Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf’s Type) in the Autonomous Case

  • Gérard Iooss
  • Daniel D. Joseph
Part of the Undergraduate Texts in Mathematics book series (UTM)


In Chapters IX and X we considered the problems of stability and bifurcation of the solution u = 0 of the evolution problem reduced to local form, = f(t,µ,u) = f(f + T, µ,u). In §1.3 we showed how the reduced problem arises from the study of forced T-periodic solutions U(t) = U(t + T) of evolution problems in the form
$${\dot U} = \text{F}(t,\mu ,\text{U}) = \text{F}(t + T,\,\mu ,\,\text{U}) $$
where U = 0 is not a solution because
$$\text{F}(t,\,\mu ,0) = \text{F}(t + T,\,\mu ,\,0)\not \equiv 0 $$

In this type of problem the outside world communicates with the dynamical system governed by (XI.l)! through the imposed data (XI.1)2. The dynamical system sees the outside world as precisely T-periodic and it must adjust its own evolution to fit this fact.


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Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • Gérard Iooss
    • 1
  • Daniel D. Joseph
    • 2
  1. 1.Faculté des Sciences, Institut des Mathématiques et Sciences PhysiquesUniversité des NiceParc Valrose, NiceFrance
  2. 2.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolisUSA

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