Symmetries of functions on networks and of mappings on Minkowski spaces
Planar phase portraits and generic bifurcations of reversible vector fields
On limit cycles in piecewise linear vector fields with algebraic discontinuity var...
| Full text | |
| Author(s): |
Buzzi, Claudio Aguinaldo
;
Lamb, Jeroen S. W.
Total Authors: 2
|
| Document type: | Journal article |
| Source: | Archive for Rational Mechanics and Analysis; v. 175, n. 1, p. 39-84, Jan. 2005. |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics |
| Abstract | |
In this paper we study codimension-one Hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields. Such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point. The eigenvalue movements near such a degeneracy typically follow one of three scenarios: splitting (from two pairs of imaginary eigenvalues to a quadruplet on the complex plane), passing (on the imaginary axis), or crossing (a quadruplet crossing the imaginary axis). We give a complete description of the behaviour of reversible periodic orbits in the vicinity of such a bifurcation point. For non-reversible periodic solutions, in the case of Hopf bifurcation with crossing eigenvalues, we obtain a generalization of the equivariant Hopf Theorem. (AU) | |