Dynamics of a chain of unidirectionally coupled non-identical Van der Pol oscillators with a sinusoidal input at the first node

This paper investigates the dynamics of a chain of unidirectionally coupled non-identical Van der Pol
(VdP) oscillators driven by a sinusoidal signal at the first node. Starting with a random generation of frequencies in the chain of standard VdP systems, we numerically study the influence of a small and a large value of disorder. It can be seen that disorder reduces the signal amplitude propagating along the chain and induces a coexistence of different dynamical states, namely periodic, quasiperiodic and chaotic states in the chain, which tend to disappear by suitably adjusting the coupling strength. A mathematical justification of the obtained dynamics is provided using multiple time-scale methods. Taking into account the cubic nonlinearity, the chain remains more or less coherent, no generation of the plateau of amplitude is observed for large disorder and the signal amplitude fluctuates greatly along the chain. Considering a chirp distribution of frequencies, the signal amplitude is characterised by a parabolic decrease followed by an irregular series of jumps back to a certain value. These effects are largely reduced by a moderate coupling strength and the presence of cubic nonlinearity.