GENERALIZED HOPF BIFURCATION
1981; Elsevier BV; Linguagem: Inglês
10.1016/b978-0-12-186280-0.50008-6
AutoresStephen R. Bernfeld, L. Salvadori,
Tópico(s)Advanced Differential Equations and Dynamical Systems
ResumoThis chapter focuses on generalized Hopf bifurcation. Chafee has considered the problem of determining the number of non-zero periodic orbits of the equation x = f(x) lying near the origin and having period T close to 2π for each f close to f0. This chapter considers the case n = 2 and shows that two properties proved by Chafee occur if and only if the origin of x = f0(x) is either (2k + 1) asymptotically stable or (2k + 1) completely unstable; that is, the origin is asymptotically stable in the future or in the past and this property is recognizable by the terms of f0 of degree ≤ 2k + 1. This result gives a complete answer to the problem because there is a constructive procedure to determine k. The chapter highlights this as the classical method of Poincaré.
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