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In this paper, the bursting oscillation phenomenon in coupled systems with two time scales is introduced. Firstly, several types of bifurcation are briefly introduced: fold bifurcation, Hopf bifurcation, fold limit cycle bifurcation, homoclinic bifurcation, etc. The bursting oscillations of the system with two excitation terms and the bifurcation delay in the bursting oscillations are considered. Secondly, some simple bursting oscillations are introduced, such as fold/fold bursting, fold/supHopf bursting, subHopf/subHopf bursting, fold/LPC bursting, Hopf/LPC bursting, fold/homoclinic bursting, Hopf/homoclinic bursting, etc. At the same time, the system also has some complex bursting oscillations, such as asymmetric bursting, delayed bursting, bursting with hysteresis loop, etc. Finally, the practical applications of bursting oscillations, such as dynamic vibration absorbers and nonlinear vibration energy harvesting technology, are introduced. [ABSTRACT FROM AUTHOR]

This paper mainly studied the analytical solutions of three types of Van der Pol-Duffing equations. For a system with parametric excitation frequency, we knew that the ordinary homotopy analysis method would be unable to find the analytical solution. Thus, we primarily used the multi-frequency homotopy analysis method (MFHAM). First, the MFHAM was introduced, and the solution of the system was expressed by constructing auxiliary linear operators. Then, the method was applied to three specific systems. We compared the numerical solution obtained using the Runge–Kutta method with the analytical solution to verify the correctness of the latter. Periodic solutions, with and without time delay, were also compared under the same parameters. The results demonstrated that it was both effective and correct to use the MFHAM to find analytical solutions to Van der Pol-Duffing systems, which were classical systems. By comparison, the MFHAM proved to be effective for time delay systems. [ABSTRACT FROM AUTHOR]

In the past few decades, the predator–prey model has played an important role in the dynamic behavior of populations. Many scholars have studied the stability of the predator–prey system. Due to the complex influence of time delay on the dynamic behavior of systems, time-delay systems have garnered wide interest. In this paper, a classical piecewise smooth slow–fast predator–prey model is considered. The dynamic properties of the system are analyzed by linearization. The existence and uniqueness of the relaxation oscillation are then proven through the geometric singular perturbation theory and entry–exit function. Finally, a stable limit cycle is obtained. A numerical simulation verifies our results for the systems and shows the effectiveness of the method in dealing with time delays. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem. [ABSTRACT FROM AUTHOR]

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV. [ABSTRACT FROM AUTHOR]