Analysis of Chaos Characteristics of a Class of Strong Nonlinear Parametric Excitation Systems

ABSTRACT This article focuses on a class of van der Pol–Duffing nonlinear dynamic systems with parameters and harmonic joint excitation and studies their resonance and chaotic behavior. First, the multiscale method is used to analyze this system and obtain an approximate analytical solution. The obtained approximate analytical solution is compared with the numerical solution through the amplitude frequency curve, and the two have a high degree of agreement, proving the correctness of the analytical solution. Second, the Melnikov method was used to obtain the conditions for the system to enter chaos in the sense of Smale horseshoe. Then, by combining bifurcation diagrams, time series diagrams, phase diagrams, and Poincar é cross‐sections, the system was analyzed to explore the occurrence of nonsingular chaotic attractors when the system enters chaos. This approach is not limited to qualitative analysis but rather detects the existence of SNAs through three‐dimensional Poincaré cross‐sections and quantitative methods. Finally, the influence of initial values on system safety, as well as the erosion phenomenon of amplitude and excitation parameters on the boundary of the system safety basin, was obtained. The evolution law of the system safety basin was analyzed, and the erosion and bifurcation mechanism of the safety basin was studied.