Bursts in oscillatory systems with broken \(D_4\) symmetry (Q1964025)

A new mechanism responsible for generating regular and irregular bursts of large dynamic range near onset of an oscillatory instability is identified. The bursts are shown to present in the system of normal form equations describing a Hopf bifurcation with broken \(D_4\) symmetry: \[ \begin{aligned} \dot z_{+}&=[\lambda +\Delta\lambda +\roman{i}(\omega +\Delta\omega)]z_{+}+ A(|z_{+}|^2+|z_{-}|^2)z_{+}+B|z_{+}|^2z_{+}+C\overline{z}_{+}z_{-}^2,\\ \dot z_{-}&=[\lambda -\Delta\lambda +\roman{i}(\omega -\Delta\omega)]z_{-}+ A(|z_{+}|^2+|z_{-}|^2)z_{-}+B|z_{-}|^2z_{-}+C\overline{z}_{-}z_{+}^2. \end{aligned}\tag{1} \] Equations (1) describe the interaction of two nearly degenerate modes of opposite parity with the complex amplitudes \(z_{\pm}\); the coefficients \(A\), \(B\), and \(C\) are complex; the degeneracy is broken by the parameters \(\Delta\lambda\) and \(\Delta\omega\). The bursts in Equations (1) are the result of codimension one heteroclinic cycles involving infinite amplitude states created when the square symmetry is broken (\(\Delta\lambda\neq 0\), \(\Delta\omega\neq 0\)). All possible cycles of this type are identified, the resulting bursts are described, and the associated dynamics are illustrated for several particular cases. It is shown that global connections involving finite amplitude states are also present in the system. The sequence of bifurcations that result is described in several cases. The robustness of the obtained results as the magnitude of the symmetry-breaking terms increases and the effects of higher-order terms are studied. An account of the results obtained in the paper is available in \textit{J.~Moehlis} and \textit{E.~Knobloch} [Forced symmetry breaking as a mechanism for bursting, Phys. Lett. 80, 5329-5332 (1998), not yet available for Zbl].