# Multiple Alternatives of Offset Boosting in a Symmetric Hyperchaotic Map

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A new control knob is found, which can control the interval and direction of attractor self-reproduction.
- (2)
- Various alternatives of offset boosting are systematically explored, which include initially controlled offset boosting, parameter-oriented offset boosting and competitive offset boosting.
- (3)
- Competitive offset boosting is firstly discussed, where the newly introduced constants show the function of offset boosting determining the direction of the shifted attractor in the phase space, and as a result, based on the constant, the bipolar and unipolar chaotic signals can be switched accordingly.
- (4)
- Periodic windows are caught by the plot of the bifurcation diagram, which poses a great threat to engineering applications. However, competitive offset boosting can be easily applied to cross this interval and reach hyperchaos.
- (5)
- An STM32-based circuit implementation is constructed to prove two different offset boosting regimes. PRNG is employed to demonstrate the practical application of the proposed map.

## 2. A 2D Hyperchaotic Map

#### 2.1. Model Description

_{n}and y

_{n}respectively denote the nth states. System parameters are represented by the symbol a. When x

_{n}

_{+1}→–x

_{n}

_{+1}, y

_{n}

_{+1}→–y

_{n}

_{+1}, x

_{n}→–x

_{n}, y

_{n}→–y

_{n}, map (1) keeps its polarity balance indicating that map (1) is of inversion symmetry.

#### 2.2. Bifurcation Analysis

_{i}

_{+1}= f(x

_{i}), denoted as λf(x), is mathematically defined as

_{0}, y

_{0}) = (1, 0) and a varies in (−3.4, 2.5), several kinds of evolution including period, chaos, and hyperchaos are found in map (1) [24,25,26], as depicted in Figure 1. When a is in (−2.7, −2.248), hyperchaos shows up; when a varies in (−2.248, −2.03), map (1) is chaotic; when a increases in (−2.03, −2.01), a small range periodic window is captured; when a increases in (1.758, 1.882), quasi-periodic oscillation behavior is exhibited; when the parameter a varies in (1.892, 2.5), map (1) is chaotic, and two separate periodic windows (1.99, 2.115) and (2.333, 2.365) are embedded. Typical dynamical behaviors of map (1) are summarized in Table 1, corresponding phase orbits are shown in Figure 2. When x

_{0}→–x

_{0}, y

_{0}→–y

_{0}, the polarity in map (1) is switched and its symmetrical attractors are produced, as shown in Figure 2d–f, their symmetrical waveforms are displayed in Figure 3.

## 3. Multiple Alternatives of Offset Boosting

#### 3.1. Initially Controlled Offset Boosting

_{n}receives the offset boosting, and the properties of the attractor are not disturbed by the value of offset boosting. For the periodicity of the sine function, the attractor is shifted in phase space with inconsistent steps, and the coexisting attractors are arranged by the polarity of x

_{0}and a.

_{0}> 0, a < 0: when x

_{0}> 0, a < 0, attractor self-reproducing is heading in the negative direction of y, as shown in Figure 4a,b.

_{0}> 0, a > 0: when x

_{0}> 0, a > 0, coexisting attractors are arranged in the positive direction of y, as plotted in Figure 4c,d.

_{0}< 0, a < 0: when x

_{0}< 0, a < 0, the direction of attractor self-reproduction is in the negative direction of y, which is the same as that of case I.

_{0}< 0, a > 0: when x

_{0}< 0, a > 0, self-reproducing attractors are extracted in the phase space in the positive direction of y, as in case II.

_{0}controls the position of coexisting attractors in two-dimensional space. As depicted in Figure 5a, the plot of Lyapunov exponents with many periodic windows indicates the nonsmooth switch from one state to another when a = −2.6 and x

_{0}varies in [0, 15]. When a = −2.6 and x

_{0}varies on the negative side of the coordinate axis, in pace with the change in x

_{0}in the positive direction, the dynamic behavior is also accompanied by mutations. Correspondingly, when a = 2.5 and x

_{0}varies in [0, 8], Figure 5b shows that the dynamic evolution is relatively smooth without a change in state. Moreover, when a = 2.5 and x

_{0}> 0, a similar smooth dynamic evolution will also appear. The basins of attraction of the coexisting attractors plotted in Figure 6 also prove this phenomenon. When a = −2.6, the separate basins are nonsmooth, while when a = 2.5, the boundary of the basins is smooth. The rule of attractor reproduction agrees with the evolution of the square plot of bifurcation.

#### 3.2. Parameter-Oriented Offset Boosting

_{n}has the offset shift l while sequence y

_{n}obtains offset boosting with −l, map (1) turns out to be,

_{n}is given a positive offset boosting (PX), and the sequence y

_{n}is offered a negative offset boosting (NY).

_{n}is assigned with negative offset boosting (NX), and the sequence y

_{n}is equipped with positive offset boosting (PY).

_{n}is shifted in the positive direction, and the signal y

_{n}is moved in the negative direction. The dynamical evolution of d is accompanied by many periodic windows, indicating the threat to engineering can be effectively diagnosed by competitive offset boosting. Different from the above-mentioned offset boosting, when a > −1, d > 0, and l > 0, the attractor moves in the negative direction of x and the positive direction of y. The dynamical behavior of d is non-bifurcation without revising Lyapunov exponents, as plotted in Figure 8b. From Figure 8, it is obvious that the evolution of parameter d makes the offset boosting of the signals x

_{n}and y

_{n}different according to the period of 2π. Further explanation will be discussed later.

_{n}receives offset boosting with p while the offset of y

_{n}is revised by q, map (1) turns out to be,

_{n}is given a negative offset boosting, and the sequence y

_{n}is offered a positive offset boosting. Conversely, when a > 0 and h > 0, then p > 0 and q > 0, the sequence x

_{n}and y

_{n}are assigned with positive offset boosting. Six different modes of competitive offset boosting are revealed in Table 3. As shown in Figure 9, the attractors are arranged in a certain interval on the x-axis and y-axis.

_{n}and y

_{n}have migrations in the opposite direction, as shown in Figure 10b–d.

_{n}oscillates periodically with the interval of 2π, but that of the signal y

_{n}climbs periodically, as shown in Figure 8. According to parameter h in Equation (6), the offset boosting with signals x

_{n}and y

_{n}oscillates periodically with a period of 2π, as shown in Figure 10. The multiple alternatives of offset boosting controlled by parameters are summarized in Table 4. Average values of chaotic sequences x

_{n}and y

_{n}are shown in Figure 11.

_{n}climbs globally but falls locally with a period of 2π, the offset of signal x

_{n}oscillates periodically with a period of 2π, as plotted in Figure 8b and Figure 11b. That is to say, there are two different offset evolutions in a large scope of initial conditions. Specifically, when a < −1, the offset of x

_{n}increases in the period but totally remains at the same level according to parameter d in Equation (4). The offset of y

_{n}decreases in the period but grows globally, as shown in Figure 8a and Figure 11a. The evolution of offset boosting according to h in Equation (6) seems to be similar. Here, the offset of y

_{n}decreases periodically while the offset of x

_{n}increases periodically, as shown in Figure 10c,d and Figure 11c.

#### 3.3. Competitive Offset Boosting

_{0}influence the offset of coexisting attractors, as shown in Figure 12, which means that to obtain a desired offset, the parameter and the initial condition should match each other. From Figure 12, we also know that in map (7) the power of offset boosting of parameter d seems stronger than parameter h, which is because parameter h is in the sinusoidal function. More demonstrations based on Lyapunov exponents can verify this phenomenon further. As shown in Figure 13, when a < −1 and x

_{0}varies with constants d and h, respectively, in different ranges, the largest 2D Lyapunov exponent under the mixture control of a parameter and initial condition is given, showing the strength of offset boosting under the evolutions of a parameter or an initial condition.

## 4. Hardware Implementation and Application

#### 4.1. Hardware Implementation Based on STM32

_{0}, y

_{0}) = (1, 0) is shown in Figure 15. Coexisting phase orbits of map (1) observed from the oscilloscope under various initial conditions are presented in Figure 16. Phase trajectories of map (1) observed from the oscilloscope with competitive offset boosting under (x

_{0}, y

_{0}) = (1, 0) are illustrated in Figure 17. Phase orbits of map (1) with different offset constant h under (x

_{0}, y

_{0}) = (1, 0) are shown in Figure 18. All figures from the hardware implementation based on the STM32 microcontroller and TLV5618 module are consistent with the numerical simulation.

#### 4.2. Application in PRNG

_{1}, x

_{2}, …, x

_{n}} or Y = {y

_{1}, y

_{2}, …, y

_{n}} [29,30]. The quantization function P

_{i}employed in this experiment can be expressed as follows

^{7}and N = 256 in this paper.

_{0}, y

_{0}) = (1, 0), the National Institute of Standards and Technology (NIST) test suite is used to measure its performance. To ensure high precision in detection, each sequence set should consist of no less than 1 × 10

^{6}bits. By setting the number of test groups as m = 128, the proportion (Prop) of passing test groups can indicate the pseudo-randomness of the sequence, thereby demonstrating the potential value for engineering applications of map (1). The results of the NIST statistical test of PRNG are illustrated in Table 5.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sheng, S.; Wen, H.; Xie, G.; Li, Y. The Reappearance of Poetic Beauty in Chaos. Symmetry
**2022**, 14, 2445. [Google Scholar] [CrossRef] - Wang, N.; Li, C.; Bao, H.; Chen, M.; Bao, B. Generating multi-scroll Chua’s attractors via simplified piecewise-linear Chua’s diode. IEEE Trans. Circuits Syst. I Reg. Pap.
**2019**, 66, 4767–4779. [Google Scholar] [CrossRef][Green Version] - Li, Y.; Li, C.; Zhao, Y.; Liu, S. Memristor-type chaotic mapping. Chaos
**2022**, 32, 021104. [Google Scholar] [CrossRef] [PubMed] - Li, C.; Lei, T.; Liu, Z. Offset parameter cancellation produces countless coexisting attractors. Chaos
**2022**, 32, 121104. [Google Scholar] [CrossRef] [PubMed] - Chen, M.; Ren, X.; Wu, H.; Xu, Q.; Bao, B. Periodically varied initial offset boosting behaviors in a memristive system with cosine memductance. Front. Inf. Technol. Electron. Eng.
**2019**, 20, 1706–1716. [Google Scholar] [CrossRef] - Gu, H.; Li, C.; Li, Y.; Ge, X.; Lei, T. Various patterns of coexisting attractors in a hyperchaotic map. Nonlinear Dyn.
**2023**, 1–12. [Google Scholar] [CrossRef] - Ma, M.; Xiong, K.; Li, Z.; Sun, Y. Dynamic behavior analysis and synchronization of memristor-coupled heterogeneous discrete neural networks. Mathematics
**2023**, 11, 375. [Google Scholar] [CrossRef] - Zhang, Y.; Xu, Y.; Yao, Z.; Ma, J. A feasible neuron for estimating the magnetic field effect. Nonlinear Dyn.
**2020**, 102, 1849–1867. [Google Scholar] [CrossRef] - Wang, S.; Wang, C.; Xu, C. An image encryption algorithm based on a hidden attractor chaos system and the Knuth–Durstenfeld algorithm. Opt. Lasers Eng.
**2020**, 128, 105995. [Google Scholar] [CrossRef] - Panda, A.K.; Ray, K.C. A coupled variable input LCG method and its VLSI architecture for pseudorandom bit generation. IEEE Trans. Instrum. Meas.
**2019**, 69, 1011–1019. [Google Scholar] [CrossRef] - Kong, S.; Li, C.; Jiang, H.; Zhao, Y.; Wang, Y. Asymmetry Evolvement and Controllability of a Symmetric Hyperchaotic Map. Symmetry
**2021**, 13, 1039. [Google Scholar] [CrossRef] - Lei, T.; Zhou, Y.; Fu, H.; Huang, L.; Zang, H. Multistability dynamics analysis and digital circuit implementation of entanglement-chaos symmetrical memristive system. Symmetry
**2022**, 14, 2586. [Google Scholar] [CrossRef] - Bao, H.; Hua, Z.; Wang, N.; Zhu, L.; Chen, M.; Bao, B. Initials-boosted coexisting chaos in a 2D sine map and its hardware implementation. IEEE Trans. Ind. Informat.
**2020**, 17, 1132–1140. [Google Scholar] [CrossRef] - Sprott, J.C.; Jafari, S.; Khalaf, A.J.M.; Kapitaniak, T. Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping. Eur. Phys. J.-Spec. Top.
**2017**, 226, 1979–1985. [Google Scholar] [CrossRef][Green Version] - Zhang, S.; Li, C.; Zheng, J.; Wang, X.; Zeng, Z.; Peng, X. Generating any number of initial offset-boosted coexisting Chua’s double-scroll attractors via piecewise-nonlinear memristor. IEEE Trans. Ind. Electron.
**2021**, 69, 7202–7212. [Google Scholar] [CrossRef] - Jafari, S.; Pham, V.T.; Golpayegani, S.M.R.H.; Moghtadaei, M.; Kingni, S.T. The relationship between chaotic maps and some chaotic systems with hidden attractors. Int. J. Bifurc. Chaos
**2016**, 26, 1650211. [Google Scholar] [CrossRef] - Panahi, S.; Sprott, J.C.; Jafari, S. Two simplest quadratic chaotic maps without equilibrium. Int. J. Bifurc. Chaos
**2018**, 28, 1850144. [Google Scholar] [CrossRef] - Lin, H.; Wang, C.; Sun, J.; Zhang, X.; Sun, Y.; Iu, H.H. Memristor-coupled asymmetric neural networks: Bionic modeling, chaotic dynamics analysis and encryption application. Chaos Solitons Fractals
**2023**, 166, 112905. [Google Scholar] [CrossRef] - Chen, M.; Ren, X.; Wu, H.; Xu, Q.; Bao, B. Interpreting initial offset boosting via reconstitution in integral domain. Chaos Solitons Fractals
**2020**, 131, 109544. [Google Scholar] [CrossRef] - Iskakova, K.; Alam, M.M.; Ahmad, S.; Saifullah, S.; Akgül, A.; Yılmaz, G. Dynamical study of a novel 4D hyperchaotic system: An integer and fractional order analysis. Math. Comput. Simul.
**2023**, 208, 219–245. [Google Scholar] [CrossRef] - Zhou, X.; Li, C.; Li, Y.; Lu, X.; Lei, T. An amplitude-controllable 3-D hyperchaotic map with homogenous multistability. Nonlinear Dyn.
**2021**, 105, 1843–1857. [Google Scholar] [CrossRef] - Liang, Z.; Sun, K.; He, S. Design and dynamics of the multicavity hyperchaotic map based on offset boosting. Eur. Phys. J. Plus
**2022**, 137, 51. [Google Scholar] [CrossRef] - Li, Y.; Li, C.; Zhang, S.; Chen, G.; Zeng, Z. A Self-reproduction hyperchaotic map with compound lattice dynamics. IEEE Trans. Ind. Electron.
**2022**, 69, 10564–10572. [Google Scholar] [CrossRef] - Lin, H.; Wang, C.; Yu, F.; Xu, C.; Hong, Q.; Yao, W.; Sun, Y. An ex-tremely simple multiwing chaotic system: Dynamics analysis, encryption application, and hardware implementation. IEEE Trans. Ind. Electron.
**2021**, 68, 12708. [Google Scholar] [CrossRef] - Peng, Y.; Sun, K.; He, S. A discrete memristor model and its application in henon map. Chaos Solitons Fractals
**2020**, 137, 109873. [Google Scholar] [CrossRef] - Zhang, S.; Li, C.; Zheng, J.; Wang, X.; Zeng, Z.; Chen, G. Memristive Autapse-Coupled Neuron Model With External Electromagnetic Radiation Effects. IEEE Trans. Ind. Electron.
**2022**, 1–9. [Google Scholar] [CrossRef] - Lai, Q.; Lai, C. Design and implementation of a new hyperchaotic memristive map. IEEE Trans. Circuits Syst. II Express Briefs
**2022**, 69, 2331–2335. [Google Scholar] [CrossRef] - He, S.; Natiq, H.; Banerjee, S.; Sun, K. Complexity and chimera states in a network of fractional-order laser systems. Symmetry
**2021**, 13, 341. [Google Scholar] [CrossRef] - Hua, Z.; Zhou, B.; Zhou, Y. Sine chaotification model for enhancing chaos and its hardware implementation. IEEE Trans. Ind. Electron.
**2018**, 66, 1273–1284. [Google Scholar] [CrossRef] - Zhou, Y.; Hua, Z.; Pun, C.M.; Chen, C.P. Cascade chaotic system with applications. IEEE Trans. Cybern.
**2014**, 45, 2001–2012. [Google Scholar] [CrossRef] - Ahmad, M.; Al Solami, E.; Wang, X.Y.; Doja, M.N.; Beg, M.S.; Alzaidi, A.A. Cryptanalysis of an image encryption algorithm based on combined chaos for a BAN system, and improved scheme using SHA-512 and hyperchaos. Symmetry
**2018**, 10, 266. [Google Scholar] [CrossRef][Green Version] - Xu, C.; Zhang, W.; Aouiti, C.; Liu, Z.; Yao, L. Bifurcation insight for a fractional-order stage-structured predator–prey system incorporating mixed time delays. Math. Meth. Appl. Sci.
**2023**, 1–16. [Google Scholar] [CrossRef] - Gao, X.; Mou, J.; Banerjee, S.; Cao, Y.; Xiong, L.; Chen, X. An effective multiple-image encryption algorithm based on a 3D cube and hyperchaotic map. J. King Saud Univ.-Comput. Inf. Sci.
**2022**, 34, 1535–1551. [Google Scholar] [CrossRef] - Xu, C.; Liu, Z.; Li, P.; Yan, J.; Yao, L. Bifurcation mechanism for fractional-order three-triangle multi-delayed neural networks. Neural Process. Lett.
**2022**, 1–27. [Google Scholar] [CrossRef] - Xu, C.; Mu, D.; Liu, Z.; Pang, Y.; Liao, M.; Aouiti, C. New insight into bifurcation of fractional-order 4D neural networks incorporating two different time delays. Commun. Nonlinear Sci.
**2023**, 118, 107043. [Google Scholar] [CrossRef]

**Figure 1.**Dynamical evolution of hyperchaotic map (1) under the initial condition (x

_{0}, y

_{0}) = (1, 0) when a varies in (−3.4, 2.5): (

**a**) Lyapunov exponents, (

**b**) bifurcation diagram.

**Figure 2.**Typical phase trajectories of map (1): (

**a**) a = −2.6, (

**b**) a = −2.2, (

**c**) a = −1.96, (

**d**) a = 1.887, (

**e**) a = 2.336, (

**f**) a = 2.5, where (

**a**–

**c**) under the initial condition (x

_{0}, y

_{0}) = (1, 0).

**Figure 4.**Coexisting phase trajectories of map (1) under various initial conditions: (

**a**) a = −2.6, (

**b**) a = −2.2, (

**c**) a = 1.887, (

**d**) a = 2.5.

**Figure 7.**Phase trajectories of map (4) under (x

_{0}, y

_{0}) = (1, 0): (

**a**) a = −2.6, (

**b**) a = −2.2, (

**c**) a = 1.887, (

**d**) a = 2.5.

**Figure 9.**Phase trajectories of map (6) with a different offset constant h under (x

_{0}, y

_{0}) = (1, 0), where gold: h = 0, violet red: h = 1.5, turquoise: h = 3: (

**a**) a = −2.6, (

**b**) a = −2.2, (

**c**) a = 1.887, (

**d**) a = 2.5.

**Figure 10.**Feature of competitive offset boosting of map (6) with a = −2.6 under (x

_{0}, y

_{0}) = (1, 0): (

**a**) discrete sequences, (

**b**) Lyapunov exponents, (

**c**,

**d**) bifurcation diagram.

**Figure 11.**Average values of x

_{n}and y

_{n}: (

**a**) a = −2.6 and d varies in [0, 10π], (

**b**) a = 2.5 and d varies in [0, 10π], (

**c**) a = −2.6 and h varies in [0, 10π].

**Figure 12.**Competitive offset boosting of map (7) when a = −2.6 and y

_{0}= 0: (

**a**) d = 0, 12.5 and 25, x

_{0}= 1, 3 and 7, (

**b**) h = −0.5π, 0 and 0.5π, x

_{0}= −7, 1 and 7.

**Figure 13.**The largest Lyapunov exponent of map (7) when a = −2.6 and y

_{0}= 0: (

**a**) x

_{0}varies in [−5, 10] and d varies in [0, 30], (

**b**) x

_{0}varies in [−15, 15] and h varies in [−π, π].

**Figure 15.**The hardware-implemented waveform of in map (1) under (x

_{0}, y

_{0}) = (1, 0): (

**a**) hyperchaos: a = −2.6, (

**b**) chaos: a = 2.5.

**Figure 16.**Coexisting phase trajectories of map (1) observed from the oscilloscope under different initial conditions: (

**a**) a = −2.6, (

**b**) a = −2.2, (

**c**) a = 1.887, (

**d**) a = 2.5.

**Figure 17.**Phase trajectories of map (4) from the oscilloscope under (x

_{0}, y

_{0}) = (1, 0): (

**a**) a = −2.6, (

**b**) a = −2.2, (

**c**) a = 1.887, (

**d**) a = 2.5.

**Figure 18.**Phase trajectories of map (6) with different offset constant h under (x

_{0}, y

_{0}) = (1, 0), where gold: h = 0, violet red: h = 1.5, turquoise: h = 3: (

**a**) a = −2.6, (

**b**) a = −2.2, (

**c**) a = 1.887, (

**d**) a = 2.5.

a | Attractor Type | Lyapunov Exponents |
---|---|---|

−2.6 | Hyperchaos | 0.2256, 0.09648 |

−2.2 | Chaos | 0.1034, −0.0136 |

−1.96 | Quasi-period | 0, −0.2493 |

1.877 | Quasi-period | 0, −0.3204 |

2.336 | Periodic points | −0.1134, −0.2626 |

2.5 | Chaos | 0.3369, −0.1136 |

Parameters | a < −1 | a > −1 |
---|---|---|

d < 0 | l > 0 | l > 0 |

PX | NX | |

NY | PY | |

d > 0 | l < 0 | l > 0 |

NX | PX | |

PY | NY |

Parameters | a < −1 | −1 < a < 0 | a > 0 |
---|---|---|---|

h < 0 | p > 0, q < 0 | p < 0, q > 0 | p < 0, q < 0 |

PX | NX | NX | |

NY | PY | NY | |

h > 0 | p < 0, q > 0 | p > 0, q < 0 | p > 0, q > 0 |

NX | PX | PX | |

PY | NY | PY |

x | y | |

d | Periodic Offset Boosting | Oscillatory Offset Boosting |

h | Periodic Offset Boosting | Periodic Offset Boosting |

No. | Statistical Test Terms | PRNG Generated by x | PRNG Generated by y | ||
---|---|---|---|---|---|

Prop | p-Value | Prop | p-Value | ||

01 | Frequency | 0.992 | 0.534146 | 0.984 | 0.689019 |

02 | Block frequency | 0.984 | 0.804337 | 0.976 | 0.654467 |

03 | Cumulative sums | 0.992 | 0.585209 | 0.984 | 0.204076 |

04 | Runs | 0.992 | 0.392456 | 1 | 0.178278 |

05 | Longest run | 1 | 0.057146 | 1 | 0.437274 |

06 | Rank | 0.992 | 0.723129 | 0.992 | 0.242986 |

07 | FFT | 0.984 | 0.253551 | 0.992 | 0.090936 |

08 | Non-overlapping template | 1 | 0.991468 | 1 | 0.980885 |

09 | Overlapping template | 1 | 0.134686 | 0.984 | 0.551026 |

10 | Universal | 0.984 | 0.170294 | 1 | 0.324108 |

11 | Approximate entropy | 0.968 | 0.204076 | 0.992 | 0.848588 |

12 | Random excursions | 1 | 0.162606 | 1 | 0.602458 |

13 | Random excursions variant | 1 | 0.275709 | 1 | 0.213309 |

14 | Serial | 0.992 | 0.739918 | 0.992 | 0.500934 |

15 | Linear complexity | 0.992 | 0.452799 | 1 | 0.324108 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ge, X.; Li, C.; Li, Y.; Zhang, C.; Tao, C. Multiple Alternatives of Offset Boosting in a Symmetric Hyperchaotic Map. *Symmetry* **2023**, *15*, 712.
https://doi.org/10.3390/sym15030712

**AMA Style**

Ge X, Li C, Li Y, Zhang C, Tao C. Multiple Alternatives of Offset Boosting in a Symmetric Hyperchaotic Map. *Symmetry*. 2023; 15(3):712.
https://doi.org/10.3390/sym15030712

**Chicago/Turabian Style**

Ge, Xizhai, Chunbiao Li, Yongxin Li, Chuang Zhang, and Changyuan Tao. 2023. "Multiple Alternatives of Offset Boosting in a Symmetric Hyperchaotic Map" *Symmetry* 15, no. 3: 712.
https://doi.org/10.3390/sym15030712