Narrow Results

Phase models represent the ideal framework to investigate the synchronization of a nonlinear oscillator with an external forcing. While many researches focused the attention to their analysis, little work has been done about the reduction of a physical system to the corresponding phase model. In this paper we show how, resorting to averaging techniques, it is possible to obtain the phase model corresponding to a given set of state equations. As examples, we derive the phase equations and investigate the synchronization properties of two popular nonlinear oscillators.

In this paper, a novel method of energy analysis is developed for dynamical systems with time delays that are slightly perturbed from undamped SDOF/MDOF vibration systems. Being served frequently as the mathematical models in many applications, such systems undergo Hopf bifurcation including the classic "Hopf bifurcation" for SDOF systems and "multiple Hopf bifurcation" for MDOF systems, under certain conditions. An interesting observation of this paper is that the local dynamics near a Hopf bifurcation, including the stability of the trivial equilibrium and the bifurcating periodic solutions, of such systems, can be justified simply by the change of the total energy function. The key idea is that for the systems of concern, the total power (the total derivative of the energy function) can be estimated along an approximated solution with harmonic entries, the main part of the solution near the Hopf bifurcation. It shows that the present method works effectively for stability prediction of the trivial equilibrium and the bifurcating periodic solutions, and that it provides a high accurate estimation of the amplitudes of the bifurcating periodic solutions. Compared with the current methods such as the center manifold reduction which involves a great deal of symbolic computation, the energy analysis features a clear physical intuition and easy computation. Two illustrative examples are given to demonstrate the effectiveness of the present method.

In this paper, a novel method named pseudo-oscillator analysis is developed for the local dynamics near a Hopf bifurcation of scalar nonlinear dynamical systems with time delays. For this purpose, a pseudo-oscillator that is slightly perturbed from an undamped oscillator is firstly constructed, its fundamental frequency is the same as the frequency at the bifurcation point, and the disturbance is associated with the original system. Next, the pseudo-power function, defined as the power function of the pseudo-oscillator, is estimated along a harmonic function. Then we conclude that the local dynamics near the Hopf bifurcation can be justified from the variation of the averaged pseudo-power function. The new method features a clear physical intuition and easy computation, and it yields very accurate prediction for the periodic solution resulted from the Hopf bifurcation, as shown in three illustrative examples.

The synchronization of an oscillator with an external stimulus or between coupled elements is the subject of intense research in many areas of applied sciences. The most successful approach is based on phase modeling, founded on the idea to represent each oscillator by a phase variable. Phase models have been analyzed with a wealth of details and in a plethora of different variants, but little research has been done in view of the reduction of a physical system to the corresponding phase model. In this paper we investigate the possibility, at least within the context of analytical approximations, to obtain the phase model corresponding to a given nonlinear system. Both periodically driven and coupled oscillators are considered, and examples based on Stuart–Landau and van der Pol systems are given.