Symmetry-Break in a Minimal Lorenz-Like System

Starting from the classical Saltzman 2D convection equations, we derive via spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled, generalized Lorenz system. The consideration of this process breaks an important symmetry, couples the dynamics of fast and slow variables, and modifies the structural properties of the attractor. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of O(Ec^-1) is introduced in the system. Moreover, the system is ergodic and hyperbolic, the slow variables feature long term memory with 1/f^3/2 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics.