Bifurcation theory

In most examples of the preceding chapters, the equations which we have studied are containing parameters. For different values of these parameters, the behaviour of the solutions can be qualitatively very different. Consider for instance equation 7.12 in example 7.3 (population dynamics). When passing certain critical values of the parameters, a saddle changes into a stable node. The van der Polequation which we have used many times, for instance in example 5.1, illustrates another phenomenon. If the parameter μ in this equation equals zero, all solutions are periodic, the origin of the phase-plane is a centre point. If the parameter is positive with 0 < μ < 1, the origin is an unstable focus and there exists an asymptotically stable periodic solution, corresponding with a limit cycle around the origin. Another important illustration of the part played by parameters is the forced Duffing-equation in section 10.3 and example 11.8.