On the dynamics of a simplified canonical Chua’s oscillator with smooth hyperbolic sine nonlinearity: Hyperchaos, multistability and multistability control

Abstract

A simplified hyperchaotic canonical Chua’s oscillator (referred as SHCCO hereafter) made of only seven terms and one nonlinear function of type hyperbolic sine is analyzed. The system is found to be self-excited, and bifurcation tools associated with the spectrum of Lyapunov exponents reveal the rich dynamical behaviors of the system including hyperchaos, torus, period-doubling route to chaos, and hysteresis when turning the system control parameters. Wide ranges of hyperchaotic dynamics are highlighted in various two-parameter spaces based on two-parameter Lyapunov diagrams. The analysis of the hysteretic window using a basin of attraction as argument reveals that the SHCCO exhibits three coexisting attractors. Laboratory measurements further confirm the performed numerical investigations and henceforth validate the mathematical model. Of most/particular interest, multistability observed in the SHCCO is further controlled based on a linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the asymmetric pair of coexisting attractors. For higher values of the coupling strength, only a unique symmetric periodic attractor survives.

Image encryption using a novel quintic jerk circuit with adjustable symmetry

“…We present in Figure 12 the phase portrait of the only remaining attractor and its corresponding basin of attraction computed when . We would like to stress that the successful control of the quintic jerk oscillator from the state of four coexisting attractors to a monostable state based on the linear augmentation scheme achieved in this work is reported for the first time according to the relevant state of the art 14–18 and thus deserves dissemination.…”

“…It is known that multistability (or the coexistence of multiple attractors for the same parameters setting; obtained by changing only the initial states) often leads to disastrous performances of the investigated system by spoiling its reliability and reproducibility 14–18 . This has been observed, for example, in laser systems where multistability intracavity second harmonic generation leads to the well‐known green problem 19,20 .…”

“…This has been observed, for example, in laser systems where multistability intracavity second harmonic generation leads to the well‐known green problem 19,20 . Thus, in order to address this issue, a control method called linear augmentation has recently been introduced and described in previous studies 14–18 . It makes it possible to control the dynamics of a nonlinear system without disturbing the system parameters.…”

“…It makes it possible to control the dynamics of a nonlinear system without disturbing the system parameters. According to this approach, a nonlinear dynamic system is coupled to a linear one 14–18 . The motivations of this coupling scheme are twofold: (1) to stabilize the steady state in a given nonlinear oscillator and (2) to reduce the number of coexisting attractors for a multistable system.…”

Control of Coexisting Attractors with Preselection of the Survived Attractor in Multistable Chua’s System: A Case Study

“…In the study of nonlinear dynamic systems, the simultaneous existence of attractors (finite or infinite), also known as multistability [1][2][3][4][5][6][7][8][9][10][11][12][13], extreme multistability [14][15][16], or megastability [17], is now in the forefront. Recall that the famous Chua's circuit is among the widely studied electronic circuits capable to display chaos [18].…”