Bifurcations Revisited

This chapter extends our earlier work on bifurcations (Chapter 3), As we move up from one-dimensional to two-dimensional systems, we still find that fixed points can be created or destroyed or destabilized as parameters are varied—but now the same is true of closed orbits as well. Thus we can begin to describe the ways in which oscillations can be turned on or off. 

In this broader context, what exactly do we mean by a bifurcation The usual definition involves the concept of topological equivalence (Section 6.3) if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred. Examples include changes in the number or stability of fixed points, closed orbits, or saddle connections as a parameter is varied. 

This chapter is organized as follows for each bifurcation, we start with a simple prototypical example, and then graduate to more challenging examples, either briefly or in separate sections. Models of genetic switches, chemical oscillators, driven pendula and Josephson junctions are used to illustrate the theory. 

The bifurcations of fixed points discussed in Chapter 3 have analogs in two dimensions (and indeed, in all dimensions). Yet it turns out that nothing really new happens when more dimensions are added—all the action is confined to a one-dimensional subspace along which the bifurcations occur, while in the extra dimensions the flow is either simple attraction or repulsion from that subspace, as weTl see below. 

The saddle-node bifurcation is the basic mechanism for the creation and destruction of fixed points. Here s the prototypical example in two dimensions  

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