Bifurcations of Cycles in Systems of Differential Equations with a Finite Symmetry Group – I

Abstract

One-parameter bifurcations of periodic solutions of differential equations in ℝⁿ with a finite symmetry group Γ are studied. The following three types of periodic solutions x(t) with the symmetry group H $\subseteq$ Γ are considered separately.

• F-cycles: H consists of transformations that do not change the periodic solution, h(x(t)) ≡ x(t);

• S-cycles: H consists of transformations that shift the phase of the solution,

$$h\left( {x\left( t \right)} \right) \equiv x\left( {t + {\tau }\left( h \right)} \right)\quad \left( {{\tau }\left( h \right) \ne 0\,{if}\,h \ne e} \right)$$

• FS-cycles: H consists of transformations of both F and S types. In the present paper bifurcations of F-cycles at double real multipliers are studied in detail; and all codimension one bifurcations of S-cycles are described. In the next paper a more complicated case of a double pair of complex multipliers for F-cycles is considered and bifurcations of FS-cycles are shortly discussed.