In this chapter we present a number of typical bifurcations that occur for reversible systems in dimensions 2, 3, and 4. We focus on bifurcations of codimension 1, which involve only one bifurcation parameter. Reversible systems are first order systems in which the vector field anticommutes with a linear symmetry. We already met reversible systems in Section 2.3.3 of Chapter 2, and in Section 3.3.2 of Chapter 3, where we have seen that the reversibility property is preserved by both the center manifold reduction and the normal form transformation (Theorem 3.15 in Chapter 2 and Theorem 3.4 in Chapter 3, respectively). We discuss in this chapter reversible systems for which the linearization at the origin has a spectrum lying on the imaginary axis, including in this way the reduced systems provided by the center manifold theorem. In all cases, the analysis relies upon the normal form transformation for reversible systems in Theorem 3.4 in Chapter 3.
KeywordsPeriodic Orbit Normal Form Phase Portrait Unstable Manifold Homoclinic Orbit
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