The damped Kuramoto-Sivashinsky equation has emerged as a fundamental tool for the understanding of the onset and evolution of secondary instabilities in a wide range of physical phenomena. Most existing studies about this equation deal with its asymptotic states on one-dimensional settings or on periodic square domains. We utilize a large-scale numerical simulation to investigate the asymptoti...

The importance of networks of coupled oscillators is widely recognized. Such networks occur in biological systems like the heart, in chemical systems, in computational problems, and in engineering systems. Systems of coupled oscillators can also be used as an abstract model for synchronisation in organisations. Here we show that synchronisability in a specific coupledoscillator model, the Kuram...

Maintaining the stability of synchronization state is crucial for the functioning of many natural and artificial systems. In this study, we develop methods to optimize the synchronization stability of the Kuramoto model by minimizing the dominant Lyapunov exponent. Using the recently proposed cut-set space approximation of the steady states, we greatly simplify the objective function, and furth...

We present a complete classification of attractors for networks of coupled identical Kuramoto oscillators. In such networks, each oscillator is driven by the same first-order trigonometric function, with coefficients given by symmetric functions of the entire oscillator ensemble. For [Formula: see text] oscillators, there are four possible types of attractors: completely synchronized fixed poin...

We propose a dynamical gradient network approach to consider the synchronization in the Kuramoto model. Our scheme to adaptively adjust couplings is based on the dynamical gradient networks, where the number of links in each time interval is the same as the number of oscillators, but the links in different time intervals are also different. The gradient network in the (n+1)th time interval is d...

We study the Kuramoto model with attractive sine coupling. introduce a complex-valued matrix formulation whose argument coincides original dynamics. derive an exact solution for model, which permits analytical insight into individual realizations of model. The existence form provides key demonstration that, in some cases, reformulations nonlinear dynamics higher-order number fields may provide ...