Floquet theory and stability of nonlinear integro-differential equations

Abstract

One of the classical topics in the qualitative theory of differential equations is the Floquet theory. It provides a means to represent solutions and helps in particular for stability analysis. In this paper first we shall study Floquet theory for integro-differential equations (IDE), and then employ it to address stability problems for linear and nonlinear equations.

Multiple stability and instability of Cohen–Grossberg neural network with unbounded time-varying delays

“…In some practical applications and hardware implementations of artificial neural networks, time delays are inevitable due to the finite switching speed of the amplifiers and the interneurons conduction distances, and they are even time-varying and unbounded in some cases such as the memory activation function of the human brain neural network model. Therefore, it is more suitable to introduce unbounded time-varying delays to the neural network, especially to Cohen-Grossberg models, and some results have been reported recently, for example, [19][20][21][22][23][24][25][26][27][28].…”

Floquet theory for integral and integro-differential equations

“…This is the case because y 1 ðt þ TÞ ¼ expðfðt þ TÞÞz 1 ðt þ TÞ ¼ expðfTÞ expðftÞz 1 ðtÞ ¼ expðfTÞy 1 ðtÞ. Conversely, if a solution y 1 satisfies y 1 ðt þ TÞ ¼ expðfTÞy 1 ðtÞ, then y 1 has the form y 1 ðtÞ ¼ expðftÞz 1 ðtÞ where z 1 is periodic with period T. Indeed, in that case, if we define z 1 ðtÞ :¼ expðÀftÞy 1 ðtÞ, then z 1 ðt þ TÞ ¼ expðÀfðt þ TÞÞ y 1 ðt þ TÞ ¼ expðÀftÞ expðÀfTÞ expðfTÞy 1 ðtÞ ¼ expðÀftÞy 1 …”

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

“…An extension of the Floquet theory to the systems with memory has been studied in [11]. In [7], Floquet theory has been employed for stability analysis of nonlinear integro-differential equations. Moreover, a generalization of Floquet theory in continuous case is studied in [28].…”