Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions

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


In Chapter IX we determined the conditions under which subharmonic solutions, nT-periodic solutions with integers n ≥1, could bifurcate from forced T-periodic solutions. That is to say, we looked for the conditions under which nonautonomous, T-periodic differential equations give rise to subharmonic solutions when the Floquet exponents at criticality lie in the set of rational points (ω 0 = 2πm/nT,0m/n <1) or, equivalently, when the Floquet multipliers at criticality are the nth roots of unity, \(\lambda _{0}^{n} = {{({{e}^{{i{{\omega }_{0}}T}}})}^{n}} = 1\). We found that unless certain very special (weak resonance) conditions were satisfied such subharmonic solutions could bifurcate only when n = 1, 2, 3, 4. (The case n = 4 is special in that there are in general two possibilities depend­ing on the parameters; see §IX.15.) So we now confront the problem of finding out what happens for all the values of ω 0 , 0 ≤ ω 0 < 27π/T such that
$$\frac{{{{\omega }_{0}}T}}{{2\pi }} \ne 0,\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{3}{4}.$$


Periodic Solution Closed Curve Rotation Number Invariant Torus Amplitude Equation 
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© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Gérard Iooss
    • 1
  • Daniel D. Joseph
    • 2
  1. 1.Institut Non Lineaire, de NiceValbonneFrance
  2. 2.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolisUSA

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