A simple non-autonomous system with complicated dynamics

In this paper, we consider a non-autonomous system of the form x(n+1) = a(n)x(n), where a(n) is a two periodic perturbation of a constant a > 1. This system can be used to see the stability properties of limit cycles of non-linear oscillators modelled by second-order non-linear differential equations under the same type of perturbations. The difference equation x(n+1) = ax(n) has a simple dynamics since all orbits are unbounded and do not exhibit sensitive dependence on initial conditions, while the non-autonomous system x(n+1) = a(n)x(n) (for some ranges of the parameters) has non-trivial dynamics since in such cases all orbits have sensitive dependence on initial conditions. The tool to see it is a natural extension of the notion of Lyapunov exponents from autonomous to non-autonomous systems. In particular, we prove that such complicated behaviour can be obtained when all parameters are fixed and only the initial phase of the perturbation is changed. It also proves that sensitive dependence on initial conditions can be independent of the waveform of the perturbation which depends on the elliptic modulus value. This case is found relevant in the setting of differential equations.