On the dynamics of single amplifier biquad based inductor-free hyperchaotic oscillators: a case study

We investigate the dynamics of an inductor-free hyperchaotic oscillator introduced by Giannakopoulos and Deliayannis (Int J Electron 92:143–159, 2005). This oscillator is constructed by coupling an LC oscillator to the Deliyannis single amplifier biquad, which has been turned into an oscillator, by means of a semiconductor diode. Based on the Shockley exponential diode equation, we propose a smooth mathematical model for a better description of both the regular and chaotic dynamics of the hyperchaos generator. Various bifurcation diagrams and corresponding graphs of Lyapunov exponents are plotted to summarize different transitions to chaos/hyperchaos. One of the most striking and interesting finding of this work is the existence of two completely different families of solutions/attractors (depending on the choice of circuit parameters’ values) related to the classical period-doubling route to chaos or to the torus breakdown transition to chaos (which finally gives rise to a hyperchaotic attractor) when monitoring a single circuit component (i.e. the negative resistor). The physical implementation of the oscillator is considered to validate the theoretical analysis through several comparisons between experimental and numerical results. The results of present investigations may be exploited for rigorous designs of such types of oscillators in various engineering applications as well as for educational purposes.