Non-autonomous bifurcation in impulsive systems

This is the first paper which considers non-autonomous bifurcations in impulsive differential equations. Impulsive generalizations of the non-autonomous pitchfork and transcritical bifurcation are discussed. We consider scalar differential equation with fixed moments of impulses. It is illustrated by means of certain systems how the idea of pullback attracting sets remains a fruitful concept in the impulsive systems. Basics of the theory are provided

Finite-time nonautonomous bifurcation in impulsive systems

The purpose of this article is to investigate nonautonomous bifurcation in impulsive differential equations. The impulsive finite-time analogues of transcritical and pitchfork bifurcation are provided

Stability analysis for periodic solutions of fuzzy shunting inhibitory CNNs with delays

https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-019-2321-z#rightslinkWe consider fuzzy shunting inhibitory cellular neural networks (FSICNNs) with time-varying coefficients and constant delays. By virtue of continuation theorem of coincidence degree theory and Cauchy–Schwartz inequality, we prove the existence of periodic solutions for FSICNNs. Furthermore, by employing a suitable Lyapunov functional we establish sufficient criteria which ensure global exponential stability of the periodic solutions. Numerical simulations that support the theoretical discussions are depicted

Perturbed Li-Yorke homoclinic chaos

It is rigorously proved that a Li–Yorke chaotic perturbation of a system with a homoclinic orbit creates chaos along each periodic trajectory. The structure of the chaos is investigated, and the existence of infinitely many almost periodic orbits out of the scrambled sets is revealed. Ott–Grebogi–Yorke and Pyragas control methods are utilized to stabilize almost periodic motions. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical results are depicted

SATURATED AND ASYMMETRIC SATURATED IMPULSIVE CONTROL SYNCHRONIZATION OF COUPLED DELAYED INERTIAL NEURAL NETWORKS WITH TIME-VARYING DELAYS

This paper considers control systems with impulses that are saturated and asymmetrically saturated which are used to examine the synchronization of inertial neural networks (INNs) with time-varying delay and coupling delays. Under the theoretical discussions, mixed delays, such as transmission delay and coupling delay are presented for inertial neural networks. The addressed INNs are transformed into first order differential equations utilizing variable transformation on INNs and then certain adequate conditions are derived for the exponential synchronization of the addressed model by substituting saturation nonlinearity with a dead-zone function. In addition, an asymmetric saturated impulsive control approach is given to realize the exponential synchronization of addressed INNs in the leader-following synchronization pattern. Finally, simulation results are used to validate the theoretical research findings

İmpalsif/hibrid sistemlerde otonom olmayan transkritik ve dirgen çatallanma.

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.Ph.D. - Doctoral Progra