Part of the Applied Mathematical Sciences book series (AMS, volume 74)
Applications of Morse Theory to Second Order Systems
The Liapunov center theorem is the classical result which follows easily from the equivariant Crandall-Rabinowitz bifurcation theorem. Consider the second order autonomous system
and assume that 0 is a solution and that
$$ \ddot u + f(u) = 0 $$
(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.
$$ \beta _1^2 < \beta _2^2 < \ldots < \beta _s^2 $$
KeywordsPeriodic Solution Order System Minimal Period Morse Index Morse Theory
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© Springer Science+Business Media New York 1989