Nonautonomous transcritical and pitchfork bifurcations in impulsive/hybrid systems

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2016

Tezin Dili: İngilizce

Öğrenci: ARDAK KASHKYNBAYEV

Danışman: Marat Akhmet

Özet:

The main purpose of this thesis is to study nonautonomous transcritical and pitchfork bifurcations in continuous and discontinuous dynamical systems. Two classes of discontinuity, impulsive differential equations and differential equations with an alternating piecewise constant argument of generalized type, are addressed. Moreover, the Bernoulli equation in impulsive as well as hybrid systems is introduced. For the former one, the corresponding jump equation is chosen so that after Bernoulli transformation the original system is reduced to a linear non-homogeneous system. For the latter, this is achieved by constructing a special type of transformation. Sufficient conditions are obtained for the existence of bounded solutions of the Bernoulli equations. Next, it is shown that different types of convergence analysis, such as pullback and forward remain as a fruitful idea in impulsive and hybrid systems. Furthermore, bifurcation scenarios are obtained depending on the sign of Lyapunov exponent by using these convergence analysis. Attraction and transition approaches are used to study bifurcation patterns in impulsive systems which cannot be solved explicitly. In other words, qualitative change in the attractor/reppeller pair is observed as a parameter goes though bifurcation value. Besides, finite-time analogues of nonautonomous transcritical and pitchfork bifurcations are investigated in impulsive systems. Illustrative examples with numerical simulations are provided to demonstrate the theoretical results.