Global continuation of forced oscillations of retarded motion equations on manifolds. (English) Zbl 1321.34091
Summary: We investigate \(T\)-periodic parametrized retarded functional motion equations on (possibly) noncompact manifolds; that is, constrained second order retarded functional differential equations. For such equations we prove a global continuation result for \(T\)-periodic solutions. The approach is topological and is based on the degree theory for tangent vector fields as well as on the fixed point index theory.{
}Our main theorem is a generalization to the case of retarded equations of an analogous result obtained by the last two authors for second order differential equations on manifolds. As corollaries we derive a Rabinowitz-type global bifurcation result and a Mawhin-type continuation principle. Finally, we deduce the existence of forced oscillations for the retarded spherical pendulum under general assumptions.
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 70K40 | Forced motions for nonlinear problems in mechanics |
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 47N20 | Applications of operator theory to differential and integral equations |
| 37C60 | Nonautonomous smooth dynamical systems |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
Keywords:
retarded functional differential equations; global bifurcation; periodic solutions; fixed point index theory; forced motion on manifoldsReferences:
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