Admissible symmetry increasing bifurcations

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Abstract
Dynamical systems with symmetry can possess different conjugate attractors simultaneously. If a parameter is varied then these may collide at reflection hyperplanes. Since such a collision typically leads to an increase in the symmetry of the dynamics, this phenomenon is called a symmetry increasing bifurcation. We show that there are restrictions on the type of symmetry increasing bifurcations that can occur. Moreover, we give an algebraic criterion which guarantees that certain collisions of attractors are admissible in the sense that they can be constructed in an appropriate dynamical system. Finally, for two-dimensional mappings we classify all the admissible symmetry increasing bifurcations at reflection hyperplanes.

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