General synchronization dynamics of coupled Van der Pol–Duffing oscillators

This paper considers the general synchronization dynamics of coupled Van der Pol–Duffing oscillators. The linear and nonlinear stability analysis on the synchronization process is derived through the Whittaker method and the Floquet theory in addition to the multiple time scales method. A stability map displaying different dynamical states of the system is performed. Numerical simulation is carried out to support and to complement the accuracy of the analytical treatment.     

Synchronization of two coupled self-excited systems with multi-limit cycles

We analyze the stability and optimization of the synchronization process between two coupled self-excited systems modeled by the multi-limit cycles van der Pol oscillators through the case of an enzymatic substrate reaction with ferroelectric behavior in brain waves model. The one-way and two-way couplings synchronization are considered. The stability boundaries and expressions of the synchronization time are obtained using the properties of the Hill equation. Numerical simulations validate and complement the results of analytical investigations.     

Effects of a locally injected signal on phase synchronization in a network of self-excited cells

We consider in this paper the problem of phase synchronization in a network of coupled self-excited cells with free-end boundary conditions. The eigenvalues properties are used to perform the stability boundaries of the synchronization process for pure sinusoidal and quasi-sinusoidal oscillatory states. In the case where the cells’ oscillatory states are quasi-periodic, more energy and time are required for the network to become fully synchronized, compared to the case where each cell exhibits a periodic oscillation. The influence on the stability boundaries of a periodic external excitation locally injected in the network is tackled. For both cases (with and without local injection), numerical simulations are used to check the accuracy and to complement the results derived from the analytical treatment. In this configuration, the local injection helps to drastically increase the range of the coupling parameter for which all cells become synchronized.     

Passive aerodynamics control of plasma instabilities

The possibility of using a smart-damping scheme to modify the dynamic responses of plasma oscillations governed by a two-fluid model is considered. The passive aerodynamics control strategy is used to address this issue. The control efficiency is found by analyzing the conditions satisfied by the control gain parameters for which, the amplitude of oscillations is reduced both in the harmonic and chaotic states. In the regular state, the analytical stability analysis uses for linear oscillations the Routh-Hurwitz criterion while the Whittaker method and Floquet theory are utilized for nonlinear harmonic oscillations. The stability boundaries in the control gain parameter space is derived. The agreement between the analytical and numerical results is good. In the chaotic states, numerical simulations are used to perform quenching of chaotic oscillations for an appropriate set of control parameters.     

Stability of the synchronization manifold in nearest neighbor nonidentical van der Pol-like oscillators

We investigate the stability of the synchronization manifold in a ring and in an open-ended chain of nearest neighbor coupled self-sustained systems, each self-sustained system consisting of multi-limit cycle van der Pol oscillators. Such a model represents, for instance, coherent oscillations in biological systems through the case of an enzymatic-substrate reaction with ferroelectric behavior in a brain waves model. The ring and open-ended chain of identical and nonidentical oscillators are considered separately. By using the Master Stability Function approach (for the identical case) and the complex Kuramoto order parameter (for the nonidentical case), we derive the stability boundaries of the synchronized manifold. We have found that synchronization occurs in a system of many coupled modified van der Pol oscillators, and it is stable even in the presence of a spread of parameters.     

Spatiotemporal dynamics in a ring of N mutually coupled self-sustained systems

In this paper, we consider the spatiotemporal dynamics in a ring of N mutually coupled self-sustained oscillators in the regular state. When there are no parameter mismatches, the good coupling parameters leading to full, partial, and no synchronization are derived using the properties of the variational equations of stability. The effects of the spatial dimension of the ring on the stability boundaries of the synchronized states are performed. Numerical simulations validate and complement the results of analytical investigations. The influences of coupling parameter mismatch on the forecasted stability boundaries are also highlighted.