A simple chaotic and hyperchaotic time-delay system: design and electronic circuit implementation

Abstract

The present paper reports the design, analysis and experimental implementation of a simple first-order nonlinear time-delay dynamical system, which is capable of showing chaotic and hyperchaotic oscillations even at low values of intrinsic time delay. The system consists of a nonlinearity that has a closed-form mathematical function, which enables one to derive exact stability and bifurcation conditions. A detailed stability and bifurcation analyses reveal that a limit cycle is born via a supercritical Hopf bifurcation. Also, we observe both mono-scroll and double-scroll chaotic and hyperchaotic attractors even for a small value of time delay. The complexity of the system is characterized by bifurcation diagram and Lyapunov exponent spectrum. We implement the system in an electronic circuit, and a data acquisition (DAQ) system with LabView environment is used to visualize the experimental bifurcation diagram along with the other characteristic diagrams, namely time series waveform, phase plane plots and frequency spectra. Experimental observations agree well with the analytical and numerical results.