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Iterates of maps with symmetry. (English) Zbl 0694.58030

The elementary aspects of bifurcation of fixed points, period doubling, and Hopf bifurcation for iterates of equivariant mappings are discussed. An algebraic formulation of the hypotheses of Ruelle’s theorem on Hopf bifurcation in the presence of symmetry [D. Ruelle, Arch. Ration. Mech. Anal. 51, 136-152 (1973; Zbl 0259.58009)] is given.

In the last sections it is shown that Hopf bifurcation from standing waves in a system of ordinary differential equations with O(2) symmetry can lead directly to motion on an invariant 3-torus.

Reviewer: Y.Asoo
37G99Local and nonlocal bifurcation theory
37C55Periodic and quasiperiodic flows and diffeomorphisms
37-99Dynamic systems and ergodic theory (MSC2000)