Archive for February, 2010

The fold bifurcation

February 24, 2010

In this blog I have already discussed a number of aspects of bifurcation theory for dynamical systems. In a previous post I mentioned that there are two generic ways in which a stationary solution can lose hyperbolicity. There I discussed one of these, the Hopf bifurcation. Here I want to discuss the other, the fold bifurcation. I follow the treatment in the book of Kuznetsov. This bifurcation can occur in dimension one and more general situations can be reduced to the one-dimensional case by centre manifold techniques. For this reason I only discuss the case of a one-dimensional dynamical system here. An example of a fold bifurcation is given by the equation \dot x=\alpha+x^2, where the dot stands for a derivative with respect to time. In fact it can be shown that any fold bifurcation in one dimension is topologically equivalent to this one. (Here I am implicitly allowing the freedom to reverse the direction of time.) The bifurcation occurs at \alpha=0 and the stationary point x=0 is non-hyperbolic. There are solutions which approach it and solutions which go away from it. For \alpha>0 there are no stationary points. For \alpha<0 there are two stationary points, one stable and one unstable. This can also be described by saying that as \alpha increases through negative values the two stationary points approach each other, collide and annihilate.

What are the conditions which characterize a fold bifurcation in one dimension? Suppose without loss of generality that the bifurcation occurs at x=0 for the parameter value \alpha=0. Then f(0,0) and the derivative f_x(0,0) vanish. For a fold bifurcation we need the genericity conditions f_{xx}(0,0)\ne 0 and f_\alpha (0,0)\ne 0. The reduction can be done in two steps. In the first the system is reduced to the form \dot x=\alpha+x^2+O(x^3) and in the second the remainder term is eliminated. The proof is elementary, using nothing more complicated than the implicit function theorem. As already mentioned, centre manifold reduction can be used to extend the analysis of the fold bifurcation to arbitrary dimensions. It remains, however, to have a direct formulation of the genericity conditions in the higher-dimensional case . This can for instance be found as Theorem 3.4.1 in the book of Guckenheimer and Holmes (Nonlinear oscillations, dynamical systems and bifurcations of vector fields).


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