Elementary Stability and Bifurcation Theory pp 208-255 | Cite as

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

Chapter

## Abstract

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*,*0*≤*m/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$$\frac{{{{\omega }_{0}}T}}{{2\pi }} \ne 0,\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{3}{4}.$$

## Keywords

Periodic Solution Closed Curve Rotation Number Invariant Torus Amplitude Equation
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## Notes

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© Springer-Verlag Berlin Heidelberg 1990