Dynamical Behavior of Chua's Circuit With Lossless Transmission Line

Recently, by replacing the parallel LC “resonator” in Chua's circuit with a lossless transmission line that is terminated by a short circuit, a nonlinear transmission line oscillator is proposed and is named as a “time-delayed Chua's circuit”. In a transmission line oscillator, a linear wave travels along a piece of the transmission line and interacts with terminating electrical components, i.e., a boundary-value problem for a hyperbolic partial differential equation with boundary conditions. Furthermore, a fixed time delay arises due to the propagation time through the transmission line. Delay-induced dynamics are particularly important in technologies where high-speed computers must take into account transmission line effects including propagation delay to prevent undesirable self-oscillations. In this paper, the equation describing Chua's circuit with a lossless transmission line is reduced to a differential-difference equation with one delay (i.e., neutral-type differential equation). By using suitable Lyapunov functionals, we reduce the problem of stability and uniform ultimate boundedness to a scalar ordinary differential inequality. Soon afterwards, we derive conditions for global exponential stability of an equilibrium point. The oscillation of small amplitude in the system are studied by perturbation theory, and a general approach is presented for equations of this type for finding the expansion of the oscillation to any order in terms of the coefficient of the fundamental frequency. The frequency-amplitude relations are derived to second order. Finally, we also give some numerical examples to verify the correctness of our analytic results by computer simulations.