Controlling Hidden Dynamics and Multistability of a Class of Two-Dimensional Maps via Linear Augmentation

This paper reports the complex dynamics of a class of two-dimensional maps containing hidden attractors via linear augmentation. Firstly, the method of linear augmentation for continuous dynamical systems is generalized to discrete dynamical systems. Then three cases of a class of two-dimensional maps that exhibit hidden dynamics, the maps with no fixed point and the maps with one stable fixed point, are studied. Our numerical simulations show the effectiveness of the linear augmentation method. As the coupling strength of the controller increases or decreases, hidden attractor can be annihilated or altered to be self-excited, and multistability of the map can be controlled to being bistable or monostable.

[1]  Wei Lin,et al.  Stick-slip suppression and speed tuning for a drill-string system via proportional-derivative control , 2020, Applied Mathematical Modelling.

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

[3]  G. Leonov,et al.  Hidden attractors in dynamical systems , 2016 .

[4]  Zhouchao Wei,et al.  Hidden Hyperchaotic Attractors in a Modified Lorenz-Stenflo System with Only One Stable Equilibrium , 2014, Int. J. Bifurc. Chaos.

[5]  Awadhesh Prasad,et al.  Perpetual points and periodic perpetual loci in maps. , 2016, Chaos.

[6]  Joseph Páez Chávez,et al.  Controlling multistability in a vibro-impact capsule system , 2017 .

[7]  Yang Liu,et al.  Controlling coexisting attractors of an impacting system via linear augmentation , 2017 .

[8]  Zhouchao Wei,et al.  Dynamics and delayed feedback control for a 3D jerk system with hidden attractor , 2015 .

[9]  J. Kengne,et al.  Coexistence of firing patterns and its control in two neurons coupled through an asymmetric electrical synapse. , 2020, Chaos.

[10]  B C Bao,et al.  Initial-switched boosting bifurcations in 2D hyperchaotic map. , 2020, Chaos.

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

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

[13]  Akif Akgul,et al.  Control, electronic circuit application and fractional-order analysis of hidden chaotic attractors in the self-exciting homopolar disc dynamo , 2018, Chaos, Solitons & Fractals.

[14]  Saleh Mobayen,et al.  Design of novel adaptive sliding mode controller for perturbed Chameleon hidden chaotic flow , 2018 .

[15]  Julien Clinton Sprott,et al.  Simple Chaotic Flow with Circle and Square Equilibrium , 2016, Int. J. Bifurc. Chaos.

[16]  Lequan Min,et al.  Discrete Chaotic Systems with One-Line Equilibria and Their Application to Image Encryption , 2017, Int. J. Bifurc. Chaos.

[17]  Julien Clinton Sprott,et al.  A Simple Chaotic Flow with a Plane of Equilibria , 2016, Int. J. Bifurc. Chaos.

[18]  Phase-flip in relay oscillators via linear augmentation , 2018 .

[19]  J. Sprott,et al.  A New Category of Three-Dimensional Chaotic Flows with Identical Eigenvalues , 2020, Int. J. Bifurc. Chaos.

[20]  Qigui Yang,et al.  A new Lorenz-type hyperchaotic system with a curve of equilibria , 2015, Math. Comput. Simul..

[21]  Julien Clinton Sprott,et al.  Elementary quadratic chaotic flows with no equilibria , 2013 .

[22]  Guanrong Chen,et al.  A chaotic system with only one stable equilibrium , 2011, 1101.4067.

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

[24]  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..

[25]  Jian Ma,et al.  Simplified hyper-chaotic systems generating multi-wing non-equilibrium attractors , 2016 .

[26]  Guanrong Chen,et al.  Constructing a chaotic system with any number of equilibria , 2012, 1201.5751.

[27]  Jacques Kengne,et al.  Control of Multistability in a Self-Excited Memristive Hyperchaotic Oscillator , 2019, Int. J. Bifurc. Chaos.

[28]  Bocheng Bao,et al.  Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability , 2017 .

[29]  Julien Clinton Sprott,et al.  Simple chaotic flows with a line equilibrium , 2013 .

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

[31]  P. Brzeski,et al.  Sample-Based Methods of Analysis for Multistable Dynamical Systems , 2018, Archives of Computational Methods in Engineering.

[32]  Kehui Sun,et al.  SF-SIMM high-dimensional hyperchaotic map and its performance analysis , 2017 .

[33]  G. D. Leutcho,et al.  Multistability Control of Hysteresis and Parallel Bifurcation Branches through a Linear Augmentation Scheme , 2019 .

[34]  Julien Clinton Sprott,et al.  Bistability in a hyperchaotic system with a line equilibrium , 2014 .

[35]  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.

[36]  Awadhesh Prasad,et al.  Controlling bistability by linear augmentation , 2013 .

[37]  Yang Liu,et al.  Torsional stick-slip vibrations and multistability in drill-strings , 2019 .

[38]  Viet-Thanh Pham,et al.  The Relationship Between Chaotic Maps and Some Chaotic Systems with Hidden Attractors , 2016, Int. J. Bifurc. Chaos.

[39]  Ping Zhou,et al.  Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points , 2014 .

[40]  Awadhesh Prasad,et al.  Controlling dynamical behavior of drive-response system through linear augmentation , 2014 .

[41]  Viet-Thanh Pham,et al.  Entropy Analysis and Neural Network-Based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors , 2019, Entropy.

[42]  Haibo Jiang,et al.  A New Class of Three-Dimensional Maps with Hidden Chaotic Dynamics , 2016, Int. J. Bifurc. Chaos.

[43]  Danylo Dantsev.  A Novel Type of Chaotic Attractor for Quadratic Systems Without Equilibriums , 2018, Int. J. Bifurc. Chaos.

[44]  Irene M. Moroz,et al.  Degenerate Hopf bifurcations, hidden attractors, and control in the extended Sprott E system with only one stable equilibrium , 2014 .

[45]  G. D. Leutcho,et al.  On the dynamics of a simplified canonical Chua's oscillator with smooth hyperbolic sine nonlinearity: Hyperchaos, multistability and multistability control. , 2019, Chaos.

[46]  Zhouchao Wei,et al.  Hidden chaotic attractors in a class of two-dimensional maps , 2016 .

[47]  Guang Zeng,et al.  Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation , 2014, Int. J. Circuit Theory Appl..

[48]  Qiang Lai,et al.  Coexisting attractors and circuit implementation of a new 4D chaotic system with two equilibria , 2018 .

[49]  Julien Clinton Sprott,et al.  Two Simplest Quadratic Chaotic Maps Without Equilibrium , 2018, Int. J. Bifurc. Chaos.

[50]  Zhouchao Wei,et al.  Delayed feedback control and bifurcation analysis of the generalized Sprott B system with hidden attractors , 2015 .

[51]  Rajat Karnatak,et al.  Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application , 2015, PloS one.

[52]  Awadhesh Prasad,et al.  Targeting fixed-point solutions in nonlinear oscillators through linear augmentation. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Sen Zhang,et al.  Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability. , 2018, Chaos.

[54]  Haibo Jiang,et al.  A New Class of Two-Dimensional Chaotic Maps with Closed Curve Fixed Points , 2019, Int. J. Bifurc. Chaos.

[55]  Christos Volos,et al.  Systems with Hidden Attractors , 2017 .