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Ordinary Dynamical Systems

In: Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

Author

Listed:
  • Murat Adıvar

    (Fayetteville State University, Broadwell College of Business and Economics)

  • Youssef N. Raffoul

    (University of Dayton, Department of Mathematics)

Abstract

Summary In this chapter, we analyze boundedness and stability of solutions of ordinary dynamical systems. We will develop general theorems in which Lyapunov functions play an important role. We begin by considering some of the boundedness results of Peterson and Tisdell (J Differ Equ Appl 10(13–15):1295–1306, 2004) and then the results on exponential stability of Peterson and Raffoul (Adv Differ Equ 2005(2):133–144, 2005). We define suitable Lyapunov-type functions on time scales and then formulate appropriate inequalities that guarantee the boundedness of solutions and the exponential stability of the zero solution. In the second part of the chapter, we turn our attention to finite delay dynamical systems, linear and nonlinear, where Lyapunov functionals are the main tools for proving stability, uniform stability, and asymptotic stability of the zero solution. Some of the results of the chapter are new and the rest can be found in Raffoul and Ünal (J Nonlinear Sci Appl 7(6):422–428, 2014) and Ünal and Raffoul (Int J Differ Equ 2013:764389, 10, 2013; Int J Math Comput 26(4):62–73, 2015) and the references therein. For more on the existence and uniqueness of solutions for ordinary dynamic equations on time scales, we refer to Dai and Tisdell (Int J Differ Equ 1(1):1–17, 2006), Erbe and Peterson (Nonlinear Anal 69(7):2303–2317, 2008), and Tisdell (Cubo 8(3):11–24, 2006).

Suggested Citation

  • Murat Adıvar & Youssef N. Raffoul, 2020. "Ordinary Dynamical Systems," Springer Books, in: Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales, chapter 0, pages 51-90, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-42117-5_2
    DOI: 10.1007/978-3-030-42117-5_2
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