Abstract
We study bifurcation of 2πq-periodic solutions in one-parameter families of 2π-periodic time-reversible systems. We obtain generically satisfied conditions which imply the bifurcation of 2q branches of such subharmonic solutions. Whenq≧5 the solutions alongq of these branches are unstable, while the solutions along the otherq branches are stable in a weak sense. Special results hold forq=3 andq=4. We also describe a situation in which there is secundary bifurcation and give a brief discussion of what happens under a perturbation which breaks the time-reversibility.
Zusammenfassung
Wir untersuchen die Verzweigung von 2πq-periodischen Lösungen in einer einparametrigen Familie von 2π-periodischen reversiblen Differentialgleichungssystemen. Wir erhalten Bedingungen, welche die Verzweigung von 2q solcher subharmonischer Lösungen garantieren und die generisch erfüllt sind. Fürq≧5 sindq Lösungen instabil undq Lösungen (linear) stabil. Spezielle Resultate gelten fürq=3 undq=4. Wir untersuchen auch einen Fall in dem sekundäre Verzweigung eintritt und diskutieren kurz die Wirkung einer Störung, die die Reversibilität zerstört.
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Vanderbauwhede, A. Bifurcation of subharmonic solutions in time-reversible systems. Z. angew. Math. Phys. 37, 455–478 (1986). https://doi.org/10.1007/BF00945425
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DOI: https://doi.org/10.1007/BF00945425