Differential Equations
Differential Equations
4th Edition
ISBN: 9780495561989
Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall
Publisher: Cengage Learning
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Chapter 5.1, Problem 18E

If a nonlinear system depends on a parameter, then the equilibrium points can change as the parameter varies. In other words, as the parameter changes, a bifurcation can occur. Consider the one-parameter system family of systems

d x d t = x 2 a d y d t = y ( x 2 + 1 )

where a is the parameter.

(a) Show that the system has no equilibrium points if a < 0

(b) Show that the system has two equilibrium points if a > 0.

(c) Show that the system has exactly one equilibrium point if a = 0

(d) Find the linearization of the equilibrium point for a = 0 and compute the eigenvalues of this linear system.

Remark: The system changes from having no equilibrium points to having two equilibrium points as the parameter a is increased through a = 0. We say that the system has a bifurcation at a = 0 , and that a = 0 is a bifurcation value of the parameter.

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