Nonsmooth Bifurcations, Transient Hyperchaos and Hyperchaotic Beats in a Memristive Murali–lakshmanan–chua Circuit

Abstract

In this paper, a memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuity boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise-smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity-induced nonsmooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor-aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyperchaotic nature. In addition, the circuit admits a codimension-5 bifurcation and transient hyperchaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0–1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincaré Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.

Bursting mechanism in a memristive Lorenz based system and function projective synchronization in its-fractional-order form: Digital implementation under ATmega328P microcontroller

“…Transient chaos is an usual dynamics existing in many nonlinear dynamical systems [53,54] where in an orbit, it behaves alternatively for a finite time interval between chaotic and periodic state by setting into a final nonchaotic state. This phenomenon can also be seen as intermittent chaos when multiple alternative sequences of periodic and chaotic states are observed [53,54].…”

“…Transient chaos is an usual dynamics existing in many nonlinear dynamical systems [53,54] where in an orbit, it behaves alternatively for a finite time interval between chaotic and periodic state by setting into a final nonchaotic state. This phenomenon can also be seen as intermittent chaos when multiple alternative sequences of periodic and chaotic states are observed [53,54]. When system parameters are fixed as : = a 0.5; = a 1.89; f m , c, b, d)=(0.5, 1.89, 0.6, 40.9, 5.9, 0.2, −15.5) and initial conditions given by (x 1 , x 2 , x 3 , x 4 )=(0.0034, 0.003, 0.004, 1.4).…”

A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization

“…In recent years, generation and nonlinear analyses of memristor-based chaotic systems have attracted more and more attention, and become a research hotspot. By replacing Chua's diodes with memristors, adding memristors to existing nonlinear systems, and designing new systems with memristors, some new chaotic and hyperchaotic systems have been constructed, such as a memristor-based Chua's circuit [1][2][3][4][5][6], memristor-based van der Pol oscillator [7], memristor-based Lorenz system [8][9][10], memristor-based Chen system [11], memristor-based Lü system [12], etc. Theoretical analyses, numerical simulations and circuit experiments consistently indicate that such memristor-based systems can exhibit complex dynamical behaviours including various bifurcation [1][2][3][4][5][6][7][8][9][10][11][12], chaos [1][2][3][4][5][6][7][8][9][10][11][12], hyperchaos [12][13][14], coexistence of attractors [15], transient dynamical behaviors [16], and initial state dependent dynamical behaviors [17].…”

Memristor Based van der Pol Oscillation Circuit

“…In [Ishaq Ahamed et al, 2011] andLakshmanan, 2013], nonautonomous systems, namely the driven memristive Chua's circuit and the Murali-Lakshmanan-Chua (MLC) circuit were built by replacing the Chua's diode with a threesegment piecewise-linear active flux-controlled memristor, respectively. Systems containing periodically forcing terms are a class of nonautonomous systems.…”