# A Hidden Chaotic System with Multiple Attractors

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Descriptions

#### 2.1. The Description of the New System’s Mathematical Model

#### 2.2. Description of Chaotic Behavior

_{1}= 0.053671, LE

_{2}= −0.0049675, LE

_{3}= −0.099401 and LE

_{4}= −3.0757. LE

_{1}is positive, LE

_{3}and LE

_{4}are negative, LE

_{2}is nearly equal to zero and sum of the LEs is negative, that is, the whole phase volume of the studied system is exponentially shrinking. Figure 2b is the Poincaré map [38,39,40] of proposed system that contains random location of dots indicates the state of chaos. In brief, it is a dissipative chaotic system.

## 3. Dynamical Properties of the System

#### 3.1. The Impacts of Parameters

_{k}can be expressed as p(k)/p

_{tot}(k). Based on the concept of Shannon entropy, the value of Spectral Entropy is equal to the sum of p

_{k}ln(1/p

_{k}) (k = 0, 1, 2…N/2−1). Because of the value of Spectral Entropy converging to ln(N/2), the normalized Spectral Entropy can be expressed as

^{−1}s and setting simulation time T = 500s. First, by respectively comparing the largest Lyapunov exponents (LLE) (see Figure 3b, Figure 5b and Figure 7b) and the Spectral Entropy (SE) (see Figure 3c, Figure 5c and Figure 7c), we can observe that the changing trend of the two calculation results are almost same. Moreover, when the hardware devices are selected as 32GB memory, Core i9-10900 CPU, the operating system is Windows 10, the calculation time of SE is about 1/160 of LLE, as listed in Table 2. The different parameters a, b and k are selected and different types of attractors from type I to type VII have been summarized in Table 1. Thus, based on the calculated results, the proposed system has the characteristics that the normalized average value of SE in the chaotic state is about 0.5, in the quasiperiodic state is about 0.2 and in the periodic state is about 0.1.

#### 3.2. Coexistence of Hidden Attractor

#### 3.3. Controllability of Attractor

#### 3.4. Transient Behaviour

## 4. Circuit Design

#### 4.1. Improved Modular Circuit Design

_{0}= 1/RC = 10

^{4}. The value of the resistor R and the capacitor C is, respectively, 10 kΩ and 10 nF. Then we define a new time variable τ instead of t, and t = τ

_{0}τ, then dt = τ

_{0}dτ. Thus, the new equation has been obtained as follows:

_{DC}). Figure 15 shows the schematics of the chaotic system in which all the resistors R

_{i}= R = 10 kΩ (i = 1,2,3…19) and all the capacitors C

_{i}= C = 10 nF (i = 1,2,3,4).

#### 4.2. Multisim Results

_{a}= 1.35 V, are almost the same as the numerical simulation results shown in Figure 1. Moreover, when setting V

_{a}= 0.1V, the circuit system has a transition from chaos (see Figure 17a) to period (see Figure 17b), which corresponds with the numerical simulation results shown in Figure 13c. Thus, we can reach the conclusion that the proposed 4-D system can be achieved physically.

## 5. Conclusions

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**2-D phase portraits of the proposed system for a = 1.35, b = 1, k = 1 and the initial condition of (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0) in (

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

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

**c**) x-w plane, (

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

**e**) y-w plane, and (

**f**) z-w plane.

**Figure 2.**The dynamics of LEs (

**a**) and Poincaré map (

**b**) of the proposed system for a = 1.35, b = 1, k = 1 and the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0).

**Figure 3.**The evolution of the state descriptors of the proposed system when the parameter a changes in [1.2, 1.6] for the selected set of b = 1, k = 1 and the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0): (

**a**) Bifurcation diagram; (

**b**) Largest Lyapunov spectrum; (

**c**) Spectral Entropy (Normalized).

**Figure 4.**Projections of chaotic attractors in the x-w plane for the selected set of b = 1, k = 1, the initial conditions of (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0), and (

**a**) a = 1.3; (

**b**) a = 1.4; (

**c**) a = 1.55.

**Figure 5.**The evolution of the state descriptors of the proposed system when the parameter b changes in [0, 0.4] for the selected set of a = 1.35, k = 1 and the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0): (

**a**) Bifurcation diagram; (

**b**) Lyapunov Exponents; (

**c**) Spectral Entropy(Normalized).

**Figure 6.**Projections of chaotic attractors in the x-w plane for the selected set of a = 1.35, k = 1, the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0), and (

**a**) b = 0.1; (

**b**) b = 0.3.

**Figure 7.**The evolution of state descriptors of the proposed system when the parameter k increases from 0.1 to 2.0 for the selected set of a = 1.35, b = 1 and the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0): (

**a**) Bifurcation diagram; (

**b**) Largest Lyapunov spectrum; (

**c**) Normalized Spectral Entropy.

**Figure 8.**Projections of chaotic attractors in the x-w plane for the selected set of a = 1.35, b = 1, the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0), and (

**a**) k = 0.3; (

**b**) k = 1.8.

**Figure 9.**The chaotic characteristic diagram based on Spectral Entropy of the proposed system with the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0) (

**a**) a–b plane with the k = 1, (

**b**) a–k plane with the b = 1, (

**c**) b–k plane with the a = 1.35.

**Figure 10.**Projections of hidden attractors on x-w plane under the initial conditions of (1, 0, 0, 0) (blue) and (−1, 0, 0, 0) (red) with the different set of parameters: (

**a**) a = 1.38, b = 1 and k = 1; (

**b**) a = 1.55, b = 1 and k = 1.

**Figure 11.**(

**a**) Attractive basins with the parameter a = 1.55, b = 1, k = 1 and in the cross section of z (0) = 0 and w (0) = 0. (

**b**) Bifurcation diagram under the initial condition of (1, 0, 0, 0) (red) and (−1, 0, 0, 0) (blue).

**Figure 12.**Controllability of chaotic attractor: (

**a**) location controllable when adjusting the parameter p in x-w plane for p = 0 (red), p = 1 (purple), and p = −1 (blue); (

**b**) shape controllable when adjusting the parameter q in x-w plane for q = 1 (red), q = 2 (purple), and q = 0.5 (blue).

**Figure 13.**Transient behaviour from chaos to period with the selected set of a = 0.1, b = 1 and k = 1 and the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0): (

**a**) time-domain diagram in the time interval [0 s, 5000 s]; (

**b**) Largest Lyapunov exponent spectrum in the time interval [0 s, 5000 s]; (

**c**) phase portrait from chaos (blue) to period (red); (

**d**) time-domain diagram in the time interval [1850 s, 2000 s].

**Figure 14.**The time-domain diagram in the interval [0 s, 5000 s] with the selected set of a = 0.1, b = 1, k = 1, and the initial conditions of (x(0), y(0), z(0), w(0)) are set as (

**a**) (1, 1, 1, 1), (

**b**) (0, 0.8, −0.5, 0).

**Figure 16.**Experimental phase portraits with V

_{a}= 1.35 V displayed by using an oscilloscope: (

**a**) Ux-Uy plane, (

**b**) Ux-Uz plane, (

**c**) Ux-Uw plane, (

**d**) Uy-Uz plane, (

**e**) Uy-Uw plane and (

**f**) Uz-Uw plane.

**Figure 17.**Experimental phase portraits with V

_{a}= 0.1 V displayed by using an oscilloscope: (

**a**) chaotic attractor on Ux-Uw plane, (

**b**) periodic attractor on Ux-Uw plane.

**Table 1.**States and SE values of the proposed system when the parameters a, b and k varying individually with the initial conditions (x(0), y(0), z(0), w(0)) = (−1, 0, 0, 0) and a = 1.35, b = 1, k = 1 when they are constant.

Range | State | Average Value of SE | Attractor Type | Corresponding Figure |
---|---|---|---|---|

a∈[1.23–1.35] | chaos | 0.493 | Ⅰ | Figure 4a |

a∈[1.36–1.43] | chaos | 0.502 | Ⅱ | Figure 4b |

a∈[1.55–1.6] | quasi-period | 0.272 | Ⅲ | Figure 4c |

b∈[0.06–0.16] | chaos | 0.513 | Ⅳ | Figure 6a |

b∈[0.17–0.40] | chaos | 0.518 | Ⅴ | Figure 6b |

k∈[0.10–0.52] | quasi-period | 0.208 | Ⅵ | Figure 8a |

k∈[1.65–2.00] | period | 0.111 | Ⅶ | Figure 8b |

Algorithm | a (s) | b (s) | k (s) |
---|---|---|---|

LLE | 1384.948954 | 1383.054144 | 1353.828976 |

SE | 8.153505 | 7.886318 | 9.132727 |

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**MDPI and ACS Style**

Zhang, X.; Tian, Z.; Li, J.; Wu, X.; Cui, Z.
A Hidden Chaotic System with Multiple Attractors. *Entropy* **2021**, *23*, 1341.
https://doi.org/10.3390/e23101341

**AMA Style**

Zhang X, Tian Z, Li J, Wu X, Cui Z.
A Hidden Chaotic System with Multiple Attractors. *Entropy*. 2021; 23(10):1341.
https://doi.org/10.3390/e23101341

**Chicago/Turabian Style**

Zhang, Xiefu, Zean Tian, Jian Li, Xianming Wu, and Zhongwei Cui.
2021. "A Hidden Chaotic System with Multiple Attractors" *Entropy* 23, no. 10: 1341.
https://doi.org/10.3390/e23101341