Emergence of collective self-oscillations in minimal lattice models with feedback

Abstract

The emergence of collective oscillations and synchronization is a widespread phenomenon in complex systems. While widely studied in dynamical systems theory, this phenomenon is not well understood in the context of out-of-equilibrium phase transitions. Here we consider classical lattice models, namely the Ising, the Blume-Capel and the Potts models, with a feedback among the order and control parameters. With linear response theory we derive low-dimensional dynamical systems for mean field cases that quantitatively reproduce many-body stochastic simulations. In general, we find that the usual equilibrium phase transitions are taken over by complex bifurcations where self-oscillations emerge, a behavior that we illustrate by the feedback Landau theory. For the case of the Ising model, we obtain that the bifurcation that takes over the critical point is non-trivial in finite dimensions. We provide numerical evidence that in 2D the most probable value of the amplitude follows the Onsager law. We illustrate multi-stability for the case of discontinuously emerging oscillations in the Blume-Capel model, whose tricritical point is substituted by the Bautin bifurcation. For the Potts model with q = 3 colors we highlight the appearance of two mirror stable limit cycles at a bifurcation line and characterize the onset of chaotic oscillations that emerge at low temperature through either the Feigenbaum cascade of period doubling or the Aifraimovich-Shilnikov scenario of a torus destruction. We show that entropy production singularities as a function of the temperature correspond to change in the spectrum of Lyapunov exponents. Our results show that mean-field behaviour can be described by the bifurcation theory of low-dimensional dynamical systems, which paves the way for the definition of universality classes of collective oscillations.