Elsevier B.V. ...
Bifurcation of a non-smooth planar limit cycle from a vertex Akhmet, Marat (2009-12-15) We investigate non-smooth planar systems of differential equations with discontinuous right-hand sides. Discontinuity sets intersect at a vertex, and are of quasilinear nature. By means of the B-equivalence method, which was introduced in [M. Akhmetov, Asymptotic representation of solutions of regularly perturbed systems of differential equations with a nonclassical right-hand side, Ukrainian Math. J. 43 (1991) 1209-1214; M. Akhmetov, On the expansion of solutions to differential equations with discontinuou...
Bifurcation in a 3D Hybrid system Akhmet, Marat; Turan, Mehmet (2010-07-01) In this paper, we study a 3 dimensional Hybrid system which involves a switching mechanism such that at the moment of switching the differential equation that governs the mode] is changing. We first show that there is a center manifold and based on the results in [2] we show that the system under investigation has a Hopf bifurcation. An appropriate example is constructed to illustrate the theory. © Dynamic Publishers, Inc.
Bifurcation Analysis of Wilson-Cowan Model with Singular Impulses Çağ, Sabahattin; Akhmet, Marat (2021-01-01) The paper concerns with Wilson-Cowan neural model. The main novelty of the study is that besides the traditional singularity of the model, we consider singular impulses. A new technique of analysis of the phenomenon is suggested. This allows to consider the existence of solutions of the model and bifurcation in ultimate neural behavior observed through numerical simulations. The bifurcations are reasoned by impulses and singularity in the model and they concern the structure of attractors, which consist of ...

Citation Formats

M. Akhmet and A. Kashkynbayev, “Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities Introduction,” BIFURCATION IN AUTONOMOUS AND NONAUTONOMOUS DIFFERENTIAL EQUATIONS WITH DISCONTINUITIES, pp. 1–9, 2017, Accessed: 00, 2022. [Online]. Available: https://hdl.handle.net/11511/99056.