A New Hyperchaotic Map and Its Manifold of Conditional Symmetry

Abstract

In this work, the polarity balance of a novel two-dimensional hyperchaotic map is considered, and thus the corresponding manifold of conditional symmetry is coined. The unique map has a simple structure but provides direct 2-D offset boosting, which brings the possibility for the construction of conditional symmetry by introducing an absolute value function. The corresponding evolution of the discrete sequences from the system is verified by the circuit implementation based on the microcontroller of CH32V307. The pseudorandom data from the map increases its adaptability for applications in information security. The hyperchaotic sequence-injected Ant Colony Optimization (ACO), Grey Wolf Optimizer (GWO), and Sparrow Search Algorithm (SSA) show their improved performance in the optimization algorithm. Robot path planning experiments confirm that all three algorithms exhibit superior convergence performance, global search capability, and path smoothness compared with the original algorithms.

1. Introduction

Chaos is ubiquitous in natural and technological systems, as its irregular, unpredictable, and intricate dynamic characteristics serve as a critical theoretical underpinning for studies in diverse scientific domains. Since Lorenz [1] presented the classical three-dimensional chaotic system in the 1960s, chaos theory [2,3,4,5] has progressively emerged as a crucial framework for analyzing complex systems. Recently, hyperchaotic systems—dynamical systems exhibiting greater complexity than classical chaotic systems—have become a focal point of research, owing to their multiple positive Lyapunov exponents and the resulting enhanced dynamic instability and behavioral diversity.

Much research has been conducted in-depth in continuous systems focusing on the exploration of new dynamical properties [6,7,8,9,10], including amplitude control [11,12,13] and offset boosting [14,15,16]. However, with the rapid development of artificial intelligence and powerful calculation, discrete chaotic systems [17,18,19] have gradually emerged as another key focus of contemporary research.

In 1990, Ott, Grebogi, and Yorke [20] first proposed chaos control theory, which is now widely applied to the control of discrete systems, such as the Logistic and Henon maps. Boccaletti [21] systematically reviewed control and synchronization issues in discrete chaotic systems, suggesting the introduction of synchronization control into coupled map lattices, thereby promoting the extension of discrete chaotic systems from single-unit models to networked and coupled frameworks.

To enhance the stability, accuracy, and robustness of discrete chaotic systems in practical applications, increasing attention has been devoted to exploring features, such as offset boosting and conditional symmetry. Kong et al. [22] demonstrated that the polarity balance of discrete chaotic systems can be achieved by introducing periodic functions while simultaneously inverting constant terms. Ostrovskii et al. [23] proposed a novel method to induce multistability in discrete systems, whereby multiple coexisting attractors can be identified in the discrete model by applying numerical integration to a continuous monostable system and determining the corresponding range of symmetric coefficients. Ren et al. [24] constructed a new three-dimensional hyperchaotic map by coupling discrete memristor models and experimentally validated its implementability in circuits. Jin et al. [25] proposed a time-varying fuzzy-parameter zeroing neural network (ZNN) model to address the synchronization of chaotic systems under external noise disturbances. Comparative studies demonstrated that the proposed model exhibits superior performance over existing approaches. Liu et al. [26] developed a noise-tolerant fuzzy-type zeroing neural network by integrating a zeroing neural network with a fuzzy logic system, and comparative analyses verified its enhanced predefined-time convergence and noise-tolerance capability.

Currently, discrete chaotic systems are widely applied in fields such as information security, data encryption, and optimization algorithms. Pourasad et al. [27] combined chaotic sequences with wavelet transforms to design a novel digital image encryption technique. Yang et al. [28] improved a one-dimensional Iterative Map with Infinite Collapses (ICMIC) map to control a two-dimensional Henon map and applied it to image encryption, scrambling pixel positions, and diffusing pixel values, thereby demonstrating the optimization benefits of incorporating chaotic theory into the algorithm. Alhadawi et al. [29] integrated discrete chaotic maps with the Cuckoo Search (CS) algorithm to design an 8 × 8 S-box with specific cryptographic properties, enhancing its resistance to cryptanalysis. Yu et al. [30] conducted a comprehensive analysis of four discrete hyperchaotic systems constructed using dynamic models, confirming their high complexity, and further applied them to image encryption, where their strong security performance was successfully demonstrated. Giuseppe Grassi [31] has explored the practical applications of chaos theory in various real-world scenarios, including but not limited to distributed sensing, hardware implementation of cryptographic systems, and robotic motion. Onder Tutsoy et al. [32] proposed a novel intelligent model-free adaptive control approach based on chaotic theory and applied it to the control of underactuated robotic manipulators. Dayal R. Parhi et al. [33] introduced chaotic dynamics into the RGSA model and, based on this framework, proposed a novel path planning method for humanoid robots.

Although hyperchaotic systems [34] and the integration of chaos theory with artificial intelligence have become one of the current research hotspots, most existing studies focus on continuous hyperchaotic systems or conventional chaotic systems combined with artificial intelligence. Tian et al. [35] proposed a novel four-dimensional chaotic system accompanied by detailed dynamical analysis and circuit implementation, while Chong et al. [36] introduced a dual-memristor-based hyperchaotic map with analog circuit realization. In contrast, discrete hyperchaotic systems and their integration with artificial intelligence remain relatively underexplored and offer substantial potential for further investigation. Therefore, this paper proposes a two-dimensional discrete hyperchaotic system and applies it to robot path optimization as a deterministic, high-complexity sequence generator to enhance the exploration capability of robot path optimization algorithms. Compared with conventional chaotic systems, hyperchaotic systems possess multiple positive Lyapunov exponents, indicating higher dynamical complexity and stronger state-space exploration capability. Introducing such dynamical characteristics into robot path optimization allows the search process to benefit from increased diversity, enhanced global exploration, and reduced sensitivity to local optima. Moreover, the discrete formulation is naturally compatible with iterative optimization algorithms and digital implementations, making it particularly suitable for robot path planning in complex environments.

It should be noted that, in the classical theory of dynamical systems, hyperchaos is strictly defined for systems with phase-space dimensions no less than three and at least two positive Lyapunov exponents. From this strict perspective, the proposed two-dimensional map does not constitute a standard hyperchaotic system. In this work, the term “hyperchaotic” is used in a generalized, application-oriented sense to describe the enhanced chaotic behavior of the proposed map, which exhibits multiple positive Lyapunov exponents and higher dynamical complexity compared with conventional chaotic maps. Similar generalized usages [37,38,39] can be found in engineering and applied studies, where the emphasis is placed on complexity enhancement and ergodicity rather than strict dimensional classification.

In this paper, a two-dimensional discrete hyperchaotic system with a simple structure and complex dynamical behaviors is proposed, which possesses two-dimensional offset boosting and conditional symmetry characteristics. The main innovations and contributions of this work are summarized as follows:

• The proposed discrete hyperchaotic map has an elegant structure hosting two-dimensional offset boosting, where a single constant rescales the average values of all variables simultaneously.
• The proposed two-dimensional discrete hyperchaotic system satisfies the polarity balance required for conditional symmetric construction. By introducing an absolute value function, conditional symmetry is coined, and consequently, the derived version generates a pair of coexisting attractors of conditional symmetry.
• Circuit implementation based on the hardware of CH32 verifies the unique dynamics of the proposed map and confirms its derived version of conditional symmetry.
• Three classical intelligent optimization algorithms are also enhanced by introducing the hyperchaotic sequences, including Ant Colony Optimization (ACO), Grey Wolf Optimizer (GWO), and Sparrow Search Algorithm (SSA), which are subsequently employed in robot path planning, showing their effectiveness and higher performance.

The rest of this paper is organized as follows. In Section 2, we propose the two-dimensional discrete hyperchaotic system and perform basic dynamical analysis, including the stability of fixed points and bifurcation. In Section 3, the property of offset boosting is analyzed in detail. In Section 4, the conditional symmetric properties of the revised version are comprehensively investigated. In Section 5, the above finding is verified on the CH32 circuit platform. In Section 6, three chaos-enhanced optimization algorithms are proposed for the application in robot path planning. The conclusions and discussion are wrapped up in the Section 7.

2. The Model of a 2-D Hyperchaotic Map and Its Basic Dynamics

2.1. System Model

The new two-dimensional hyperchaotic system is defined as follows:

$$\left\{\begin{array}{l}{x}_{n+1}=b{y}_n+a{x}_n,\\ y_{n+1}=y_n+c{x}_n^2+h.\end{array}\right.$$

(1)

Here, n denotes a natural number, and the n-th states of the map are represented by x_n and y_n. h is the constant term, and a, b, and c are three nonzero system parameters. When the parameters are set as ℎ = 0.5, a = −0.7, b = 1.7, and c = −1, the system exhibits hyperchaotic behavior, with Lyapunov exponents of LEs = (0.1853, 0.0931). The Kaplan–Yorke dimension can be calculated according to the following formula:

$$D_{KY}=j+{\displaystyle \frac{1}{\left|L{E}_{j+1}\right|}}{\displaystyle \sum _{i=1}^{j}L{E}_i}.$$

(2)

where j denotes the largest integer that satisfies ${\displaystyle \sum _{i=1}^{j}L{E}_i\ge 0}$ and ${\displaystyle \sum _{i=1}^{j+1}L{E}_i\le 0}$. Consequently, since system (1) exhibits two positive Lyapunov exponents, the corresponding Kaplan–Yorke dimension is determined to be D_KY = 2 > 0.

A notable feature of this system is that it contains only a single quadratic nonlinear term. Compared with the two-dimensional Hénon map, the proposed discrete chaotic system has a self-feedback term and an offset constant. In the conventional Hénon map, the coupling is relatively weak, and the system possesses only one positive Lyapunov exponent, exhibiting standard chaotic behavior. In contrast, the proposed system establishes mutual interaction between x_n+1 and y_n, enabling y_n+1 to achieve integral-type self-feedback. As a result, the system exhibits two positive Lyapunov exponents, indicating that it is a hyperchaotic system. Therefore, compared with the traditional two-dimensional Hénon map, the proposed discrete system maintains a simple structural form while exhibiting more complex and diverse dynamical behaviors. Moreover, the introduction of the constant h breaks the system’s symmetry; however, further analysis reveals that the system can still exhibit symmetry under certain specific conditions. In addition, the system possesses a single-parameter two-dimensional offset boosting property, which will be discussed in detail later. Figure 1 shows the typical phase trajectory and the corresponding discrete sequences of the system.

2.2. Fixed Points and Their Stability

Let x_n+1 = x_n = x_∗ and y_n+1 = y_n = y_∗, then the fixed point (x_∗, y_∗) satisfies

$$\left\{\begin{array}{l}{x}_{*}=b{y}_{\ast }+a{x}_{\ast },\\ y_{*}=y_{\ast }+c{x}_{\ast }^2+h.\end{array}\right.$$

(3)

When the system parameters are set as h = 0.5, a = −0.7, c = −1, b = 1.7, the fixed points of the discrete system (1) can be calculated as (0.7071, 0.7071) and (−0.7071, −0.7071). The stability of the discrete system can be preliminarily assessed by examining the stability of these fixed points. For the discrete system (1), the corresponding Jacobian matrix is given by

$$J=\left[\begin{array}{cc}-0.7& 1.7\\ -2x_{*}& 1\end{array}\right].$$

(4)

Therefore, the characteristic equation can be expressed as

$$\left|\lambda E-J\right|=(\lambda +0.7)(\lambda -1)+3.4x_{*}=0.$$

(5)

The eigenvalues at the fixed point (0.7071, 0.7071) can be obtained as

$${\lambda }^2-0.3\lambda +1.70414=0.$$

(6)

The eigenvalues are calculated as λ₁ = 0.15 + 1.2968i; λ₂ = 0.15 − 1.2968i. Since |λ₁| > 1 and |λ₂| > 1, the fixed point (0.7071, 0.7071) is unstable.

Similarly, by substituting the fixed point (−0.7071, −0.7071), we obtain

$${\lambda }^2-0.3\lambda -3.10414=0.$$

(7)

The eigenvalues are calculated as λ₁ = 1.9182; λ₂ = −1.6182. Since |λ₁| > 1, |λ₂| > 1, the fixed point (−0.7071, −0.7071) is also unstable. Therefore, the hyperchaotic attractor of this system can be regarded as a self-excited attractor rather than a hidden attractor.

2.3. Basic Dynamics

2.3.1. Bifurcation Analysis Under the Parameter of a

The system exhibits two separate chaotic intervals within the negative range of the parameter a. For the initial condition IC = (0.1, 0.1) and parameters h = 0.5, b = 1.7, and c = −1, as a varies within the range [−1.5, −0.67], the system sequentially displays multiple dynamical behaviors, including periodic, quasiperiodic, chaotic, and hyperchaotic states. Specifically, when a varies within (−0.7370, −0.6700) and (−0.8112, −0.7414), the system exhibits hyperchaotic behavior; within (−0.9122, −0.8113), the system shows chaotic dynamics; within (−1.3989, −1.1138), the system captures a large chaotic interval, intermittently interspersed with small periodic windows; and within (−1.500, −1.3989), (−1.1138, −0.9123) and (−0.7413, −0.7371), the system exhibits periodic windows. Under the same initial condition and parameters, as a varies within [−1.5, −0.67], Figure 2 presents the bifurcation diagram and Lyapunov exponents of the discrete system (1), from which a quasiperiodic route to chaos is observed as the numerical value of parameter a increases along the negative axis.

Table 1. Typical solutions of map (1), with h = 0.5, b = 1.7, c = −1 under IC = (x₀, y₀) = (0.1, 0.1).

a	Attractor Type	Lyapunov Exponents
−1.3079	Quasi-periodicity	0, −0.1410
−0.888	Periodic points	−0.0036, −0.2042
−0.88	Chaos	0.0913, −0.2100
−0.8	Hyperchaos	0.1466, 0.0366
−0.7383	Periodic points	−0.0331, −0.0331
−0.7	Hyperchaos	0.1853, 0.0931

summarizes the typical dynamical behaviors of the system, with the corresponding representative trajectories illustrated in Figure 3.

2.3.2. Bifurcation Analysis Under the Parameter of b

For the initial condition IC = (0.1, 0.1) and parameters h = 0.5, a = −0.7, and c = −1, as b varies within the range [1, 1.73], the system sequentially exhibits multiple dynamical behaviors, including periodic, quasiperiodic, chaotic, and hyperchaotic states. Specifically, when b varies within (1, 1.2051), the system shows periodic behavior; within (1.2051, 1.4577), it captures a large chaotic interval with multiple small periodic windows; within (1.4577, 1.4642), (1.4745, 1.4854), (1.5029, 1.5044), and (1.6497, 1.6555), the system exhibits periodic windows; within (1.4643, 1.4745) and (1.4855, 1.5030), it displays chaotic behavior, with the interval (1.4643, 1.4745) corresponding to hyperchaotic dynamics; and within (1.5117, 1.6497) and (1.6555, 1.7300), the system enters hyperchaotic states. Under the same initial condition and parameters, as b varies within [1, 1.73], Figure 4 presents the bifurcation diagram and Lyapunov exponents of the discrete system (1), from which a quasiperiodic route to chaos is observed as the numerical value of parameter b increases. An enlarged view is provided for the interval (1.25, 1.45). Due to the LE1 values oscillating around zero, the bifurcation diagram exhibits a large chaotic interval interspersed with numerous small periodic windows.

Table 2. Typical solutions of map (1), with h = 0.5, a = −0.7, c = −1 under IC = (x₀, y₀) = (0.1, 0.1).

b	Attractor Type	Lyapunov Exponents
1.2255	Quasi-periodicity	0, −0.0362
1.4535	Quasi-periodicity	0, −0.0295
1.47	Chaos	0.0126, −0.0323
1.5	Hyperchaos	0.0466, 0.0026
1.655	Periodic points	−0.0212, −0.0187
1.7	Hyperchaos	0.1853, 0.0931

summarizes the typical dynamical behaviors of the system, and the corresponding representative trajectories are illustrated in Figure 5.

2.3.3. Bifurcation Analysis Under the Parameter of c

For the initial condition IC = (0.1, 0.1) and parameters h = 0.5, a = −0.7, b = 1.7, as c varies within the range [−1.04, −0.4], the system sequentially exhibits multiple dynamical behaviors, including periodic, quasiperiodic, chaotic, and hyperchaotic states. Specifically, when c varies within (−1.0400, −0.9491) and (−0.9408, −0.7917), the system exhibits hyperchaotic behavior; within (−0.7910, −0.7872) and (−0.7520, −0.7424), it displays chaotic states; and within (−0.7628, −0.7526), (−0.7417, −0.7340) and (−0.4998, −0.400), the system exhibits periodic windows. Under the same initial condition with a = 0.5 and b = 1.7, as c varies within [−1.04, −0.4], Figure 6 presents the bifurcation diagram and Lyapunov exponents of the discrete system (1), from which a quasiperiodic route to chaos is observed as the numerical value of parameter c increases. An enlarged view is provided for the interval (−0.8, −0.6). Since the LE1 are mostly greater than zero, these regions correspond to chaotic dynamics in the bifurcation diagram, occasionally interspersed with short periodic windows.

Table 3. Typical solutions of map (1), with h = 0.5, a = −0.7, b = 1.7 under IC = (x₀, y₀) = (0.1, 0.1).

c	Attractor Type	Lyapunov Exponents
−1	Hyperchaos	0.1853, 0.0931
−0.945	Periodic points	−0.0622, −0.0619
−0.8	Hyperchaos	0.0654, 0.0301
−0.79	Chaos	0.061, −0.0054
−0.7628	Periodic points	−0.0024, −0.005
−0.746	Chaos	0.0109, −0.043
−0.52	Quasi-periodicity	0, −0.0369
−0.49	Periodic points	−0.0086, −0.0086

summarizes the typical dynamical behaviors of the system, with the corresponding trajectories illustrated in Figure 7. Figure 7c,d shows two distinct chaotic patterns: panel (c) represents outer-envelope evolving chaos, where the system’s phase trajectory undergoes a sequence of bifurcations beyond a critical value of c, losing periodicity and entering to chaotic state; Figure 7d represents inner-envelope evolving chaos, where changes in c cause the phase trajectory to gradually diffuse inward, ultimately forming a hyperchaotic state. The coupling and coexistence of these two chaotic evolution modes contribute to the emergence of more complex and richer hyperchaotic behavior in the system.

3. Two-Dimensional Offset Boosting

In practical engineering and computational applications, chaotic and hyperchaotic attractors are often required to operate around adjustable equilibrium positions rather than being confined to a fixed region of the phase space. The ability to flexibly shift attractor locations without altering the intrinsic dynamical behavior is beneficial for signal modulation, control design, and hardware implementation. Offset boosting provides an effective mechanism to achieve such controllable phase-space translation while preserving the original dynamics of the system.

Similarly to continuous systems, a discrete system can achieve a direct shift in attractor positions, referred to as offset boosting, without altering the original dynamics, only under the changing of an independent external parameter combined with the corresponding initial conditions. In discrete systems, the equations typically relate the new states of variables on the left-hand side to the old states on the right-hand side, lacking the Dimension of Variable Differentiation (DVD) [40]. Based on the discretization formula of continuous systems $\stackrel{•}{x}={\displaystyle \frac{x\left(n+1\right)-x\left(n\right)}{\mathrm{\Delta }t}},\mathrm{\Delta }t=1$, it can be inferred that to realize offset boosting in a discrete system, an identical constant term should be added to both sides of the equation to cancel the offset parameters. After the offset cancelation, if a constant parameter remains, the attractor position can be adjusted with this parameter, thereby implementing offset boosting in the discrete chaotic system. If the attractor moves along only one direction (either x or y), it is referred to as one-dimensional offset boosting; if the attractor moves along both x and y directions, it is termed two-dimensional offset boosting.

For a dynamical system, the offset parameters influence the two-dimensional offset boosting behavior through the fixed points. Therefore, the offset conditions of the system can be inferred by analyzing its fixed points. Setting the initial condition IC = (0.1, 0.1), h = 0.5, and c = −1, and assuming that the offset constant d is introduced in both the x and y dimensions, the discrete system (1) can be rewritten as

$$\left\{\begin{array}{l}{x}_{n+1}-d=b\left(y_n-d\right)+a\left(x_n-d\right),\\ y_{n+1}-d=y_n-d+c{\left(x_n-d\right)}^2+h.\end{array}\right.$$

(8)

Let x_n − d = v_n, y_n − d = u_n, then the above equations can be rewritten as

$$\left\{\begin{array}{l}{v}_{n+1}=b{u}_n+a{v}_n,\\ u_{n+1}=u_n+{c{v}_n}^2+h.\end{array}\right.$$

(9)

Let v_n+1 = v_n = v_*; u_n+1 = u_n = u_*, then the fixed point (v_*, u_*) satisfies

$$\left\{\begin{array}{l}{v}_{\ast }=b{u}_{\ast }+a{v}_{\ast },\\ u_{\ast }=u_{\ast }+{c{v}_{\ast }}^2+h.\end{array}\right.$$

(10)

It can be calculated that

$$\left\{\begin{array}{l}{v}_{\ast }=\pm \sqrt{-{\displaystyle \frac{h}{c}}},\\ u_{\ast }={\displaystyle \frac{1-a}{b}}{v}_{\ast }.\end{array}\right.$$

(11)

That is

$$\left\{\begin{array}{l}{x}_{\ast }=\pm \sqrt{-{\displaystyle \frac{h}{c}}}+d,\\ y_{\ast }={\displaystyle \frac{1-a}{b}}\left(x_{\ast }-d\right)+d.\end{array}\right.$$

(12)

When a + b = 1, the fixed points (x_*, y_*) share the same offset constant d, and the system thus exhibits two-dimensional offset boosting. When a + b ≠ 1, the fixed points (x_∗, y_∗) still incorporate the same offset constant d, indicating that the system’s stability is not directly affected by the offset d. However, in the case of a + b ≠ 1, the offset d appears not only inside the nonlinear term of the equation for y_n+1 but also in the equation for x_n+1, which may distort the system dynamics. The following analysis is therefore divided into two specific cases.

3.1. Case 1: a + b = 1

By setting the initial condition IC = (0.1, 0.1), h = 0.5, c = −1 and varying b within the interval [1, 1.73], while assuming that an offset constant d is introduced in both the x- and y-dimensions such that x_n+1 → x_n+1 − d, x_n → x_n − d, y_n+1 → y_n+1 − d and y_n → y_n − d, the discrete system (1) can be reformulated as follows:

$$\left\{\begin{array}{l}{x}_{n+1}=b{y}_n+a{x}_n,\\ y_{n+1}=y_n+c{\left(x_n-d\right)}^2+h.\end{array}\right.$$

(13)

Unlike conventional two-dimensional offset mechanisms, the independent constant d in this system appears within the quadratic term. Nevertheless, it can still realize offset regulation of both the x- and y-variables. In particular, the sum of the coefficients of the x- and y-feedback terms in the first-dimensional difference equation equals unity. Under this condition, if other parameters remain unchanged, while only the bifurcation parameter b is varied, it is observed that introducing different independent constant d results in a corresponding shift in the attractor’s position. This indicates that, across different phase trajectories, the constant d can consistently achieve joint offset regulation in both the x- and y-dimensions, as illustrated in Figure 8. Specifically, Figure 8 presents the offset behavior under four cases: b = 1.1514, b = 1.33, b = 1.52, and b = 1.7, which demonstrates that the offset control remains robust regardless of whether the system is in quasiperiodic, chaotic, or hyperchaotic states.

When b = 1.7, the system operates in the hyperchaotic state. Figure 9a illustrates the discrete sequences offsets obtained under different independent constant d. Figure 9b reveals that as d varies, the mean values of variables x and y also shift, with the two mean trajectories coinciding. Furthermore, Figure 9c,d present the bifurcation diagram and Lyapunov exponents of system (13). It can be observed that the bifurcation diagram retains equal-width characteristics and the Lyapunov exponents remain nearly unchanged, indicating that the independent constant d enables the offset of the system while preserving its intrinsic dynamical properties.

3.2. Case 2: a + b ≠ 1

By setting the initial condition IC = (0.1, 0.1), h = 0.5, c = −1, b varies within the range [1, 1.73], and a varies within the range [−1.5, −0.67]. Suppose that an offset constant d is introduced in both the x and y dimensions, such that x_n+1 → x_n+1 − d, x_n → x_n − d, y_n+1 → y_n+1 − d and y_n → y_n − d. Under this modification, the discrete system (1) is transformed into

$$\left\{\begin{array}{l}{x}_{n+1}=b{y}_n+a{x}_n-d\left(a+b-1\right),\\ y_{n+1}=y_n+c{\left(x_n-d\right)}^2+h.\end{array}\right.$$

(14)

It is not difficult to observe that, compared with the previous case, the equation for x_n+1 now contains an additional nonzero offset term –d(a + b − 1). Since the offset parameter d simultaneously appears in multiple terms, it is practically difficult to achieve identical offset values in different locations, whether in physical circuit implementation or numerical simulation. Therefore, under this condition, the system is not considered to satisfy the two-dimensional offset property. At this point, the offset behavior of system (13) is influenced by two pathways: the term –d(a + b − 1) and the nonlinear term (x_n − d)². When a + b ≈ 1, the term –d(a + b − 1) ≈ 0, meaning its influence is almost negligible, and the system offset is mainly dominated by the nonlinear term (x_n − d)². When |a + b − 1| is relatively large, both pathways jointly affect the system’s offset dynamics. If a + b > 1, then –(a + b − 1) < 0, and an increase in d suppresses the growth of x_n+1, acting as a negative feedback mechanism. In this case, the offset term –d(a + b − 1) and the nonlinear feedback term (x_n − d)² compete with each other: the nonlinear term tends to induce oscillations and chaos, whereas the offset term tends to pull the system back toward stability. Their interaction may lead the system to maintain a more stable state, requiring stronger excitation to trigger bifurcation. Conversely, if a + b < 1, then –(a + b − 1) > 0, and an increase in d facilitates the growth of x_n+1, forming a positive feedback mechanism. In this case, the terms –d(a + b − 1) and (x_n − d)² act synergistically: the former pushes the system away from equilibrium, while the latter provides a restoring effect. Their combined influence intensifies the system’s instability, making it more likely to undergo bifurcations and enter a chaotic state.

4. Manifold of Conditional Symmetry

Polarity inversion is a common approach for generating symmetric dynamical behaviors; however, in discrete systems, a direct polarity inversion of state variables generally breaks the polarity balance of the system, leading to asymmetric attractor distributions. This imbalance motivates the introduction of an additional mechanism to restore polarity balance. Take x_n+1 → –x_n+1, x_n → –x_n, y_n+1 → y_n+1 + d and y_n → y_n + d and a transformation y_n → F(y_n) with the following condition: F(y_n+1 + d) = –F(y_n+1) and F(y_n + d) = –F(y_n), then the polarity balance of the system can be recovered. To satisfy the above antisymmetric conditions in a simple and structure-preserving manner, the absolute value function is adopted. The absolute value operation naturally provides polarity inversion with respect to a shifted reference point while maintaining continuity and boundedness of the transformed variables. Compared with other nonlinear functions, it offers a minimal and effective way to couple polarity inversion with offset boosting, making the resulting transformation intuitive and robust. Typically, an absolute value function is employed to give the polarity inversion from offset boosting. As in a continuous system, if the polarity balance arises jointly from polarity inversion and offset boosting, the system is said to have conditional symmetry. Therefore, to realize conditional symmetry in a discrete system, after performing a polarity inversion in any of the variables, offset boosting should be applied to the other variables to restore polarity balance.

Let us take x_n+1 → –x_n+1, x_n → –x_n, y_n+1 → y_n+1 + e and y_n → y_n + e in a discrete system (1). According to the above definition, to maintain the polarity balance of the system, a function F(y) must be introduced such that F(y + e) = –F(y). Therefore, by introducing the function F(y) = |y| − e, the transformed system exhibits conditional symmetry. Accordingly, the conditional symmetric version of the discrete system (1) can be expressed as

$$\left\{\begin{array}{l}{x}_{n+1}=bF\left(y_n\right)+a{x}_n,\\ y_{n+1}=y_n+c{x}_n^2+h.\end{array}\right.$$

(15)

In Case 1, with initial conditions IC = (0.1, 0.1), h = 0.5, c = −1, and a = 1 − b while b varies in the range [1, 1.73], it can be observed that, across different phase trajectories corresponding to various bifurcation parameters b, system (15) consistently exhibits conditional symmetry with respect to x. Moreover, by adjusting the constant e, the attractor positions along the y-dimension can be shifted. The corresponding phase portraits are presented in Figure 10. Figure 11a shows the discrete sequences for b = 1.7 under two sets of initial conditions: IC = (0.1, 0.1 + e) and IC = (−0.1, 0.1 − e). As evidenced by the uniform-width bifurcation diagrams, invariant Lyapunov exponents, and parallel mean lines in Figure 11b–d, system (15) satisfies conditional symmetry. Figure 12 illustrates the basins of attraction for the two coexisting attractors in system (15), where the green region corresponds to the green attractor and the purple region corresponds to the purple attractor. Due to the extreme sensitivity of the hyperchaotic system to initial conditions, a green ring appears around the boundary of the purple region. Nevertheless, this does not affect the conditional symmetry properties of the system.

According to the above definition, to achieve conditional symmetry in a discrete system, it is necessary to introduce an offset boosting to the variable not undergoing polarity inversion, following the incorporation of an absolute value or sign function, in order to restore polarity balance. In contrast, the construction of a two-dimensional offset boosting in a discrete system requires adding the same constant term to both sides of the system equation to achieve offset cancelation, leaving only a constant term on the right-hand side. By adjusting this single constant parameter, offsets along both dimensions can be realized. In this study, the absolute value function is employed. However, after adding or subtracting a constant term within the absolute value function, the sign of the constant cannot be determined, preventing the offset cancelation on both sides of the equation. Consequently, it is not possible to achieve a two-dimensional offset in the discrete system while simultaneously maintaining conditional symmetry.

5. Circuit Implementation Based on CH32

The hardware implementation of the above discrete hyperchaotic map is based on the CH32V307, an interconnected microcontroller designed on a 32-bit RISC-V architecture. This platform features a hardware stack and fast interrupt entry, significantly improving interrupt response speed compared to standard RISC-V implementations. The CH32V307 operates at a maximum frequency of 144 MHz and is equipped with 256 KB of Flash memory and 64 KB of SRAM. It supports single-cycle multiplication, hardware division, and hardware floating-point operations, providing enhanced computational performance. In addition to a 2-unit, 16-channel, 12-bit ADC and 16 TouchKey inputs, the microcontroller is equipped with two 12-bit DAC converters, facilitating analog-to-digital and digital-to-analog signal conversion. The system converts chaotic digital signals into analog voltage signals in the range of 0–4095, which are cyclically written into the DAC_DORx registers via the main program loop. For each iteration, the next cycle uses the previously generated chaotic data as the initial condition. By inputting these analog voltage signals into the X/Y axes of an oscilloscope, the corresponding phase trajectories of the system can be observed. Figure 13 shows the experimental setup. The two-dimensional offset property and conditional symmetry of the discrete system can be visualized on the oscilloscope. Figure 14, Figure 15, Figure 16 and Figure 17 illustrate the offset phase trajectories and discrete sequences of chaotic map (13) and the conditionally symmetric phase trajectories and discrete sequences of chaotic map (15) under four different values of b: 1.1514, 1.33, 1.52, and 1.7.

6. Application in Path Optimization

Existing studies indicate that traditional intelligent optimization algorithms are prone to being trapped in local optima and often exhibit relatively slow convergence. It is well known that chaotic systems possess pseudo-randomness and good ergodicity, which can be leveraged in intelligent optimization algorithms to improve search quality and algorithmic robustness. Using chaotic sequences to generate the initial population, compared with conventional random initialization, produces a more dispersed population and enhances population diversity. In addition, by introducing chaotic perturbations to replace the standard random function in intelligent optimization algorithms, the search strategy of the algorithms can be further optimized.

The physical significance of hyperchaotic dynamics lies in the coexistence of multiple positive Lyapunov exponents, which indicate exponential divergence along more than one independent direction in the phase space. Compared with chaotic systems characterized by a single positive Lyapunov exponent, hyperchaotic systems exhibit stronger ergodicity and higher-dimensional instability, leading to more complex and less predictable trajectories. From an optimization perspective, this enhanced instability translates into improved exploration capability. The presence of two positive Lyapunov exponents enables the generated sequences to traverse the search space more uniformly, thereby increasing the probability of escaping local optima. In robotic path planning, this property is particularly beneficial in complex or cluttered environments, where conventional chaotic sequences may still exhibit limited exploration due to lower-dimensional dynamics.

Based on this idea, three classical intelligent optimization algorithms are selected in this work: Ant Colony Optimization (ACO) [41], Grey Wolf Optimization (GWO) [42], and Sparrow Search Algorithm (SSA) [43]. Chaotic map (15) is introduced during the initialization phase of each algorithm, resulting in the Hyperchaotic Ant Colony Optimization (HCACO), Hyperchaotic Grey Wolf Optimization (HCGWO), and Hyperchaotic Sparrow Search Algorithm (HCSSA), which are then applied to robot path planning. The convergence performance and optimality of these six algorithmic variants are investigated to validate the reliability of the algorithms and the feasibility of the system. Taking HCGWO as an example, the algorithm flowchart is illustrated in Figure 18.

The simulation platform used in this study is MATLAB R2021a. On this platform, the convergence performance and optimality of six intelligent optimization algorithms—ACO, HCACO, GWO, HCGWO, SSA, and HCSSA—are tested for solving mobile robot path planning problems. To ensure a consistent experimental environment, all tests are performed on an AMD Ryzen 5 5600X (6-Core) processor. For environment modeling, the grid method is employed, representing the robot’s workspace as a two-dimensional plane composed of uniformly sized grids with binary occupancy information. Considering that most real-world objects are irregularly shaped, a dilation method is applied during modeling: obstacles that occupy less than one grid are expanded to occupy a full grid, ensuring that each obstacle occupies at least one grid. In this study, a 20 × 20 high-density obstacle grid environment is simulated. To guarantee fairness and reliability, each of the six algorithms is independently executed 15 times in the same environment, and the most frequently occurring path among the 15 runs is selected as the final path result. The population size is set to 50, and the maximum number of iterations is 20.

Figure 19 presents the comparison of convergence curves of six intelligent optimization algorithms under the same robotic environment, and the corresponding path maps are shown in Figure 20. When applied to robotic path optimization, the shortest path lengths obtained by HCGWO, GWO, HCSSA, SSA, HCACO, and ACO are 29.0351, 30.3623, 29.9547, 31.0219, 32.5498, and 32.8832, respectively. Analyzing Figure 20, it is evident that the paths generated by the three hyperchaos-enhanced algorithms are smoother than those produced by the original algorithms. It can be concluded that the shortest paths obtained using the chaos-enhanced algorithms designed with the introduced system (15) are shorter than those of the original algorithms. Compared with the original algorithms, the hyperchaos-enhanced algorithms exhibit better convergence performance and stronger global search capability, confirming the reliability of the algorithms and the feasibility of the system. This demonstrates that incorporating hyperchaotic systems into intelligent optimization algorithms can improve robustness, enhance search capability, and increase convergence quality.

7. Conclusions

This paper introduces a novel two-dimensional discrete hyperchaotic system, characterized by a simple formulation yet rich and complex dynamical behaviors. The system exhibits two-dimensional offset boosting and has the polarity balance for conditional symmetry. Its feasibility and theoretical validity were confirmed through circuit implementation based on the CH32 platform. Integrating chaotic theory into intelligent optimization algorithms has been shown to enhance resistance to premature convergence and improve global search performance. Accordingly, three classical algorithms—Ant Colony Optimization (ACO), Grey Wolf Optimization (GWO), and Sparrow Search Algorithm (SSA)—were combined with the proposed hyperchaotic sequences to develop three chaos-enhanced variants: HCACO, HCGWO, and HCSSA. To evaluate both the reliability of these algorithms and the feasibility of the hyperchaotic system, extensive robot path-planning experiments were conducted under identical environments for all six algorithms. The results demonstrate that the hyperchaos-enhanced algorithms outperform their conventional counterparts in convergence speed, global optimization capability, and path smoothness. These findings confirm that embedding hyperchaotic systems into intelligent optimization algorithms can substantially improve performance, highlighting the significant potential of merging chaos theory with intelligent optimization. Future research will further explore the application of diverse hyperchaotic structures in intelligent optimization, as well as potential integrations with deep learning, adaptive control, and other AI techniques, thereby expanding the practical scope of chaos-enhanced methods. In addition, inspired by recent advances in coupled dynamical systems, an important future direction is the development of structure-preserving iterative algorithms [44,45] for hyperchaotic modeling. In coupled systems such as hub–flexible beam systems, structure-preserving iteration methods employ symplectic schemes to handle rigid-body rotational (or planar) motion and generalized multi-symplectic schemes to describe flexible vibrations, thereby preserving the intrinsic coupling structure and stability of the system [46,47]. Incorporating such structure-preserving schemes may enable more faithful reproduction of hyperchaotic features, reduce numerical distortion in long-term iterations, and further enhance the robustness and reliability of hyperchaos-driven path planning and optimization algorithms.