Oscillations and hysteresis: from simple harmonic oscillator and unusual unbounded increasing amplitude phenomena to the van der Pol oscillator and chaos control

In this work we investigate various oscillating systems under hysteretic action. For a simple harmonic oscillator under hysteretic force an unusual unbounded increasing amplitude phenomena (growing rate for the amplitude is proportional to square root of time) are obtained and discussed. For a system of coupled van der Pol oscillators (taking into account the hysteretic coupling) we investigate various dynamic regimes (both regular and chaotic) and show that the hysteresis in the link between two van der Pol oscillators may serve as an effective mechanism for chaos control. Also we consider the effect of hysteresis on synchronisation processes in this system.