Introduction to Stability and Boundedness in Dynamical Systems

Abstract

In this chapter, we provide a brief introduction to time scale calculus and introduce fundamental concepts that we need throughout this book. In Sect. 1.2 based on the work of Peterson and Tisdell (J Differ Equ Appl 10(13–15):1295–1306, 2004), we introduce the concept of Lyapunov functions for ordinary dynamical equations on time scales and prove general theorems in terms of wedges in which we derive specific conditions for stability of the zero solution and the boundedness of all solutions. We proceed to prove new and general theorems in terms of wedges and the existence of Lyapunov functions that satisfy certain conditions and give necessary conditions for stability and instability of the zero solution. In Sect. 1.4 we furnish all the details in proving the existence of the resolvent of Volterra integral dynamic equations by appealing to the results of Adivar and Raffoul (Bull Aust Math Soc 82(1):139–155, 2010). In addition, we will introduce the notion of shift that we make use of in Chaps. 5 and 8, and utilize the notion of resolvent and develop new results concerning Volterra dynamic equations. We end the chapter by introducing the notion of periodicity. In the famous paper (Kuchment, Floquet theory for partial differential equations, Birkhauser Verlag, Basel, 1993) of Raffoul and Kaufmann, the notion of periodicity on time scales was first introduced. Later on, Adivar (Math Slovaca 63(4):817–828, 2013) generalized the definition of periodicity to general time scales by introducing the concept of periodic shift operator. We end the chapter with some interesting and meaningful open problems that should be somewhat challenging.