Applications of Morse Theory to Second Order Systems

Abstract

The Liapunov center theorem is the classical result which follows easily from the equivariant Crandall-Rabinowitz bifurcation theorem. Consider the second order autonomous system $$ \ddot u + f(u) = 0 $$ and assume that 0 is a solution and that $$ \beta _1^2 < \beta _2^2 < \ldots < \beta _s^2 $$ (β r ≥ 0,0 ≤ r ≤ s) are the non-negative eigenvalues of f’(0). The Liapunov theorem insures that if the geometric multiplicity of \( \beta _i^2 \) is one and if β r /β i ∉ N for r ≠ i, then this system has a family of periodic solutions with minimal period tending to 2Π/β i and with amplitude tending to 0.KeywordsPeriodic SolutionOrder SystemMinimal PeriodMorse IndexMorse TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.