Repeated resonances to chaos in forced oscillators (Q1342155)

The periodically forced dissipative nonlinear oscillator \(\ddot x+ \alpha\dot x+ \omega^2_0 (1+ h\cos \omega t)x= cx^2- \beta x\dot x+\gamma \dot x^2\) is considered. An analytical approach is suggested allowing to approximate the subharmonic solutions of order \(q\), their bifurcation curves, and the homoclinic bifurcation curves. It is shown on the basis of approximations constructed that the analytical conditions for successive subharmonic bifurcations lead to the condition for homoclinic bifurcation when \(q\to \infty\). The approximations are based on the asymptotic small \(\varepsilon\)-expansion of the solutions with the condition \(\omega^2_0= (\omega p/q)^2+ \delta\), where \(p\), \(q\) are relatively prime, and \(\delta\) is a small real parameter. Then, for each order of approximation, the algebraic equation for the parameters determining the steady-state periodic solutions are formulated, the solvability conditions for which define the bifurcation curves. The homoclinic bifurcation curves are obtained by considering their limit as \(q\to \infty\).