Complex Dynamical Behaviors of a Fractional-Order System Based on a Locally Active Memristor

A fractional-order locally active memristor is proposed in this paper. When driven by a bipolar periodic signal, the generated hysteresis loop with two intersections is pinched at the origin. The area of the hysteresis loop changes with the fractional order. Based on the fractional-order locally active memristor, a fractional-order memristive system is constructed. The stability analysis is carried out and the stability conditions for three equilibria are listed. The expression of the fractional order related to Hopf bifurcation is given. The complex dynamical behaviors of Hopf bifurcation, period-doubling bifurcation, bistability and chaos are shown numerically. Furthermore, the bistability behaviors of the different fractional order are validated by the attraction basins in the initial value plane. As an alternative to validating our results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. The research results clarify that the complex dynamical behaviors are attributed to two facts: one is the fractional order that affects the stability of the equilibria, and the other is the local activeness of the fractional-order memristor.

[1]  Bocheng Bao,et al.  Extreme multistability in a memristive circuit , 2016 .

[2]  Leon O. Chua,et al.  Edge of Chaos and Local Activity Domain of the Brusselator CNN , 1998 .

[3]  Leon O. Chua,et al.  Edge of Chaos and Local Activity Domain of FitzHugh-Nagumo Equation , 1998 .

[4]  Leon O. Chua,et al.  Local Activity is the Origin of Complexity , 2005, Int. J. Bifurc. Chaos.

[5]  L. Chua,et al.  The double scroll family , 1986 .

[6]  Bocheng Bao,et al.  Multiple attractors in a non-ideal active voltage-controlled memristor based Chua's circuit , 2016 .

[7]  Sergey P. Kuznetsov,et al.  Co-existing hidden attractors in a radio-physical oscillator system , 2015 .

[8]  Leon O. Chua,et al.  Simplest Chaotic Circuit , 2010, Int. J. Bifurc. Chaos.

[9]  Jacques Kengne,et al.  Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit , 2016 .

[10]  Leon O. Chua,et al.  The Four-Element Chua's Circuit , 2008, Int. J. Bifurc. Chaos.

[11]  Mo Chen,et al.  Three-Dimensional Memristive Hindmarsh-Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors , 2018, Complex..

[12]  Hilaire Bertrand Fotsin,et al.  Dynamics, Analysis and Implementation of a Multiscroll Memristor-Based Chaotic Circuit , 2016, Int. J. Bifurc. Chaos.

[13]  Dalibor Biolek,et al.  Interpreting area of pinched memristor hysteresis loop , 2014 .

[14]  M. I. Shehata,et al.  On stability, persistence, and Hopf bifurcation in fractional order dynamical systems , 2008, 0801.1189.

[15]  Giuseppe Grassi,et al.  On the simplest fractional-order memristor-based chaotic system , 2012 .

[16]  E. Ahmed,et al.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models , 2007 .

[17]  Jacques Kengne,et al.  Reversals of period doubling, coexisting multiple attractors, and offset boosting in a novel memristive diode bridge-based hyperjerk circuit , 2018, Analog Integrated Circuits and Signal Processing.

[18]  Guangyi Wang,et al.  A Locally Active Memristor and Its Application in a Chaotic Circuit , 2018, IEEE Transactions on Circuits and Systems II: Express Briefs.

[19]  Qingdu Li,et al.  Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria , 2015 .

[20]  Weihua Deng,et al.  Remarks on fractional derivatives , 2007, Appl. Math. Comput..

[21]  Z. Njitacke Tabekoueng,et al.  Periodicity, chaos, and multiple attractors in a memristor-based Shinriki's circuit. , 2015, Chaos.

[22]  M. Lakshmanan,et al.  Nonsmooth bifurcations, Transient Hyperchaos and hyperchaotic beats in a Memristive Murali-Lakshmanan-Chua Circuit , 2013, Int. J. Bifurc. Chaos.

[23]  Lili Zhou,et al.  Generating hyperchaotic multi-wing attractor in a 4D memristive circuit , 2016, Nonlinear Dynamics.

[24]  D. Stewart,et al.  The missing memristor found , 2008, Nature.

[25]  Nikolay V. Kuznetsov,et al.  Matlab Code for Lyapunov Exponents of Fractional-Order Systems , 2018, Int. J. Bifurc. Chaos.

[26]  Bocheng Bao,et al.  Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator , 2018 .

[27]  Jürgen Kurths,et al.  Optimal Synchronization of a Memristive Chaotic Circuit , 2016, Int. J. Bifurc. Chaos.

[28]  Zhonglin Wang,et al.  A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium , 2015 .

[29]  Guanrong Chen,et al.  Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses , 2016 .

[30]  Ivo Petrás,et al.  Fractional-Order Memristor-Based Chua's Circuit , 2010, IEEE Transactions on Circuits and Systems II: Express Briefs.

[31]  Huagan Wu,et al.  Coexisting infinitely many attractors in active band-pass filter-based memristive circuit , 2016 .

[32]  Huagan Wu,et al.  Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator , 2016 .

[33]  Anirban Ray,et al.  Memristive non-linear system and hidden attractor , 2015, The European Physical Journal Special Topics.

[34]  L. Chua.  Memristor-The missing circuit element , 1971 .