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Insight into Phenomena of Symmetry Breaking Bifurcation

We show that symmetry-breaking (SB) bifurcation is just a transition of different forms of symmetry, while still preserving system's symmetry SB bifurcation always associates with a periodic saddle-node bifurcation, identifiable by a zero maximum of the top Lyapunov exponent of the system. In addition, we show a significant phase portrait of a newly born periodic saddle and its stable and unstable invariant manifolds, together with their neighbouring flow pattern of Poincare mapping points just after the periodic saddle-node bifurcation, thus gaining an insight into the mechanism of SB bifurcation.