Dynamical Systems and Their Bifurcations

Abstract

In this chapter we summarize the basic definitions and tools of analysis of dynamical systems, with particular\ud emphasis on the asymptotic behavior of continuous-time autonomous systems. In particular, the possible\ud structural changes of the asymptotic behavior of the system under parameter variation, called bifurcations,\ud are presented together with their analytical characterization and hints on their numerical analysis. The\ud literature on dynamical systems is huge and we do not attempt to survey it here. Most of the results on\ud bifurcations of continuous-time systems are due to Andronov and Leontovich [see Andronov et al., 1973].\ud More recent expositions can be found in Guckenheimer & Holmes [1997] and Kuznetsov [2004], while less\ud formal but didactically very effective treatments, rich in interesting examples and applications, are given\ud in Strogatz [1994] and Alligood et al. [1996]. Numerical aspects are well described in Allgower & Georg\ud [1990] and in the fundamental papers by Keller [1977] and Doedel et al. [1991a,b], but see also Beyn et al.\ud [2002] and Kuznetsov [2004]. This chapter mainly combines material from two previous contributions of the\ud authors, the first part of the book Biosystems and Complexity [Rinaldi, 1993, in Italian] and the Appendix A\ud of a recent book on evolutionary dynamics [Dercole & Rinaldi, 2008]

Bifurcations in Ratra–Peebles quintessence models and their extensions

“…We discuss local bifurcations of codimension 1, as they appear in models under consideration. For getting more information about the bifurcation theory we refer readers to [6,8,16,20,22].…”

“…Bifurcation diagrams corresponding to the normal forms(5)(6)(7)(8). Bifurcations take place in the origin O.…”

The effect of seasonal strength and abruptness on predator–prey dynamics

“…Arnol'd tongues are parameter regions where the system displays cyclic behaviours whose periodicities are rational numbers of the period of the forcing signal. They are usually rooted in Neimark-Sacker bifurcations and framed by fold bifurcations (Dercole and Rinaldi, 2011;Taylor et al, 2013a). We have thus encountered some of these parameter regions in our numerical simulations (see periodic windows in Fig.…”

“…the system crosses a (subcritical) period-doubling bifurcation by which the stable cycle of period one year collides with the unstable cycle of period two to become unstable Dercole and Rinaldi, 2011). This gives way to the stable two-year cycle which previously emerged at the fold bifurcation ( Fig.…”

A feedback linearisation algorithm for single-track models with structural stability properties

“…This analysis is conducted using MatCont software tool [14] and its results are summarised in Figure 8. Subcritical Hopf bifurcations [22] are detected at negative values of dl, but these values are not physically meaningful, since they fall outside the range [−l f , l r ] for all considered velocity values. Therefore, the considered equilibrium point is structurally asymptotically stable.…”