Qualitative analysis of the dynamics of the time-delayed Chua´s circuit

Several variants of the Chua´s circuit have been recently proposed in order to enlarge the class of nonlinear phenomena that can be generated by relatively simple circuits. In particular, Sharkovsky et al. proposed the so called time-delayed Chua´s circuit (TDCC), where the original lumped LC resonator is substituted by an ideal transmission line, thereby generating an infinite dimensional system, The TDCC has been studied in details in the absence of the capacitor C, the only lumped dynamic element left in the circuit. This paper studies the effects of the presence of C on the dynamics of the circuit. After recasting the circuit equations in a suitable normalized form, their characteristic equation is theoretically investigated and the regions in the parameter space where all the eigenvalues have negative real part are exactly evaluated along with all the possible qualitative eigenvalue distributions. This analysis allows for a qualitative description of the TDCC dynamics in presence of the capacitor C. In particular, it is shown that, for particular sets of circuit parameters, an even small value of C, e.g., a parasitic element, may completely change the behavior of TDCC and that, on the other hand, any TDCC, exhibiting the period-adding phenomenon for C=0, stilt continues to present this phenomenon even if a small capacitor C is added to the circuit