Coalescence of Attractors in Dynamical Systems

We study a class of bifurcations generically occurring in dynamical systems with Z₂ symmetry in which limit cycles, limit tori, or strange attractors coalesce with their mirror image. Mathematically, the bifurcations are characterized by the coalescence of covariant Lyapunov vectors, generalizing the notion of exceptional points to nonlinear dynamical systems. A generic way to construct these bifurcations for arbitrary attractors is presented, and we show that it is a normal form in the case of limit cycles. Breaking the Z₂ symmetry explicitly leads to the formation of memory observable through the presence of hysteresis cycles. Our results apply to systems including minimal models of coupled neurons, ecological systems, and open quantum systems.