Singularity Theory and Equivariant Symplectic Maps
by Thomas J Bridges and Jacques E Furter
Table of Contents
- 1. Introduction
- 2. Generic bifurcation of periodic points
- 2.1 Lagrangian variational formulation
- 2.2 Linearization and unfolding
- 2.3 Symmetries
- 2.4 Normal form for bifurcating period-q points
- 2.5 Reduced stability of bifurcating periodic points
- 2.6 4D-symplectic maps and the collision singularity
- 3. Singularity theory for equivariant gradient bifurcation problems
- 3.1 Contact equivalence and gradient maps, 35}
- 3.2 Fundamental results, 38}
- 3.3 Potentials and paths, 40}
- 3.4 Equivalence for paths, 43}
- 3.5 Proofs
- 4. Classification of Z_q-equivariant gradient bifurcation problems
- 4.1 {\cal A}^{Z_q}-classification of potentials
- 4.2 Classification of $\openZ_q$-equivariant bifurcation problems
- 4.3 Bifurcation diagrams for the unfolding (4.9)
- 5. Period-3 points of the generalized standard map
- 5.1 Computations of the bifurcation equation
- 5.2 Analysis of the bifurcation equations
- 6. Classification of D_q-equivariant gradient maps on R^2
- 6.1 D_q-normal forms when q\neq4
- 6.2 D_4-invariant potentials with a distinguished parameter path
- 7. Reversibility and degenerate bifurcation of period-q points
of multiparameter maps
- 7.1 Period-3 points with reversibility in multiparameter maps
- 7.2 Period-4 points with reversibility in multiparameter maps
- 7.3 Generic period-5 points in the generalized standard map
- 8. Periodic points of equivariant symplectic maps
- 8.1 Subharmonic bifurcation in equivariant symplectic maps
- 8.2 Subharmonic bifurcation when \Sigma acts absolutely
irreducibly on R^n
- 8.3 O(2)-equivariant symplectic maps
- 8.4 Parametrically forced spherical pendulum
- 8.5 Reduction to the orbit space
- 8.6 Remarks on linear stability for equivariant maps
- 9. Collision of multipliers at rational points for symplectic maps
- 9.1 Generic theory for nonlinear rational collision
- 9.2 Collision at third root of unity: \theta=2\pi/3
- 9.3 Collision of multipliers at +/- i
- 9.4 Collision at rational points with q\geq5
- 9.5 Reduced instability for the bifurcating period-$q$ points
- 9.6 Reduced stability for bifurcating period-4 points
- 9.7 Remarks on the collision at irrational points
- 10. Equivariant maps and the collision of multipliers
- 10.1 Reversible symplectic maps on R^4
- 10.2 Symplectic maps on R^4 with spatial symmetry