# A Symmetric Controllable Hyperchaotic Hidden Attractor

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- I)
- There exists a parameter to control amplitude and frequency of signals in a small range.
- II)
- Amplitude of x and y can be controlled simultaneously.
- III)
- There is an offset boosting controller.
- IV)
- A special parameter can realize both amplitude and offset control of one system variable.

## 2. Model Description

_{KY}= 3.3256 under initial conditions (1, −1, −1, 1), as shown in Figure 2.

## 3. Basic Dynamic Analysis

#### 3.1. Anylis of Equilibria

^{2}, which means that there is no real solution, correspondingly the hyperchaotic attractor of system (2) is hidden.

#### 3.2. Bifurcation Analysis

#### 3.3. Amplitude Control

#### 3.4. Offset Boosting

#### 3.5. Mixed Geometric Control

## 4. Bistability Analysis

## 5. Circuit Implementation

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Li, C.; Sprott, J.C. Amplitude control approach for chaotic signals. Nonlinear Dyn.
**2013**, 73, 1335–1341. [Google Scholar] [CrossRef] [Green Version] - Li, C.; Sprott, J.C. Finding coexisting attractors using amplitude control. Nonlinear Dyn.
**2014**, 78, 2059–2064. [Google Scholar] [CrossRef] - Chen, H.; Bayani, A.; Akgul, A.; Jafari, M.-A.; Pham, V.-T.; Wang, X.; Jafari, S. A flexible chaotic system with adjustable amplitude, largest Lyapunov exponent, and local Kaplan–Yorke dimension and its usage in engineering applications. Nonlinear Dyn.
**2018**, 92, 1791–1800. [Google Scholar] [CrossRef] - Wang, C.; Liu, X.; Xia, H. Multi-piecewise quadratic nonlinearity memristor and its 2N-scroll and 2N + 1-scroll chaotic attractors system. Chaos Interdiscip. J. Nonlinear Sci.
**2017**, 27, 033114. [Google Scholar] [CrossRef] [PubMed] - Hu, W.; Akgul, A.; Li, C.; Zheng, T.; Li, P. A switchable chaotic oscillator with two amplitude-frequency controllers. J. Circuits Syst. Comput.
**2017**, 26, 1750158. [Google Scholar] [CrossRef] - Li, C.; Sprott, J.C.; Akgul, A.; Herbert, H.C.; Iu, H.H.C.; Zhao, Y. A new chaotic oscillator with free control. Chaos
**2017**, 27, 083101. [Google Scholar] [CrossRef] [PubMed] - Li, C.; Sprott, J.C.; Yuan, Z.; Li, H. Constructing chaotic systems with total amplitude control. Int. J. Bifurc. Chaos
**2015**, 25, 1530025. [Google Scholar] [CrossRef] - Li, C.; Sprott, J.C.; Xing, H. Constructing chaotic systems with conditional symmetry. Nonlinear Dyn.
**2017**, 87, 1351–1358. [Google Scholar] [CrossRef] - Li, C.; Sprott, J.C. Variable-boostable chaotic flows. Optik—Int. J. Light Electron Opt.
**2016**, 127, 10389–10398. [Google Scholar] [CrossRef] - Leonov, G.A.; Vagaitsev, V.I.; Kuznetsov, N.V. Localization of hidden Chua’s attractors. Phys. Lett. A
**2011**, 375, 2230. [Google Scholar] [CrossRef] - Leonov, G.A.; Vagaitsev, V.I.; Kuznetsov, N.V. Hidden attractor in smooth Chua systems. Phys. D
**2012**, 241, 1482. [Google Scholar] [CrossRef] - Rocha, R.; Ruthiramoorthy, J.; Kathamuthu, T. Memristive oscillator based on Chua’s circuit: stability analysis and hidden dynamics. Nonlinear Dyn.
**2017**, 88, 2577–2587. [Google Scholar] [CrossRef] - Bao, B.; Xu, Q.; Bao, H.; Chen, M. Extreme multistability in a memristive circuit. Electron. Lett.
**2016**, 52, 1008–1010. [Google Scholar] [CrossRef] - Lai, Q.; Nestor, T.; Kengne, J.; Zhao, X. Coexisting attractors and circuit implementation of a new 4D chaotic system with two equilibria. Chaos Solitons Fractals
**2018**, 107, 92–102. [Google Scholar] [CrossRef] - Wang, G.Y.; Zheng, Y.; Liu, J.B. A hyperchaotic Lorenz attractor and its circuit implementation. Acta Phys. Sin.
**2007**, 56, 3113–3120. [Google Scholar] - Jafari, S.; Sprott, J.C. Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals
**2013**, 57, 79–84. [Google Scholar] [CrossRef] - Bao, H.; Wang, N.; Bao, B.C.; Chen, M.; Jin, P.P.; Wang, G.Y. Initial condition dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria. Commun. Nonlinear Sci.
**2018**, 57, 264–275. [Google Scholar] [CrossRef] - Jafari, S.; Sprott, J.C.; Molaie, M. A simple chaotic flow with a plane of equilibria. Int. J. Bifurc. Chaos
**2016**, 26, 1650098. [Google Scholar] [CrossRef] - Jafari, S.; Sprott, J.C. Elementary quadratic chaotic flows with no equilibria. Phys. Lett. Sect. A Gen. Atomic Solid State Phys.
**2013**, 377, 699–702. [Google Scholar] [CrossRef] - Bao, B.C.; Bao, H.; Wang, N.; Chen, M.; Xu, Q. Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals
**2017**, 94, 102–111. [Google Scholar] [CrossRef] - Munmuangsaen, B.; Srisuchinwong, B. A hidden chaotic attractor in the classical lorenz system. Chaos Solitons Fractals
**2018**, 107, 61–66. [Google Scholar] [CrossRef] - Lai, Q.; Chen, S.M. Research on a new 3d autonomous chaotic system with coexisting attractors. Optik—Int. J. Light Electron Opt.
**2016**, 127, 3000–3004. [Google Scholar] [CrossRef] - Wang, C.; Wei, Z.; Yu, P.; Zhang, W.; Yao, M. Study of hidden attractors, multiple limit cycles from hopf bifurcation and boundedness of motion in the generalized hyperchaotic rabinovich system. Nonlinear Dyn.
**2015**, 82, 131–141. [Google Scholar] - Zhou, L.; Wang, C.; Zhou, L. A novel no-equilibrium hyperchaotic multi-wing system via introducing memristor. Int. J. Circ. Theor. App.
**2018**, 46, 84–98. [Google Scholar] [CrossRef] - Wang, Z.; Cang, S.; Ochola, E.O.; Sun, Y. A hyperchaotic system without equilibrium. Nonlinear Dyn.
**2012**, 69, 531–537. [Google Scholar] [CrossRef] - Chlouverakis, K.E.; Sprott, J.C. Chaotic hyperjerk systems. Chaos Solitons Fractals
**2006**, 28, 739–746. [Google Scholar] [CrossRef] - Yuan, F.; Wang, G.; Wang, X. Extreme multistability in a memristor- based multi-scroll hyperchaotic system. Chaos
**2016**, 26, 073107. [Google Scholar] [CrossRef] - Ruan, J.Y.; Sun, K.H.; Mou, J. Memristor-based Lorenz hyper-chaotic system and its circuit implementation. Acta Phys. Sin.
**2016**, 65, 190502. [Google Scholar] - Lai, Q.; Guan, Z.; Wu, Y.; Liu, F.; Zhang, D. Generation of multi-wing chaotic attractors from a lorenz-like system. Int. J. Bifurc. Chaos
**2013**, 23, 1650177. [Google Scholar] [CrossRef] - Si, G.; Cao, H.; Zhang, Y. A new four dimensional hyperchaotic Lorenz system and its adaptive control. Chin. Phys. B
**2011**, 20, 010509. [Google Scholar] [CrossRef] - Wang, H.; Cai, G.; Miao, S.; Tian, L. Nonlinear feedback control of a novel hyperchaotic system and its circuit implementation. Chin. Phys. B
**2010**, 19, 030509. [Google Scholar] - Zhou, L.; Wang, C.; Zhou, L.L. Generating Four-Wing Hyperchaotic Attractor and Two-Wing, Three-Wing, and Four-Wing Chaotic Attractors in 4D Memristive System. Int. J. Bifurc. Chaos
**2017**, 27, 1750027. [Google Scholar] [CrossRef] - Pham, V.-T.; Volos, C.; Gambuzza, L.V. A memristive hyperchaotic system without equilibrium. Sci. World J.
**2014**, 2014, 368986. [Google Scholar] [CrossRef] [PubMed] - Xiao, J.; Ma, Z.Z.; Yang, Y.H. Dual synchronization of fractional-order chaotic systems via a linear controller. Sci. World J.
**2013**, 2013, 159194. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhou, P.; Bai, R. One adaptive synchronization approach for fractional-order chaotic system with fractional-order. Sci. World J.
**2014**, 2, 490364. [Google Scholar] [CrossRef] [PubMed] - Zhang, X.; Wang, C.H. Multiscroll hyperchaotic system with hidden attractors and its circuit implementation. Int. J. Bifurc. Chaos
**2019**, 29, 1950117. [Google Scholar] [CrossRef] - Zhang, S.; Zeng, Y.C.; Li, Z.J.; Wang, M.J.; Xiong, L. Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability. Chaos
**2018**, 28, 013113. [Google Scholar] [CrossRef] - Wang, Z.L.; Ma, J.; Cang, S.J.; Wang, Z.H.; Chen, Z.Q. Simplified hyper-chaotic systems generating multi-wing non-equilibrium attractor. Optik
**2016**, 127, 2424–2431. [Google Scholar] [CrossRef] - Cang, S.; Wang, Z.; Chen, Z.; Jia, H. Analytical and numerical investigation of a new lorenz-like chaotic attractor with compound structures. Nonlinear Dyn.
**2014**, 75, 745–760. [Google Scholar] [CrossRef] - Li, C.; Wang, X.; Chen, G. Diagnosing multistability by offset boosting. Nonlinear Dyn.
**2017**, 90, 1334–1341. [Google Scholar] [CrossRef]

**Figure 2.**Hyperchaotic attractor of system (2) with a = 5, b = 4, c = 1, k = 0.5, m = 1 and initial conditions [1, −1, −1, 1]: (

**a**) x-y plane, (

**b**) x-z plane, (

**c**) y-z plane, (

**d**) x-u plane.

**Figure 4.**Dynamical behavior of system (2) with b = 4, c = 1, k = 0.5, m = 1 under initial conditions [1, −1, −1, 1]: (

**a**) Lyapunov exponents, (

**b**) bifurcation diagram.

**Figure 5.**Typical phase trajectories of system (2) with b = 4, c = 1, k = 0.5, m = 1 under initial condition [1, −1, −1, 1] in the plane x-u: (a) a = −5 (period), (

**b**) a = −0.6 (chaos), (

**c**) a = 3 (chaos), (

**d**) a = 5 (hyperchaos).

**Figure 6.**Dynamical behavior of system (2) with a = 5, c = 1, k = 0.5, m = 1under initial condition [1, −1, −1, 1]: (

**a**) Lyapunov exponents, (

**b**) bifurcation diagram.

**Figure 7.**Chaotic oscillations of system (2) with c = 1, k = 0.5, m = 1 under initial condition [1, −1, −1, 1]: (

**a**) phase trajectory in x-z (b = 4), (

**b**) signal x(t), (

**c**) phase trajectory in y-u plane (a = 5), (

**d**) signal y(t).

**Figure 8.**Dynamical behavior of system (2) with a = 5, b = 4, m = 1 under initial conditions [1, −1, −1, 1]: (

**a**,

**b**): Lyapunov exponents and bifurcation diagram of c when k = 0.5, (

**c**,

**d**): Lyapunov exponents and bifurcation diagram of k when c = 1.

**Figure 9.**Rescaled variables in system (2) with a = 5, b = 4, c = 1, k = 0.5 under initial condition [1, −1, −1, 1]: (

**a**) signal x(t), (

**b**) signal y(t), (

**c**) signal u(t), (

**d**) signal z(t).

**Figure 10.**Phase trajectories of system (2) with a = 5, b = 4, c = 1, k = 0.5 under initial condition [1, −1, −1, 1]: (

**a**) x-u, (

**b**) y-z.

**Figure 11.**Dynamical evolution of system (2) with a = 5, b = 4, c = 1, k = 0.5 and initial condition [1, −1, −1, 1]: (

**a**) average values of the absolute value of chaotic signals, (

**b**) invariable Lyapunov exponents.

**Figure 12.**Typical chaotic oscillation of system (6) with a = 5, b = 4, c = 1, k = 0.5, m = 1 under initial condition [1, −1, −1, 1]: (

**a**) phase trajectory in the plane of x-u, (

**b**) waveform u(t).

**Figure 13.**Dynamical evolution of system (6) with a = 5, b = 4, c = 1, k = 0.5, m = 1 under initial conditions [1, −1, −1, 1]: (

**a**) Lyapunov exponent spectra of n, (

**b**) average values of the hyperchaotic signal.

**Figure 14.**Typical chaotic oscillation of system (2) with a = 5, b = 4, k = 0.5, m = 1 under initial conditions [1, −1, −1, 1]: (

**a**) phase trajectory in x-z, (

**b**) signal z(t), (

**c**) phase trajectory in x-y, (

**d**) signal x(t).

**Figure 15.**Dynamical evolution of system (2) with a = 5, b = 4, k = 0.5, m = 1 under initial conditions [1, −1, −1, 1]: (

**a**) Lyapunov exponent spectra of c, (

**b**) average values of the signals x, y and z.

**Figure 16.**Dynamical behaviors of system (7) with a = 5, b = 4, k = 0.5, m = 1 and initial condition [1, −1, −1, 1], when parameter d varies in [−5, 5]: (

**a**) average values of the signals, (

**b**) Lyapunov exponents.

**Figure 17.**Coexisting symmetrical chaotic attractors of system (2) with a = 5, b = 4, c = 1.3, k = 0.5, m = 1 with initial conditions IC1 = (1, −1, −1, 1) (green); IC2 = (−1, 1, −1, −1) (red).

**Figure 18.**Basins of attraction of system (2) with a = 5, b = 4, c = 1.3, k = 0.5, m = 1 in plane of z(0) = −1 and u(0) = 0.

**Figure 19.**Dynamical behavior of system (2) with a = 5, b = 4, k = 0.5, m = 1 under different initial condition (

**a**) c = 1; (

**b**) c = 1.3.

**Figure 21.**Circuit simulation of system (8) with a = 5, b = 4, c = 1.3, k = 0.5, m = 1 (green), m = 4 (red) under initial condition [1, −1, −1, 1]: (

**a**) x-u plane, (

**b**) y-z plane.

**Figure 22.**Circuit simulation of symmetric attractors in system (8) with a = 5, b = 4, c = 1.3, k = 0.5, m = 1 under initial conditions IC1= (1, −1, −1, 1)(green), IC2= (−1, 1, −1, −1)(red): (

**a**) x-y plane, (

**b**) x-z plane, (

**c**) y-z plane, (

**d**) x-u plane.

Reference | Number of Terms | Number of Equilibrium | Amplitude/Frequency Control | Amplitude/Offset Control |
---|---|---|---|---|

[15] | 9 | one | no | not mentioned |

[28] | 10 | line equilibrium | no | not mentioned |

[30] | 10 | one | no | not mentioned |

[31] | 9 | one | no | not mentioned |

[38] | 9 | no | no | not mentioned |

this work | 9 | no | yes | yes |

Parameters | Execution Interval | Amplitude Control | Frequency Control | Offset Control |
---|---|---|---|---|

a | [6.9, 23.3] | positive control with x | positive | no |

b | [3, 13] | positive control with x, y, z, u | positive | no |

c | [10, 50] | positive control with y negative control with x | no | z |

m | [0.1, 5] | Positive control with x, y, u | no | no |

n | [−30, 30] | no | no | u |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Li, C.; Lei, T.; Liu, Z.; Tao, C.
A Symmetric Controllable Hyperchaotic Hidden Attractor. *Symmetry* **2020**, *12*, 550.
https://doi.org/10.3390/sym12040550

**AMA Style**

Zhang X, Li C, Lei T, Liu Z, Tao C.
A Symmetric Controllable Hyperchaotic Hidden Attractor. *Symmetry*. 2020; 12(4):550.
https://doi.org/10.3390/sym12040550

**Chicago/Turabian Style**

Zhang, Xin, Chunbiao Li, Tengfei Lei, Zuohua Liu, and Changyuan Tao.
2020. "A Symmetric Controllable Hyperchaotic Hidden Attractor" *Symmetry* 12, no. 4: 550.
https://doi.org/10.3390/sym12040550