document: Bistability of harmonically forced relaxation oscillations Relaxation oscillations appear in processes which involve transitions between two states characterized by fast and slow time scales . When a relaxation oscillator is coupled to an external periodic force its entrainment by the force results in a response which can include multiple periodicities and bistability . The prototype of these behaviors is the harmonically driven van der Pol equation which displays regions in the parameter space of the driving force amplitude where stable orbits of periods 2n + or-1 coexist , flanked by regions of periods 2n +1 and 2n-1 . The parameter regions of such bistable orbits are derived analytically for the closely related harmonically driven Stoker-Haag piecewise discontinuous equation . The results are valid over most of the control parameter space of the system . Also considered are the reasons for the more complicated dynamics featuring regions of high multiple periodicity which appear like noise between ordered periodic regions . Since this system mimics in detail the less analytically tractable forced van der Pol equation , the results suggest extensions to situations where forced relaxation oscillations are a component of the operating mechanisms
doc_bio_tags: B O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B O O O O O O O O O O O O O B O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O
extractive_keyphrases: bistability harmonically forced relaxation oscillations external periodic force entrainment van der pol equation control parameter space
abstractive_keyphrases: harmonically driven stoker-haag piecewise discontinuous equation nonlinear dynamics