By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation abundant types of explicit and exact solutions for the double sinh-Gordon equation are derived in a simple manner.
Bifurcation characteristics of the Langford system in a general form are systematically analysed, and nonlinear controls of periodic solutions changing into invariant tori in this system are achieved. Analytical relationship between control gain and bifurcation parameter is obtained. Bifurcation diagrams are drawn, showing the results of control for secondary Hopf bifurcation and sequences of bifurcations route to chaos. Numerical simulations of quasi-periodic tori validate analytic predictions.
This paper applies washout filter technology to amplitude control of limit cycles emerging from Hopf bifurcation of the van der Pol-Duffing system. The controlling parameters for the appearance of Hopf bifurcation are given by the Routh-Hurwitz criteria. Noticeably, numerical simulation indicates that the controllers control the amplitude of limit cycles not only of the weakly nonlinear van der Pol-Duffing system but also of the strongly nonlinear van der Pol-Duffing system. In particular, the emergence of Hopf bifurcation can be controlled by a suitable choice of controlling parameters. Gain-amplitude curves of controlled systems are also drawn.
