arXiv subjects

Murat Adivar

Publications and source records attributed to Murat Adivar.

4 recordsLinked to original sources

arXiv2015

Floquet theory based on new periodicity concept for hybrid systems involving $q$-difference equations

Using the new periodicity concept based on shifts, we construct a unified Floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous, discrete or hybrid structure. New periodicity concept based on shifts enables the construction of Floquet theory on hybrid domains that are not necessarily additive periodic. This makes periodicity and stability analysis of hybrid periodic systems possible on non-additive domains. In particular, this new approach can be useful to know more about Floquet theory for linear $q$-difference systems defined on $\overline{q^{\mathbb{Z}}}:=\{q^{n}% :n\in\mathbb{Z}\} \cup \{0\}$ where $q>1$. By constructing the solution of matrix exponential equation we establish a canonical Floquet decomposition theorem. Determining the relation between Floquet multipliers and Floquet exponents, we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts. Finally, we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts.

math.DS↗

arXiv2014

Existence of periodic solutions in shifts $δ_{\pm}$ for neutral nonlinear dynamic systems

In this study, we focus on the existence of a periodic solution for the neutral nonlinear dynamic systems with delay% \[ x^Δ(t)=A(t)x(t)+Q^Δ\left(t,x\left(δ_{-}(s,t)\right) \right) +G\left(t,x(t),x\left(δ_{-}(s,t)\right) \right) . \] We utilize the new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement $t\pm T\in\mathbb{T}$ for all $t\in\mathbb{T}$ and for a fixed $T>0,$ may not hold. More, importantly, the new concept will easily handle time scales that are not periodic in the conventional way such as; $\overline{q^{\mathbb{Z}}}$ and $\cup_{k=1}^{\infty}\left[ 3^{\pm k},2.3^{\pm k}\right] \cup\left\{0\right\} .$ Hence, we develop a tool that enables the investigation of periodic solutions of $q$-difference systems. Since we are dealing with systems, in order to convert our equation to an integral systems, we resort to the transition matrix of the homogeneous Floquet system $y^Δ(t)=A(t)y(t)$ and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point.

math.CA↗

arXiv2011

Shift operators and stability in delayed dynamic equations

In this paper, we use what we call the shift operator so that general delay dynamic equations of the form \[ x^Δ(t)=a(t)x(t)+b(t)x(δ_{-}(h,t))δ_{-}^Δ% (h,t),\ \ \ t\in\lbrack t_{0},\infty)_{\mathbb{T}}% \] can be analyzed with respect to stability and existence of solutions. By means of the shift operators we define a general delay function opening an avenue for the construction of Lyapunov functional on time scales. Thus, we use the Lyapunov's direct method to obtain inequalities that lead to stability and instability. Therefore, we extend and unify stability analysis of delay differential, delay difference, delay $h-$difference, and delay $q-$difference equations which are the most important particular cases of our delay dynamic equation. \textbf{Keywords}: Delay dynamic equation, instability, shift operators, stability, time scales.

math.CA↗

arXiv2009

Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors

In this paper, a quadratic pencil of Schrödinger type difference operator $L_λ$ is taken under investigation to give a general perspective on the spectral analysis of non-selfadjoint difference equations of second order. Introducing Jost-type solutions, structural and quantitative properties of spectrum of the operator $L_λ$ are analyzed and hence, a discrete analog of the theory in Degasperis, (\emph{J.Math.Phys}. 11: 551--567, 1970) and Bairamov et. al, (\emph{Quaest. Math.} 26: 15--30, 2003) is developed. In addition, several analogies are established between difference and $q$-difference cases. Finally, the principal vectors of $L_λ$ are introduced to lay a groundwork for the spectral expansion. Mathematics Subject Classification (2000): 39A10, 39A12, 39A13

math.SP↗