Chaos in Nonequilibrium Two-Temperature $(T_x, T_y)$ Nos\'e-Hoover Cell Models

We revisit a two-temperature Nos\'e-Hoover wanderer particle embedded in a two-dimensional periodic 2x2 cell with four smooth repulsive corners at $(x,y) = (\pm 1, \pm 1)$ to explore chaos with anisotropic thermostatting. The model employs separate thermostats in the x and y directions, enabling controlled deviations from equilibrium. By integrating the full six-dimensional equations of motion and computing the complete Lyapunov spectrum, we confirm chaos and quantify phase-space contraction with high numerical precision. The total contraction rate, interpreted as entropy production, increases nonlinearly with the thermostat anisotropy, deviating from the quadratic dependence expected from linear-response theory, $\Lambda\propto -\delta^{2}$. We compare two fits for $\Lambda$ as a function of $\delta = 0.5 -T_y$: 1) a power law, $\Lambda\propto -\delta^{2.44}$, 2) a quadratic-plus-quartic expansion. While the former captures low-driving behavior slightly better, the latter more accurately describes the strongly driven regime and remains consistent with linear response theory near equilibrium. An empirical linear relation between dissipation and phase-space dimensionality loss is also identified, $\Lambda\propto (D_{KY}-6) / 3$, where $D_{KY}$ is the approximate Kaplan-Yorke dimension. Our results demonstrate that nonlinear dissipation scaling emerges naturally even in minimal driven systems. Momentum statistics show significant non-Gaussian behavior under strong driving. Despite its dissipative nature, the model remains strictly time-reversible, offering a pedagogically rich example of microscopic reversibility coexisting with macroscopic entropy production.