A Novel 5D Chaotic System with Extreme Multi-stability and a Line of Equilibrium and Its Engineering Applications: Circuit Design and FPGA Implementation

Extreme multi-stability is a newly introduced property observed in nonlinear dynamical systems. Such systems have very rich dynamical solutions depending on both parameters and initial conditions. On the other hand, designing dynamical systems with special features related to their equilibria is of great interest. In this paper, a novel chaotic system with extreme multi-stability and a line of equilibrium is presented. Such systems are so infrequent. It also should be noted that this designed chaotic system belongs to the category of dynamical systems with hidden attractors. Complete dynamical properties of this new system are investigated. Also, by the assistance of FPGA and electronic circuit implementation, this system is implemented.

[1]  Karthikeyan Rajagopal,et al.  An Exponential Jerk System: Circuit Realization, Fractional Order and Time Delayed Form with Dynamical Analysis and Its Engineering Application , 2019, J. Circuits Syst. Comput..

[2]  N. Kuznetsov,et al.  The Lyapunov dimension and its estimation via the Leonov method , 2016, 1602.05410.

[3]  Viet-Thanh Pham,et al.  A New Four-Dimensional Chaotic System With No Equilibrium Point , 2019 .

[4]  Viet-Thanh Pham,et al.  Taking control of initiated propagating wave in a neuronal network using magnetic radiation , 2018, Appl. Math. Comput..

[5]  Viet-Thanh Pham,et al.  A New Five Dimensional Multistable Chaotic System With Hidden Attractors , 2019, Recent Advances in Chaotic Systems and Synchronization.

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

[7]  Guangyi Wang,et al.  Coexisting Oscillation and Extreme Multistability for a Memcapacitor-Based Circuit , 2017 .

[8]  Chun-Lai Li,et al.  Adaptive Sliding Mode Control for Synchronization of a Fractional-Order Chaotic System , 2013 .

[9]  E. Tlelo-Cuautle,et al.  Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs , 2016 .

[10]  Viet-Thanh Pham,et al.  Is that Really Hidden? The Presence of Complex Fixed-Points in Chaotic Flows with No Equilibria , 2014, Int. J. Bifurc. Chaos.

[11]  Julien Clinton Sprott,et al.  Simple chaotic 3D flows with surfaces of equilibria , 2016 .

[12]  Awadhesh Prasad,et al.  Perpetual Points: New Tool for Localization of Coexisting Attractors in Dynamical Systems , 2017, Int. J. Bifurc. Chaos.

[13]  Sajad Jafari,et al.  A new chaotic model for glucose-insulin regulatory system , 2018, Chaos, Solitons & Fractals.

[14]  Guangyi Wang,et al.  Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria , 2018, Commun. Nonlinear Sci. Numer. Simul..

[15]  Tomasz Kapitaniak,et al.  Evaluation of the largest Lyapunov exponent in dynamical systems with time delay , 2005 .

[16]  Nikolay V. Kuznetsov,et al.  Control of multistability in hidden attractors , 2015 .

[17]  Abbas Dandache,et al.  A new auto-switched chaotic system and its FPGA implementation , 2013, Commun. Nonlinear Sci. Numer. Simul..

[18]  Kehui Sun,et al.  Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems , 2018, Entropy.

[19]  Karthikeyan Rajagopal,et al.  FPGA implementation of fractional-order discrete memristor chaotic system and its commensurate and incommensurate synchronisations , 2017 .

[20]  J. Sprott.  Elegant Chaos: Algebraically Simple Chaotic Flows , 2010 .

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

[22]  T. N. Mokaev,et al.  Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system , 2015, 1504.04723.

[23]  Robert C. Hilborn,et al.  Chaos And Nonlinear Dynamics: An Introduction for Scientists and Engineers , 1994 .

[24]  Sajad Jafari,et al.  Fractional Order Synchronous Reluctance Motor: Analysis, Chaos Control and FPGA Implementation , 2018 .

[25]  Nikolay V. Kuznetsov,et al.  Hidden attractor in smooth Chua systems , 2012 .

[26]  Robert C. Hilborn,et al.  Chaos and Nonlinear Dynamics , 2000 .

[27]  B. Bao,et al.  Multistability in Chua's circuit with two stable node-foci. , 2016, Chaos.

[28]  Ainuddin Wahid Abdul Wahab,et al.  Synchronization in coupled Ikeda delay systems , 2014 .

[29]  Binoy Krishna Roy,et al.  New family of 4-D hyperchaotic and chaotic systems with quadric surfaces of equilibria , 2018 .

[30]  Nikolay V. Kuznetsov,et al.  Hidden attractors in Dynamical Systems. From Hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits , 2013, Int. J. Bifurc. Chaos.

[31]  Bocheng Bao,et al.  Crisis‐induced coexisting multiple attractors in a second‐order nonautonomous memristive diode bridge‐based circuit , 2018, Int. J. Circuit Theory Appl..

[32]  Nikolay V. Kuznetsov,et al.  Hidden chaotic sets in a Hopfield neural system , 2017 .

[33]  Julien Clinton Sprott,et al.  Simple Chaotic Flows with a Curve of Equilibria , 2016, Int. J. Bifurc. Chaos.

[34]  Viet-Thanh Pham,et al.  A hyperchaotic memristor oscillator with fuzzy based chaos control and LQR based chaos synchronization , 2018, AEU - International Journal of Electronics and Communications.

[35]  Mohammed F. Tolba,et al.  FPGA implementation of fractional-order integrator and differentiator based on Grünwald Letnikov's definition , 2017, 2017 29th International Conference on Microelectronics (ICM).

[36]  Przemyslaw Perlikowski,et al.  Multistability in nonlinearly coupled ring of Duffing systems , 2016 .

[37]  Nikolay V. Kuznetsov,et al.  Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity , 2015, Commun. Nonlinear Sci. Numer. Simul..

[38]  Bashir Ahmad,et al.  Chaos and multi-scroll attractors in RCL-shunted junction coupled Jerk circuit connected by memristor , 2018, PloS one.

[39]  Nikolay V. Kuznetsov,et al.  Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor , 2014, Commun. Nonlinear Sci. Numer. Simul..

[40]  Sajad Jafari,et al.  Time-delayed chameleon: Analysis, synchronization and FPGA implementation , 2017, Pramana.

[41]  Julien Clinton Sprott,et al.  Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping , 2017 .

[42]  Nikolay V. Kuznetsov,et al.  Time-Varying Linearization and the Perron Effects , 2007, Int. J. Bifurc. Chaos.

[43]  Viet-Thanh Pham,et al.  A new nonlinear oscillator with infinite number of coexisting hidden and self-excited attractors , 2018 .

[44]  Binoy Krishna Roy,et al.  Multistability and hidden chaotic attractors in a new simple 4-D chaotic system with chaotic 2-torus behaviour , 2018 .

[45]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[46]  Zhijun Li,et al.  Realization of current-mode SC-CNN-based Chua’s circuit , 2017 .

[47]  Nikolay V. Kuznetsov,et al.  Hidden and self-excited attractors in Chua circuit: synchronization and SPICE simulation , 2018, Int. J. Parallel Emergent Distributed Syst..

[48]  Awadhesh Prasad,et al.  Controlling Dynamics of Hidden Attractors , 2015, Int. J. Bifurc. Chaos.

[49]  Julien Clinton Sprott,et al.  Infinite Multistability in a Self-Reproducing Chaotic System , 2017, Int. J. Bifurc. Chaos.

[50]  Viet-Thanh Pham,et al.  A Modified Multistable Chaotic Oscillator , 2018, Int. J. Bifurc. Chaos.

[51]  Nikolay V. Kuznetsov,et al.  Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE , 2017, Commun. Nonlinear Sci. Numer. Simul..

[52]  Xiaofeng Liao,et al.  Synchronization and chaos in coupled memristor-based FitzHugh-Nagumo circuits with memristor synapse , 2017 .

[53]  Sajad Jafari,et al.  Effects of different initial conditions on the emergence of chimera states , 2018, Chaos, Solitons & Fractals.

[54]  Akif Akgul,et al.  Chaos-based engineering applications with a 3D chaotic system without equilibrium points , 2015, Nonlinear Dynamics.

[55]  Sajad Jafari,et al.  Experimental Observations and Circuit Realization of a Jerk Chaotic System With Piecewise Nonlinear Function , 2019 .

[56]  Zhouchao Wei,et al.  Dynamical behaviors of a chaotic system with no equilibria , 2011 .

[57]  Sajad Jafari,et al.  Chaotic chameleon: Dynamic analyses, circuit implementation, FPGA design and fractional-order form with basic analyses , 2017 .

[58]  Guanrong Chen,et al.  Twin birds inside and outside the cage , 2018, Chaos, Solitons & Fractals.

[59]  Julien Clinton Sprott,et al.  Simple Chaotic flows with One Stable equilibrium , 2013, Int. J. Bifurc. Chaos.

[60]  Julien Clinton Sprott,et al.  Coexisting Hidden Attractors in a 4-D Simplified Lorenz System , 2014, Int. J. Bifurc. Chaos.

[61]  Guanrong Chen,et al.  Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system , 2015, 1511.07765.

[62]  U. Feudel,et al.  Control of multistability , 2014 .

[63]  Mustapha Djeddou,et al.  An FPGA Real-time Implementation of the Chen's Chaotic System for Securing Chaotic Communications , 2009 .