Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration
The authors of this interesting paper study the collective behaviors of the Kuramoto model with frustration. It was discussed by \textit{H. Sakaguchi} and \textit{Y. Kuramoto} [``A soluble active rotator model showing phase transitions via mutual entrainment'', Progr. Theor. Phys. 76, No. 3, 576--581 (1986; \url{doi:10.1143/PTP.76.576})] as a prototype example of synchronization of phase-coupled oscillators. Here the authors study the phase concentration for the kinetic Kuramoto-Sakaguchi (K-S) type equations with frustration via detailed estimates on the dynamics of ordered parameters. The Kuramoto order parameters measure the overall degree of phase concentrations. It is introduced \(\theta_i=\theta_i(t)\in\mathbb{T}\) and \(\alpha \in (-\pi /2, \ \pi /2)\) which are the phase of the \(i\)-th Kuramoto oscillator and the uniform frustration between oscillators, respectively. \[ \begin{gathered} \dot{\theta}_i=\nu_i+(\kappa /N) \sum\limits_{j=1}^{N}\sin(\theta_j- \theta_i+\alpha ), \ \ t>0,\\ \theta (0)=\theta_i^{in}, \ \ i=1,\ldots ,N, \end{gathered}\tag{\(\ast\)} \] where \(\nu_i,\kappa >0\), \(N\in \mathbb{N}_{+}\) denote the natural frequency, the coupling strength, and the number of oscillators, respectively. Here the right-hand side of \((\ast )\) is Lipschitzean. The need of frustration is suggested from empirical facts in physics and biology. The presence of frustration causes some mathematical difficulties in the analysis of synchronization behaviors of oscillatory systems. Let \(f = f (t,\nu ,\theta )\) be an one-oscillator probability density function at phase \(\theta \) with natural frequency \(\nu \) at time \(t\). Then, the dynamics of \(f\) is governed by the following Cauchy problem to the K-S equation with frustration \begin{gather*} \partial_tf+\partial _{\theta } (V[f]f)=0, \ \ (\theta ,\nu ,t) \in \mathbb{T}\times\mathbb{R} \times\mathbb{R}_{+}, \\ V[f](t,\nu ,\theta )=\nu +\kappa \int_{\mathbb{T}\times\mathbb{R}} \sin{(\theta_{\ast }-\theta +\alpha )} f(t,\nu_{\ast },\theta_{\ast }) d\theta_{\ast }d\nu_{\ast }, \\ f(0,\nu,\theta )=f^{in} (\nu ,\theta )\geq 0, \ \ f^{in}(\nu ,\theta + 2\pi )= f^{in}(\nu , \theta ), \\ \int_{\mathbb{T}}f^{in}(\nu ,\theta ) d\theta =g(\nu ), \ \ \int_{\mathbb{T}\times \mathbb{R}}f^{in}(\nu ,\theta ) d\theta d\nu =1. \end{gather*} Next suppose that \(f(t,\nu ,\theta )\) is a classical solution to the latter under initial data \(f^{in}\) continuously differentiable and \(R_0=R(0)>0\). Also suppose the coupling strength \(\kappa \) is sufficiently large, the size of frustration \(|\alpha |\) is sufficiently small, and \(\mathrm{supp}\, g(\nu )\subset [-\ell,\ell ]\), then the following estimates hold \[ \lim\limits_{t\to\infty }\inf \, R(t)\geq R_{\infty }, \] \[ \sup\limits_{\theta\in \mathbb{T} \setminus\mathcal{D}_{\infty}(t)} |f(t,\nu ,\theta )|\leq C\|f^{in}\|_{L^{\infty}} e^{-\kappa (R_{\infty} \sqrt{1-\xi_{\infty}^2}-\epsilon)t}, \ \ t\geq 0. \] It follows that when the coupling strength is sufficiently large and the size of frustration parameter is sufficiently small, then the amplitude order parameter has a positive lower bound uniformly in time.
scientific article; zbMATH DE number 7606063
scientific article