Bifurcation and chaos in the Duffing oscillator with a PID controller. (English) Zbl 0881.70014

Cui, Fangsen; Chew, C. H.; Xu, Jianxue; Cai, Yuanli

Nonlinear Dyn. 12, No. 3, 251-262 (1997).

Summary: We discuss the title problem and establish that the Hopf bifurcation can occur. Additionally, we show that there is a global stable fixed point. The PID controller works well in some fields of the parameter space, but in other fields of the parameter space, or if the reference input is not equal to zero, chaos is common for hard spring type system. The Melnikov method is used to obtain a criterion concerning fractal basin boundary for soft spring system.

MSC:

70K50	Bifurcations and instability for nonlinear problems in mechanics
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

Hopf bifurcation; global stable fixed point; parameter space; Melnikov method; fractal basin boundary