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Rajapakse-Smale Equations: Pitchfork Bifurcation Without Symmetry 📂Dynamical Systems

Rajapakse-Smale Equations: Pitchfork Bifurcation Without Symmetry

System 1 2

$$ \begin{align*} \dot{x} =& y^{2} - \left( \varepsilon + 1 \right) y - x \\ \dot{y} =& x^{2} - \left( \varepsilon + 1 \right) x - y \end{align*} $$

Explanation

A pitchfork bifurcation usually appears in systems that have a symmetry, but the Rajapakse-Smale equations above contain squared terms and therefore have no symmetry, and yet a pitchfork bifurcation appears at $\varepsilon = 0$.

In a sense, it is significant as a counterexample to the proposition ‘a pitchfork bifurcation appears only in systems with symmetry’, and whether or not it is due to the influence of this work, these days it is often stated as ‘a pitchfork bifurcation usually appears when there is symmetry’.

  1. Rajapakse, I., & Smale, S. (2016). The pitchfork bifurcation. arXiv preprint arXiv:1609.05996. https://doi.org/10.1142/S0218127417501322 ↩︎

  2. Pujals, E., Shub, M., & Yang, Y. (2020). Stable and non-symmetric pitchfork bifurcations. Science China Mathematics, 63(9), 1837-1852. https://doi.org/10.48550/arXiv.1804.03264 ↩︎