The nonstationary effects on a softening Duffing oscillator
Object of the numerical investigations is the equation \(\ddot x(t)+ x(t)+ 0,4\dot x (t)- x^3 (t)= f(t)\cos \theta (t)\) with \(\dot \theta(t)= \Omega (t)\). First, the stationary case \((f=\text{const}\), \(\Omega= \text{const})\) is considered and a bifurcation diagram along the \(\Omega\)-line \((\Omega (t)= \Omega_0+ \alpha t)\) is obtained. This shows a flip-flop response by the period-doubling bifurcation. A second diagram shows in the \((f, \Omega)\) plane the stationary period-doubling regions and their boundaries and the chaos regions. The nonstationary effects are studied for \(\Omega= \Omega_0\pm \alpha t\) for certain \(\alpha\) values, for one fixed \(f\) value, and certain initial conditions. The result is a diagram showing bifurcations for nonstationary as well as for stationary cases. Then nonstationary bifurcations are examined for \(\Omega= \Omega_0\pm \alpha t\), \(f= f_0\pm \alpha t\), for certain numerical cases, and are visualized in diagrams. Several conclusions -- mainly of physical character -- are drawn.
scientific article; zbMATH DE number 754957