On resonances in systems of two weakly coupled oscillators

Assuming that two weakly coupled oscillators are essentially nonlinear we construct the most suitable form of a shortened 3-dimensional system which describes behavior of solutions inside non-degenerate resonance zones. We analyze a model system of that kind and establish the existence of limit cycles of different types and also the existence of nonregular attractors which are explained by the existence of saddle-focus loops.      

Number and stability of nonlinear parametric oscillations in a system of weakly coupled oscillators with pendulum-type nonlinearity

The range of parameters for which the nonlinear system under study has three stable cycles with certain symmetry properties is determined. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 369–376, March, 1999.     

Exponential synchronization of stochastic coupled oscillators networks with delays

This paper investigates the exponential synchronization problem of coupled oscillators networks with disturbances and time-varying delays. On basis of graph theory and stochastic analysis theory, a feedback control law is designed to achieve exponential synchronization. By constructing a global Lyapunov function for error network, both pth moment exponential synchronization and almost sure exponential synchronization of drive-response networks are obtained. Finally, a numerical example is given to show the effectiveness of the proposed criteria.     

Robust heteroclinic cycles

Summary One phenomenon in the dynamics of differential equations which does not typically occur in systems without symmetry is heteroclinic cycles. In symmetric systems, cycles can be robust for symmetry-preserving perturbations and stable. Cycles have been observed in a number of simulations and experiments, for example in rotating convection between two plates and for turbulent flows in a boundary layer. Theoretically the existence of robust cycles has been proved in the unfoldings of some low codimension bifurcations and in the context of forced symmetry breaking from a larger to a smaller symmetry group. In this article we review the theoretical and the applied research on robust cycles.     

Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay

A system of three coupled van der Pol oscillators with delay is considered. Hopf bifurcations at the zero equilibrium as the delay increases are exhibited. The existence and stability of multiple periodic solutions are established using a symmetric Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799-4838).     

具有Kelvin-Voigt阻尼的弱耦合系统的能量衰减估计

研究具有 Kelvin-Voigt 阻尼的弱耦合系统。首先在合适的假设条件下,应用线性算子半群理论证明了系统的适定性;进而运用线性算子半群的频域定理证明了具有Kelvin-Voigt阻尼的弱耦合梁―弦系统的能量是一致指数衰减的。     

Periodic solutions for neutral coupled oscillators network with feedback and time-varying delay

This paper is considered to be about the existence of periodic solutions for neutral coupled oscillators network with feedback control and time-varying delay (NCONFT). Based on the systematic method which is firstly applied for NCONFT and consisting of coincidence degree theory, graph theory, and Lyapunov method, some sufficient criteria are obtained to verify the existence of periodic solutions for NCONFT. What’s more, how coupling topology, feedback control, and time-varying delay affect the existence of periodic solutions for NCONFT can be shown by these sufficient criteria. Finally, a numerical simulation is offered to illustrate the effectiveness of our results.     

Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling

In this paper, network of stochastic van der Pol oscillators with time-varying delayed coupling is considered. By using graph theory and Lyapunov functional method, the asymptotic boundedness in pth moment of the network is investigated. Moreover, by constructing an appropriate Lyapunov function, sufficient principle in the form of coefficients of network which ensures the asymptotic boundedness is established. Finally, a numerical example is given to show the effectiveness of the proposed criteria.     

Approximate partial Noether operators and first integrals for cubically coupled nonlinear Duffing oscillators subject to a periodically driven force

The approximate first integrals for a system of two cubically coupled nonlinear Duffing oscillators subject to a periodically driven force are constructed with the help of approximate partial Noether approach. Firstly, approximate partial Noether operators associated with a partial Lagrangian are derived. Then approximate first integrals are obtained for both the resonant and non-resonant cases.     

The finite difference method for the three-dimensional nonlinear coupled system of dynamics of fluids in porous media

For the three-dimensional coupled system of multilayer dynamics of fluids in porous media, the second-order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in l2 norm are derived to determine the error in the second-order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources.