Hopf Bifurcation in Systems with Dihedral Group Symmetry: Broken Reflection Symmetry

In systems with dihedral [Formula: see text]-symmetry, where [Formula: see text] is the group of symmetries (i.e. rotations and reflections, of an N-gon), a symmetry-breaking Hopf bifucation can lead to three types of solutions: a traveling wave solution and two types of standing wave solutions. Generically, when [Formula: see text], these are the only types of solutions that bifurcate from the trivial solution at a double Hopf bifurcation point. In related previous work, it is found that when the rotation symmetry is broken, traveling wave solutions do not exist anymore and that the double Hopf bifurcation point splits to two standard Hopf bifurcations, resulting in two types of standing wave solutions with fixed phase differences. In this work, the consequences of forcefully breaking the reflection symmetry are analytically and computationally studied. It is proven that when reflection symmetry is broken, the double Hopf bifurcation point splits into two standard, back-to-back Hopf bifurcation points, corresponding to two types of traveling wave solutions. Standing wave solutions are found to be no longer possible. However, a secondary Hopf bifurcation is found to occur leading to modulated traveling wave solutions. This torus bifurcation and these modulated wave solutions are not found to occur in the system with perfect symmetry.