Numeric-analytic solutions of the smooth and discontinuous oscillator

• Dynamics of SD oscillator investigated in frequency domain by MHBM.
• Periodic solutions of different orders and their branches are obtained.
• Stability and types of bifurcation are investigated using Floquet theory.
• Period doubling route and sudden transition to chaos are observed.
• Discontinuous oscillator treated as Filippov system and analyzed numerically.

Earlier works on the smooth and discontinuous (SD) oscillator concentrated mainly on the time domain analysis using analytical, semi-analytical and numerical integration methods. In this paper, the frequency domain analysis of the SD oscillator subjected to harmonic excitation which is as important and giving further insight into the dynamics is carried out. Multi-Harmonic Balance Method (MHBM) in combination with arc length continuation is used to obtain the periodic solutions and their branches in the frequency domain for different values of the smoothing parameter α and exciting frequency ω  . Stability of the periodic motions and bifurcation behavior are analyzed using the Floquet theory. For the discontinuous case, the oscillator is treated as a Filippov system and an event driven numerical integration method is used to obtain the response. For α>1α>1, the dynamics of the SD oscillator is similar to that of the hardening Duffing oscillator, for α=1α=1, it is like that of the Ueda oscillator and for 0<α<10<α<1 it is like that of the Duffing oscillator with double well potential. The SD oscillator exhibits period 1 solutions, higher order periodic solutions, chaotic solutions through symmetry breaking bifurcations, period doubling and boundary crises in different parameter ranges. Chaos is observed over a larger frequency range interspersed by narrow windows of higher order periodic solutions.