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Bifurcations in two-dimensional reversible maps

Author

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  • Post, T.
  • Capel, H.W.
  • Quispel, G.R.W.
  • Van Der Weele, J.P.

Abstract

We give a treatment of the non-resonant bifurcations involving asymmetric fixed points with Jacobian J ≠ 1 in reversible mappings of the plane. These bifurcations include the saddle-node bifurcation not in the neighbourhood of a fixed point with J = 1, as well as the so-called transcritical bifurcations and generalized Rimmer bifurcations taking place at a fixed point with Jacobian J = 1. The bifurcations are illustrated by some simple examples of model maps. The Rimmer type of bifurcation, with e.g. a center point with J = 1 changing into a saddle with Jacobian J = 1, an attractor and a repeller, occurs under more general conditions, i.e. also in non-reversible mappings if only a certain order of local reversibility is satisfied. These Rimmer bifurcations are important in connection with the emergence of dissipative features in non-measure-preserving reversible dynamical systems.

Suggested Citation

  • Post, T. & Capel, H.W. & Quispel, G.R.W. & Van Der Weele, J.P., 1990. "Bifurcations in two-dimensional reversible maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 164(3), pages 625-662.
    • Handle: RePEc:eee:phsmap:v:164:y:1990:i:3:p:625-662
    DOI: 10.1016/0378-4371(90)90226-I
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    Cited by:

    1. Lamb, J.S.W. & Roberts, J.A.G. & Capel, H.W., 1993. "Conditions for local (reversing) symmetries in dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 197(3), pages 379-422.
    2. Brown, A., 1991. "Conditions for local reversibility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 173(1), pages 267-280.

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