On the spatial spread of active control in a van der Pol ring via synchronisation and its stabilisation using parameter mismatch

Abstract

This work explores spatial propagation of amplitude control in a network of self-excited oscillators having ring topology. We consider a ring of N Van der Pol oscillators, with a first order control model connected to an arbitrary kth oscillator in the network. It is shown that even such a simple one-dimensional control, applied to the kth oscillator, can be made to propagate to other oscillators due to the synchronising property of the network, bringing about the control of their amplitudes also. The slow flow equations of amplitude and phase of the complete model, obtained analytically, are used to study the spatial propagation of control in the network. The oscillator responses are approximated by quasi-harmonic functions whereas the first order control response is approximated by the superposition of relaxation and oscillatory terms. The obtained analytical approximations are validated by comparing them with numerical simulation. The effect of the synchronising property of the network on the spread of amplitude control is investigated and it is shown that the extent and speed of amplitude control increases with increase in coupling strength. It is further shown that the control strategy is not effective when the control gain parameter is greater than the square of linear natural frequency of the oscillator (γ>ωk2). We propose the use of parameter mismatch between oscillators as a tool for overcoming this limitation. Frequency detuning, brought about by parameter mismatch, is shown to stabilise the fixed point at the origin and cause amplitude control even outside the effective range of the control mechanism. This shows that slight differences between the oscillators, unavoidable during manufacturing, can be beneficial for the spatial spread of control in the network.