Nongeneric bifurcations near a nontransversal heterodimensional cycle (Q1696643)

The authors consider the following autonomous differential system \[ \dot{z}= f(z) + g(z,\mu)\tag{1} \] and its unperturbed system \[ \dot{z}= f(z), \tag{2} \] where \(z \in \mathbb{R}^3\), \(\mu \in \mathbb{R}^l\), \(l \geq 3\), \(0 < | \mu | \ll 1\), \(g(z,0)=0\), \(f(z) \) is \(C^r\) with respect to the phase variable \(z\), \(g(z,\mu)\) is \(C^r\) with respect to the phase variable \(z\) and the parameter \(\mu\) and \(r \geq 4\). \newline The authors assume that system (2) has a heteroclinic cycle connecting two hyperbolic equilibria which is heterodimensional, that is, the equilibrium points in the cycle do not have the same index (dimension of the stable manifold). The authors also assume that the two heteroclinic orbits of the heterodimensional cycle are both nontransversal and with an orbit flip. In this strong degeneracy condition, the authors provide some rich bifurcation phenomena. The techniques that they use in order to tackle with the Poincaré return map are Shilnikov coordinates and a local moving frame. \newline As an example, the authors consider bifurcations from the following system \[ \begin{aligned} \dot{z}_1 & = -(z_1-1)(z_1+1)+3(z_1^2+z_2^2-1), \\ \dot{z}_2 & = -z_1z_2, \\ \dot{z}_3 & = (7-8z_1)z_3/3, \end{aligned} \] which has an heteroclinic cycle formed by \(\Gamma_1 \cup \Gamma_2\) with \[ \Gamma_1= \left\{ z \mid z_1^2+z_2^2=1, z_3=0, z_2 \geq 0\right\}, \] \[ \Gamma_2 = \left\{ z \mid z_2=z_3=0, z_1 \in (-1,1) \right\}. \]