In this chapter we summarize the basic definitions and tools of analysis of dynamical systems, with particular emphasis on the asymptotic behavior of continuous-time autonomous systems. In particular, the possible structural changes of the asymptotic behavior of the system under parameter variation, called bifurcations, are presented together with their analytical characterization and hints on their numerical analysis. The literature on dynamical systems is huge and we do not attempt to survey it here. Most of the results on bifurcations of continuous-time systems are due to Andronov and Leontovich [see Andronov et al., 1973]. More recent expositions can be found in Guckenheimer & Holmes  and Kuznetsov , while less formal but didactically very effective treatments, rich in interesting examples and applications, are given in Strogatz  and Alligood et al. . Numerical aspects are well described in Allgower & Georg  and in the fundamental papers by Keller  and Doedel et al. [1991a,b], but see also Beyn et al.  and Kuznetsov . This chapter mainly combines material from two previous contributions of the authors, the first part of the book Biosystems and Complexity [Rinaldi, 1993, in Italian] and the Appendix A of a recent book on evolutionary dynamics [Dercole & Rinaldi, 2008].
|Titolo:||Dynamical Systems and Their Bifurcations|
|Data di pubblicazione:||2011|
|Appare nelle tipologie:||02.1 Contributo in Volume|
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