Analytical routes of period-1 motions to chaos in a periodically forced Duffing oscillator with a twin-well potential. (English) Zbl 1300.34081
Summary: Analytical routes of period-1 motions to chaos in the Duffing oscillator with a twin-potential well are investigated through the generalized harmonic balance method. The analytical solutions of period-\(m\) motions are presented by the Fourier series, and the corresponding Hopf bifurcation of periodic motions leads to new periodic motions with period-doubling. Three analytical routes of asymmetric period-1 motions to chaos are presented comprehensively. To verify approximate, analytical periodic solutions, numerical simulations are carried out.
In the analytical routes, the unstable periodic motions are presented, and such analytical routes with unstable periodic motions can help one find unstable chaos. Such unstable chaos cannot be obtained simply via the time going to infinity (i.e., \(t\to-\infty)\).
In the analytical routes, the unstable periodic motions are presented, and such analytical routes with unstable periodic motions can help one find unstable chaos. Such unstable chaos cannot be obtained simply via the time going to infinity (i.e., \(t\to-\infty)\).
MSC:
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
