Steady-State Exceptional Point Degeneracy and Sensitivity of Nonlinear Saturable Coupled Oscillators

TL;DR

This work analyzes steady-state exceptional point degeneracies in a nonlinear two-oscillator system with saturable gain using nonlinear coupled-mode theory. It derives the steady-state frequencies via a cubic condition and characterizes multiple degeneracy orders, including a unique third-order SS-EPD at ω1 = ω2 and g_s = γ = κ, around which different perturbations yield cubic-root, linear, or square-root frequency sensitivities. The authors connect energy balance and a laser-inspired saturable gain model to stability, bistability, and time-domain dynamics, then apply the framework to inductively and capacitively coupled RLC circuits, comparing predictions to full nonlinear circuit simulations. The results provide design insights on exploiting exceptional degeneracy-enhanced sensitivity while managing stability and energy, and they offer practical retrievability formulas for perturbations and broad guidance for nonlinear sensor implementations.

Abstract

Near exceptional degenerate points in parameter space, coupled oscillator systems display enhanced sensitivity of their saturated steady-state (SS) oscillation frequencies to small changes in system parameters. Linear $\mathcal{PT}$-symmetric systems made of two coupled resonators have exceptional point of degeneracy (EPD), around which square-root sensitivity is observed. However, realistic systems with gain are inherently saturable and nonlinear, thereby invalidating linear assumptions, and when $\mathcal{PT}$-symmetry is broken the coupled resonator system becomes unstable, hence it seems that the best working regime is to use such instability to make an SS-EPD-based oscillator. We study the saturated steady-state of a general system of two coupled oscillators with saturable nonlinear gain. Extending previous analyses, we find the steady-state oscillation frequency-gain pairs, and we analytically and numerically derive the sensitivity of the oscillation frequency to system's perturbations around a unique third-order degeneracy which corresponds to SS$\mathcal{PT}$ symmetry because it is the saturated gain that is symmetric to losses. In general, unlike linear systems, we find that at SS, the sensitivity of the oscillation frequency to exhibit linear, square-root, or cube-root dependence on small perturbations. We additionally study the energy and stability of each SS, and demonstrate the application and limitations of this analysis to coupled RLC circuits. We give a comprehensive outlook for exploiting exceptional degeneracy-enhanced sensitivity in nonlinear coupled oscillators and suggest the best operative conditions.

Figures (15)

• Figure 1: Generic coupled oscillator system with saturable nonlinear gain $g$.
• Figure 2: The saturated steady-state solution pair $\omega,g_s$ of the symmetric system ($\omega_2=\omega_1$) from (\ref{['eq:ReCharacteristicSimplified']}) and (\ref{['eq:ImCharacteristicSimplified']}) compared against the linear solutions of the $\mathcal{PT}$ system with equivalent gain and loss ($g=\gamma$). Both solutions are plotted around the degenerate point, $\kappa=\gamma$, varying $\hat{\kappa}$ and assuming that $\hat{\gamma}=0.1$ with the hat $\hat{\ }$ denoting a normalization to $\omega_1$. For the nonlinear steady-state case, red and gray lines indicate stable and not stable steady-state solutions, respectively.
• Figure 3: The three-dimensional steady-state solution space of $\hat{\omega}$ (a)-(b) (cusp catastrophe like geometry varying the two parameters) and $\hat{g}_s$ (c)-(d) varying $\hat{\omega}_2$, and $\hat{\kappa}$ around the third-order degenerate solution $\hat{\omega}_2=1$ and $\hat{\kappa}=\hat{\gamma}=0.1$. Both solution spaces are recorded from two different angles for better visualization, with the $\hat{\ }$ symbol denoting a normalization of the parameters and solutions to $\omega_1$. The colormap connects the steady-state pair between (a)-(b) and (c)-(d), indicating that when there are three solutions, the middle $\hat{\omega}$ will have the largest $\hat{g}_s$ value. As seen in (a)-(b), there is an inherent anti-symmetry across $\hat{\omega}_2=1$, which leads the system to have chiral dynamics Wang2019_Chiral.
• Figure 4: Steady-state oscillation frequency $\hat{\omega}$ (real-valued solutions of $p(\omega)=0$), in red, plotted varying $\hat{\kappa}$ and $\hat{\omega}_2$ (the complex blue-dotted branches are shown for a better understanding of the solutions). The parameter for each of the plots are as follows, with the $\hat{\ }$ denoting a normalization to $\omega_1$. Varying $\hat{\kappa}$, with $\hat{\gamma}=0.1$: (a) $\hat{\omega}_2=0.98$; (b) $\hat{\omega}_2=1.02$. Varying $\hat{\omega}_2$, with $\hat{\gamma}=0.1$: (c) $\hat{\kappa}=0.08$; (d) $\hat{\kappa}=\hat{\gamma}=0.1$; and (e) $\hat{\kappa}=0.13$.
• Figure 5: Saturated steady-state gain $g_s$ from (\ref{['eq:gsCubic']}), in red, plotted around the third order degenerate solution varying $\kappa$ and $\omega_2$ (the complex branches, blue dotted, are shown only for a better understanding of the solutions). The parameters for each plots are as follows, with the $\hat{\ }$ denoting a normalization to $\omega_1$. (a) Varying $\hat{\kappa}$: $\hat{\gamma}=0.1$ and either $\hat{\omega}_2=0.98$ or $\hat{\omega}_2=1.02$ (symmetry across $\omega_2=\omega_1$ creates the same plots). Varying $\hat{\omega}_2$: $\hat{\gamma}=0.1$ and (b) $\hat{\kappa}=0.08$; (c) $\hat{\kappa}=\hat{\gamma}=0.1$; (d) $\hat{\kappa}=0.13$.