Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 5 - Bifurcations

Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 5 - Bifurcations

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CHAPTER 5 Bifurcations I n practice, we often deal with ode s which contain parameters (unspecified constants) whose values can profoundly influence the dynamical behavior of the system. For example, suppose we model population changes for a species. The birth and death rates of the species would be examples of parameters whose values would substantially impact that behavior of solutions of the underlying differential equation. Example 5 . 0 . 2 . Consider the ode d x d t = μ x , where μ is a parameter. The solution of this equation is x ( t ) = x 0 e μ t , where x 0 = x ( 0 ) . Notice that if μ > 0 , the solutions exhibit exponential growth, whereas if μ < 0 we observe exponential decay. If μ = 0 , solutions are constant. The critical value μ = 0 marks the “boundary” between two very different types of dynamical behavior. Definition 5 . 0 . 3 . Consider a system of ode s of the form x ( t ) = f ( x ; μ ) , where μ is a parameter. A bifurcation is any major qualitative change in the dynamical behavior of the system in response to varying the parameter μ . In the previous example, we would say that a bifurcation occurs at μ = 0 , because the equilibrium x = 0 changes from stable ( μ < 0 ) to unstable ( μ > 0). 5.1. Three Basic Bifurcations There are many ways that the qualitative behavior of a system can be drastically altered in response to changes in parameters. Equilibria and/or periodic solu- tions can be created or destroyed, or they can change their stability. In what follows, we will survey several common types of bifurcations. For simplicity, we 140
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bifurcations 141 will deal primarily with ode s with only one dependent variable. An excellent reference for this material is provided by Strogatz [ 11 ]. Saddle-node bifurcations. First, we discuss one common mechanism for the birth or destruction of an equilibrium solution. The canonical example to keep in mind is given by the ode d x d t = μ + x 2 . ( 5 . 1 ) Suppose μ < 0 . The equilibria satisfy x 2 + μ = 0 , which implies that x = ± - μ are the two equilibria of the system. To determine whether they are stable, we compute that the “Jacobian” of f ( x ) = x 2 + μ is simply f ( x ) = 2 x . By calculating f ( - μ ) = 2 - μ > 0 , we find that - μ is an unstable equilibrium. Similarly, since f ( - - μ ) = - 2 - μ < 0 , it follows that - - μ is a stable equilibrium. Next, suppose that μ = 0 . There is only one equilibrium, namely x = 0 . Although it is non-hyperbolic, it is easy to check (via separation of variables) that this equilibrium is unstable. Finally, suppose μ > 0 . Then x 2 + μ > 0 for all x , implying that there are no equilibria. In summary, as μ increases from negative to positive, two equilibria (one stable and one unstable) merge and annihilate each other, leaving no equilibria if μ > 0 . Clearly μ = 0 is a bifurcation point, and this type of bifurcation is called a saddle-node bifurcation . A minor adjustment of Equation ( 5 . 1 ) reveals that saddle-node bifurcations can also create new equilibria as the parameter μ increases. The equation x = - μ + x 2 experiences a saddle-node bifurcation at μ = 0 , creating two equilibria for μ > 0 .
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