Introduction to Bifurcation and Stability
On several occasions when working out examples in the earlier chapters, we came across problems that had more than one solution. Such situations are not uncommon when studying nonlinear problems, and we are now going to examine them in detail. The first s
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Introduction to Bifurcation and Stability
6.1 Introduction On several occasions when working out examples in the earlier chapters, we came across problems that had more than one solution. Such situations are not uncommon when studying nonlinear problems, and we are now going to examine them in detail. The first step is to determine when multiple solutions appear. Once the solutions are found, the next step is to determine if they are stable. Thus, we will focus our attention on what is known as linear stability theory. In terms of perturbation methods, almost all the tools we need were developed in earlier chapters. For example, the analysis of steady-state bifurcation uses only regular expansions (Chap. 1), and the stability arguments will use regular and multiple-scale expansions (Chap. 3). On certain examples, such as when studying relaxation dynamics, we will use matched asymptotic expansions (Chap. 2). It is worth making a comment or two about the approach used in this chapter compared to most other textbooks on this subject. The conventional method is to first express the problem as a dynamical system and then prove results related to stability using the eigenvalues of a Jacobian matrix. The approach here concentrates more on the construction of an approximation of the solution, and stability considerations are a consequence of this analysis. Also, the objective is to develop methods that are applicable to a wide variety of problems, including nonlinear partial differential equations and integral equations. This does not mean that tidy little formulas related to the stability of particular problems are avoided, but they will not play a prominent role in the presentation.
M.H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics 20, DOI 10.1007/978-1-4614-5477-9 6, © Springer Science+Business Media New York 2013
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6 Introduction to Bifurcation and Stability
6.2 Introductory Example Before presenting the general formulation, it is worth while to briefly examine a relatively simple example. One that has a number of interesting characteristics is the nonlinear oscillator equation y + 2βy − λy + y 3 = 0 for < t.
(6.1)
This is a Duffing-type equation, and we have come across these cubically nonlinear oscillators before. We begin by investigating the steady states. From (6.1) we see that they satisfy λy − y 3 = 0.
(6.2)
The number of solutions of this equation depends on √ the parameter λ. That is, y = 0 is a solution for all values of λ, and y = ± λ are solutions if λ ≥ 0. These steady states are shown in Fig. 6.1. Given the existence of multiple steady states, it is natural to ask if any of them are achievable. In other words, will the solution of (6.1) approach one of the three steady states as t → ∞? A partial answer to this question can be inferred from Fig. 6.2, where the numerical solution of (6.1) is shown for different initial values. When starting at (y, y ) = (0, 1), the solution spirals into the steady state ys = 1, while the initial point (y, y ) = (0, 2) results
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