Double Hopf Bifurcation Analysis for Coupled van der Pol–Rayleigh System with Time Delay.

Purpose: This paper mainly studies the double Hopf bifurcation for coupled van der Pol–Rayleigh system with time delay. It aims to see how time delay and coupling factors affect the system, then avoid unstable or harmful vibrations and look for steady state in engineering. Methods: This paper mainly uses the normal form and center manifold theory and the Runge–Kutta method. First, we analyze the stability switching of the coupled van der Pol–Rayleigh system with time delay and the dynamical behavior caused by the double Hopf bifurcation. Secondly, we derive the normal form of double Hopf bifurcation by the normal form and center manifold theory. Then the dynamical phenomena of the system are investigated. Finally, we perform numerical simulations to verify the theoretical analysis by the Runge–Kutta method. Results: The stability switching region of the system with respect to the delay is obtained. And the parameter regions of amplitude death, stable periodic motion and quasi-periodic motion are obtained. And as parameter values change, the steady state of the periodic solution follows. Conclusions: In the field of engineering, it is more practical to consider systems with time delay. So we consider system with time delay, obtain the stability switching region about the time delay and discuss the dynamics of each bifurcation region. When doing some system design, it is necessary to avoid the occurrence of bifurcation behavior, because it will lead to changes in system stability. We can adjust the system parameters to avoid bifurcation according to stability region. Or some engineering problem requires the avoidance of harmful vibrations and the pursuit of steady state. With the change of time delay and coupling factors, we can find the amplitude death region and the region of stable periodic motion and quasi-periodic motion, thus avoiding harmful vibration in practical engineering by adjusting the parameters. [ABSTRACT FROM AUTHOR]