Analytical Study and Experimental Confirmation of SNA Through Poincaré Maps in a Quasiperiodically Forced Electronic Circuit

Abstract

Quasiperiodically forced series LCR circuit with simple nonlinear element is studied analytically and experimentally. To the best of our knowledge, this is the first time that strange nonchaotic attractors (SNAs) are studied analytically. From the explicit analytical solution, the bifurcation process is shown. With a single negative conduction region of the nonlinear element two routes namely, Heagy-Hammel and fractalization routes to the birth of SNA are identified. The analytical analysis are confirmed by laboratory hardware experiments. In addition, for the first time, a detailed stroboscopic Poincaré map is generated experimentally for two different frequencies, for the above two routes, which clearly confirm the presence of SNAs in these two routes. Also, from the experimental data of the corresponding attractors, we quantitatively confirm the presence of SNAs through singular-continuous spectrum analysis. The analytical results as well as experimental observations are characterized qualitatively in terms of phase portraits, Poincaré map, power spectrum, and sensitivity dependance on initial conditions.

Energy computation and multistability in a class of second-order chaotic systems with simple nonlinearities: numerical, experimental and analytical results

“…The considered simple Sprott type nonlinearities help in validating the proposed approach. Analytical solutions for the state equations of second-order non-autonomous circuits with piecewise-linear elements exhibiting chaotic, strange non-chaotic and synchronization behavior have been reported recently [32][33][34][35][36]. The evolution of chaotic dynamics in second-order systems with different simple nonlinearities discussed in the present work is studied through explicit analytical solutions.…”

Numerical studies on the synchronization of a network of mutually coupled simple chaotic systems

“…An intermediate dynamical state between periodic and chaotic motion without any sensitive dependence on initial conditions and with a fractal nature namely, the Strange Non-chaotic Attractor (SNA), has been observed by Grebogi et al [7]. Several simple chaotic systems have also been found to exhibit SNA behavior upon quasiperiodic forcing [8][9][10][11]. The application of a SNA observed in a a second-order chaotic system, for computation, has been reported recently [12].…”

An analytical study on the synchronization of strange non-chaotic attractors

“…This is achieved by finding a solution to the normalized state variables of the difference system given by Eq. (11). The solution of those equations are, [x * (t; t 0 , x * 0 , y * 0 ), y * (t; t 0 , x * 0 , y * 0 )] T for which the initial conditions are written as (t, x * , y * ) = (t 0 , x * 0 , y * 0 ).…”

“…Different routes, such as the Heagy-Hammel or torus doubling, fractalization, intermittency, blow out bifurcation routes, to SNA have been identified [2][3][4][5]. A good number of nonlinear systems and electronic circuits exhibiting SNAs in their dynamics have been studied numerically and experimentally [6][7][8][9][10][11][12] while a few systems have been studied analytically [10,11,13]. The phenomenon of chaos synchronization finding potential applications in secure communication has been enchanting researchers after the Master-Slave concept introduced by Pecora and Carroll [14,15].…”