Hysteresis in a Generalized Kuramoto Model with a Simplified Realistic Coupling Function and Inhomogeneous Coupling Strengths

Abstract

We investigate hysteresis in a generalized Kuramoto model with identical oscillators, focusing on coupling strength inhomogeneity, which results in oscillators being coupled to others with varying strength, and a simplified, more realistic coupling function. With the more realistic coupling function and the coupling strength inhomogeneity, each oscillator acquires an effective intrinsic frequency proportional to its individual coupling strength. This is analogous to the positive coupling strength-frequency correlation introduced explicitly or implicitly in some previous models with nonidentical oscillators that show explosive synchronization and hysteresis. Through numerical simulations and analysis using truncated Gaussian, uniform, and truncated power-law coupling strength distributions, we observe that the system can exhibit abrupt phase transitions and hysteresis. The distribution of coupling strengths significantly affects the hysteresis regions within the parameter space of the coupling function. Additionally, numerical simulations of models with weighted networks including a brain network confirm the existence of hysteresis due to the realistic coupling function and coupling strength inhomogeneity, suggesting the broad applicability of our findings to complex real-world systems.

Figures (8)

• Figure 1: Hysteresis with truncated Gaussian distributions for coupling strength $K_i$. (a) Truncated Gaussian distributions of Eq. (\ref{['eq:tr_gauss_dist']}) for several values of $s$. (b) Curves of critical $c_0$ ($c_0^*$) for the stability of a uniformly incoherent state as a function of $\beta$ with the distributions of (a). The curves are obtained numerically from Eqs. (\ref{['eq:uniform_state_stability_a']}) and (\ref{['eq:uniform_state_stability_b']}). $\sin\beta$ is plotted for the reference. (c) A case with no hysteresis ($s=0.5$, $\beta=0.45\pi$) : $R$ values for the forward path (cyan circles) and for the backward path (gray circles) from 10 hysteresis procedures with different $\{K_i\}$ and initial conditions. See the text for more details. (d) A case with hysteresis ($s=1.0$, $\beta=0.45\pi$). $R$ vs $c_0$ as in (c). In (c) and (d), the red solid lines and the gray dashed lines represent $r$ values obtained numerically from Eq. (\ref{['eq:self_consistency']}), corresponding to at least linearly neutrally stable and unstable states, respectively. In (d), the lines with arrows in the middle denote the theoretically obtained abrupt transitions along the hysteresis procedure, considering the stability analysis results. For the case of (d), the system shows two different states with the same parameter values ($s=1.0$, $\beta=0.45\pi$, $c_0=0.8$) along the hysteresis procedure: (e) Forward path: a partially locked state with $R \approx 0.886$ and (f) Backward path: an incoherent state with $R \approx 0.070$.
• Figure 2: Hysteresis search in $\beta$-$c_0$ plane using hysteresis procedures with truncated Gaussian distributions for every value of $\beta \in [0,~0.49\pi]$ separated by $\Delta \beta=0.01\pi$. (a) $s=0.5$: (Forward) $R_\text{forward}(\beta,c_0)$, time-averaged order parameter for the forward path, averaged over 5 hysteresis procedures with different $\{K_i\}$ and initial conditions; (Backward) $R_\text{backward}(\beta,c_0)$, $R$ values for the backward path, averaged over the 5 hysteresis procedures; (Difference) $\Delta R(\beta,c_0) \equiv R_\text{forward}(\beta,c_0) - R_\text{backward}(\beta,c_0)$. (b) $s=1.0$. (c) $s=3.0$. $c_0^{0.9}$ and $c_0^{0.1}$ denote the boundaries explained in detail in Subsec. \ref{['subsec:tr_gaussian_dist']}. White X marks indicate data points where the standard deviation(std) of the order parameter time series $r(t)$ exceeds $0.1$ for at least one simulation out $5$. See main text for other details.
• Figure 3: Hysteresis with uniform distributions for coupling strength $K_i$. (a) Uniform distributions of Eq. (\ref{['eq:uniform_dist']}) for several values of $w$. (b) Curves of critical $c_0$($c_0^*$) as a function of $\beta$ with the distributions of (a). $R$ vs $c_0$ for the cases with hysteresis: (c) $w=0.6$, $\beta=0.25\pi$, (d) $w=0.8$, $\beta=0.40\pi$, (e) $w=1.0$, $\beta=0.45\pi$ and (f) $w=0.8$, $\beta=0.47\pi$. For the case of (f), the system shows two different states with the same parameter values ($w=0.8$, $\beta=0.47\pi$, $c_0=0.77$) along the hysteresis procedure: (g) Forward path: a partially locked state with $R\approx 0.837$ and (h) Backward path: a partially locked state with $R \approx 0.296$. Other details are as in Fig \ref{['fig:01_trg']}.
• Figure 4: Hysteresis search in $\beta$-$c_0$ plane using hysteresis procedures with uniform distributions for every value of $\beta \in [0,~0.49\pi]$ separated by $\Delta \beta=0.01\pi$. (a) $w=0.6$: (Forward) $R_\text{forward}(\beta,c_0)$, time-averaged order parameter values for the forward path, averaged over 5 hysteresis procedures with different $\{K_i\}$ and initial conditions; (Backward) $R_\text{backward}(\beta,c_0)$, $R$ values for the backward path, averaged over the 5 hysteresis procedures; (Difference) $\Delta R(\beta,c_0) \equiv R_\text{forward}(\beta,c_0) - R_\text{backward}(\beta,c_0)$. (b) $w=0.8$. (c) $w=1.0$. Other details are as in Fig. \ref{['fig:02_trg']}.
• Figure 5: Hysteresis with truncated power-law distributions for coupling strength $K_i$. (a) Truncated power-law distributions of Eq. (\ref{['eq:tr_power_law_dist']}) for several values of $\gamma$. (b) Curves of critical $c_0$($c_0^*$) as a function of $\beta$ with the distributions of (a). $R$ vs $c_0$ for the cases with hysteresis: (c) $\gamma=1.3$, $\beta=0.15\pi$ and (d) $\gamma=1.1$, $\beta=0.30\pi$. For the case of (c), the system shows two different states with the same parameter values ($\gamma=1.3$, $\beta=0.15\pi$, $c_0=0.7$) along the hysteresis procedure: (e) Forward path: a fully locked state with $R \approx 0.931$ and (f) Backward path: an incoherent state with $R \approx 0.058$. For the case of (d), the system shows two different states with the same parameter values ($\gamma=1.1$, $\beta=0.30\pi$, $c_0=0.6$) along the hysteresis procedure: (g) Forward path: a partially locked state with $R \approx 0.874$ and (h) Backward path: an incoherent state with $R \approx 0.087$. Other details are as in Fig \ref{['fig:01_trg']}.