Ardak Kashkynbayev

Hopf Bifurcation in Filippov Systems

We investigate nonsmooth planar systems of differential equations with discontinuous right-hand side. Discontinuity sets intersect at a vertex and are of the quasilinear nature. By means of the B-equivalence method, which was introduced in Akhmet and Kashkynbayev (Proceedings of 10th QTDE, Electronic Journal of Qualitative Theory of Differential Equations, 2016, [46], Akhmetov and Perestyuk (Ukr Math J 43:1209–1214, 1991, [54], Akhmet and Kashkynbayev (Nonautonomous Bifurcation in Hybrid Systems, submitted [55] (see also Akhmet, Principles of Discontinuous Dynamical Systems, 2010, [1], Akhmet, Nonlinear Anal: TMA 60:163–178, 2005, [6]), these systems are reduced to impulsive differential equations. Sufficient conditions are established for the existence of foci and centers both in the noncritical and critical cases. Hopf bifurcation is considered from a vertex, which unites several curves, in the critical case. An appropriate example is provided to illustrate the results.

Hopf Bifurcation in Impulsive Systems

This chapter is organized in the following manner. In the first section, we give the description of the systems under consideration and prove the theorem of existence of foci and centers of the nonperturbed system. The main subject of Sect. 2.1.2 is the foci of the perturbed equation. The noncritical case is considered. In Sect. 2.1.3, the problem of distinguishing between the center and the focus is solved. Bifurcation of a periodic solution is investigated in Sect. 2.1.4. The last section consists of examples illustrating the bifurcation theorem.

Bifurcation in autonomous and nonautonomous differential equations with discontinuities

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Nonautonomous Bifurcations in Nonsolvable Impulsive Systems

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Nonautonomous Bifurcations in Bernoulli Differential Equations with Piecewise Constant Argument of Generalized Type

In this section, we consider impulsive analogues of nonautonomous transcritical and pitchfork bifurcations in the systems which cannot be solved explicitly. We extend the theorem on asymptotic properties of the quasilinear impulsive systems which was considered by Samoilenko and Perestyuk (Impulsive Differential Equations, 1995, [216]) to the entire time.

Nonautonomous Bifurcation in Impulsive Bernoulli Equations

In this chapter, we study existence of the bounded solutions and asymptotic behavior of the Bernoulli equations with piecewise constant argument.

Nonautonomous Bifurcations in Nonlinear Impulsive Systems

In this chapter, we discuss impulsive generalizations of the nonautonomous pitchfork and transcritical bifurcations. Scalar differential equations with fixed moments of impulses are considered. By means of certain systems, we show that how the idea of pullback attracting sets remains a fruitful concept in the impulsive systems. Basics of the theory are provided.