Symmetric Pseudo-Multi-Scroll Attractor and Its Application in Mobile Robot Path Planning

Abstract

The symmetric multi-scroll strange attractor has shown great potential in chaos-based applications due to its high complexity in phase space. Here, the approach of symmetrization is employed for attractor doubling to generate pseudo-multi-scroll attractors in a discrete map, where a carefully selected offset constant is the key to organizing coexisting attractors. By choosing the Hénon map to generate the pseudo-multi-scroll attractor and implementing the digital circuit on a microcontroller, this study fills a significant gap in the research on discrete chaotic systems. The complexity performance is further validated using a pseudo-random number generator, demonstrating substantial academic contributions to the field of chaos theory. Additionally, a pseudo-multi-scroll attractor-based squirrel search algorithm is first developed, showcasing its practical application in mobile robot path planning. This work not only advances the theoretical understanding of chaotic systems but also provides practical methods for implementation in digital systems, offering valuable insights for policy-making in advanced robotic systems and intelligent manufacturing.

1. Introduction

Attractor control has been established as a prominent and well-researched topic within the nonlinear dynamics community [1,2,3]. Some methods for modifying or transforming chaotic behavior into various useful forms, including precise amplitude control and offset boosting [4,5,6], have been explored and employed to modify chaos. The inherent unpredictability and complexity of chaos have been harnessed by effectively manipulating attractors to improve the functionality and performance of diverse applications such as secure communications, signal processing, and advanced computational systems [7,8,9,10]. Innovations have been opened for modifying chaos, which allow us to use chaos more efficiently and securely by leveraging the unique properties of chaos [11,12,13]. However, more work on this topic is associated with those continuous systems [14,15].

The discrete map is a new channel for chaos generation. For chaos enhancement, some approaches like system cascading and nesting are typically employed. As reported, even discrete memristors can be used to increase the complexity as a new kind of nonlinear feedback [16]. Although the chaos modulation in continuous systems seems to be already mature, there is a large margin for chaos regulation in a discrete map [17,18,19]. For example, attractor doubling is a promising approach for multi-scroll or pseudo-multi-scroll attractor generation. Within an expanded basin of attraction, one can explore chaotic attractors [20,21]. By selecting the appropriate offset, the symmetrization can be realized through the direct substitution of the absolute value function in a continuous system. Following this routine, a pseudo-multi-scroll attractor in a discrete map can be realized based on the regulation of offset boosting. The main achievements are summarized as follows:

(1)

Symmetry-based attractor doubling is firstly applied to pseudo-multi-scroll attractor generation by offset boosting in discrete map. By varying the offset for doubling, a pseudo-multi-scroll attractor is coined, in which case the coexisting attractors become new ones rather than coexisting ones.

(2)

Symmetrization-based attractor doubling along any desired map dimension is accomplished through offset boosting operations. Here, offset regulation is employed as an effective tool for inducing coexisting attractors.

(3)

To validate the effectiveness of pseudo-multi-scroll attractor generation, digital circuit implementation is conducted using the microcontroller platform. A pseudo-random number generator is constructed to demonstrate that as the number of pseudo-multi-scroll attractors increases, the complexity of chaotic signals increases.

(4)

Finally, a pseudo-multi-scroll attractor-based application in mobile robot path planning is explored. The shortest path can be obtained with fewer iterations and shorter duration in this work, but it also shows higher accuracy and better optimization performance.

The rest of this paper is organized as follows: Section 2 demonstrates the general approach for pseudo-multi-scroll attractor generation. Section 3 applies a chaotic map for attractor doubling in various dimensions. Circuit implementation is set up for physical verification in Section 4. In Section 5, the application of the pseudo-multi-scroll attractor is explored. The approach for pseudo-multi-scroll attractor generation is concluded in the last section.

2. Symmetric Pseudo-Multi-Scroll Attractor Generation

The primary method for positioning an attractor in any desired dimension, without altering the original dynamics, is offset boosting [22]. Based on [23], the mechanism for offset boosting in a discrete map undergoes a slight variation due to the iterative adjustment of the entire sequence [24]. Doubling the coexisting attractors within the relevant dimensions to achieve attractor doubling requires adjusting the substitution of an absolute value function accordingly [25]. Offset boosting is indeed an effective method for reorganizing attractors in phase space, allowing them to double or grow [22]. Compared to other methods, it is simple and efficient. However, it has the drawback of introducing the special function signum, which is non-differentiable, requiring a continuous equivalent treatment when solving for the system’s Lyapunov exponents. Additionally, the introduction of the absolute value function increases the complexity of feedback [23]. This mechanism is now explained in the following.

Lemma 1

: The attractors in a continuous n-dimensional dynamical system F(X), F = (f₁, f₂, …, f_i, …f_n), X = (x₁, x₂, …x_i,…, x_n), can obtain attractor doubling in the dimension of x_i by substituting an absolute value function x_i → |x_i| − d, f_i → sgn(x_i)f_i.

3. Hénon Attractor and Its Symmetric Versions of Pseudo-Multi-Scroll Attractor

A discrete sequence shows the changes in the values of x and y over the iterations. This can be understood as the time series plot of the x and y values at each iteration step. For a better demonstration, the effectiveness of the aforementioned approach is demonstrated by selecting a typical chaotic map. Here is the Hénon map [26]:

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

(1)

where a is 1.4, b is 0.3, and the initial condition (x₀, y₀) = (0.1, 0.1). Parameters a and b are system parameters, and x_n and y_n are system variables, and represent the nth-iteration of the system. Using MATLAB 2018a software, the numbers proposed in this paper are system parameters, which ensure the generation of a chaotic sequence and allow the attractor to be expanded in the phase space. The corresponding parameters are substituted into the respective equations to obtain our simulation results. Following the iteration method of Equation (1), where the current result is used as the next iteration value x_n, the values of x_n and y_n for nth-iterations are plotted separately to obtain the discrete sequence diagram, as shown in Figure 1. To facilitate checking for the presence of chaotic attractors, the x-y phase diagram is plotted, where the horizontal axis represents variable x and the vertical axis represents y, to obtain the system phase orbits. The subsequent phase orbits and discrete sequences in this paper are all obtained using the above method, but the iterative equations and equation parameters are different.

The data are obtained from the natural solution of the difference equations and are presented as a set of randomly evolving sequences [26]. These sequences are the solutions of the difference equations. Thus, the solutions are the data, and the data are the solutions of the difference equations. As a result, it is demonstrated that these solutions transform from a single set into multiple sets in phase space. These multiple sets can either be extracted using different initial values or, when forming pseudo-attractors, can be extracted using any initial value.

3.1. Symmetric Pseudo-Multi-Scroll Attractor in the x-Dimension

To achieve attractor doubling, introducing a direct absolute value function in the x-dimension is key. This preserves the integrity of the difference equation by maintaining x_n on the right-hand side while designating other terms involving x_n as F(x_n) = |x_n| − d [23]:

$$\left\{\begin{array}{l}{x}_{n+1}=x_n+\mathrm{sgn}(x_n)(1-a{F}^2(x_n)+y_n-F(x_n))\\ y_{n+1}=bF(x_n)\end{array}\right.$$

(2)

where a is 1.4, b is 0.3, d is 1.28, and the initial condition (x₀, y₀) = (0.1, 0.1). Parameters a, b, and d are system parameters, and x_n and y_n are system variables, and represent the nth-iteration of the system (2). Balancing polarity, the sgn(x_n) function is introduced, derived from the derivative of the absolute value function. This facilitates the doubling of the original attractor, with the constant d controlling the distance between them. In continuous systems, a small value of d causes the doubled attractors to closely approach each other, resulting in a pseudo-double-scroll attractor. As seen in Figure 2, pseudo-attractors are produced by a proper offset booster d. Chaotic sequences oscillate between positive and negative intervals in the x-dimension, as shown in Figure 2b.

The above operation can be repeated anytime for more coexisting phase orbits. For example, if the algorithm is executed twice, the number of coexisting attractors reaches four. The modified equation is:

$$\left\{\begin{array}{l}{x}_{n+1}=x_n+\mathrm{sgn}(x_n)(\mathrm{sgn}(F^{\prime }(x_n))(1-a{F}^2(F^{\prime }(x_n))+y_n-F(F^{\prime }(x_n))))\\ y_{n+1}=bF(F^{\prime }(x_n))\end{array}\right.$$

(3)

where a is 1.4, b is 0.3. Parameters a, b, and h are system parameters, and x_n and y_n are system variables, and represent the nth-iteration of the system (3). F’(x_n) = |x_n| − h, and h, d control the distances on one side and both sides separately. When the coexisting two attractors get too close and controlled by the offset controllers h, d, they link together as a whole. As shown in Figure 3, chaotic oscillation of four-scroll attractors is plotted. To obtain enough space for coexisting attractors, here h > d. This case is quite like the one in continuous systems. Separately, coexisting attractors hug together, becoming pseudo-multi-scroll attractors, which depend on the two offsets. A pseudo-four-scroll attractor with d = 1.28, h = 2.55 is plotted in Figure 3a,b. Switching oscillation with positive or negative polarity can be further verified from the sequences. Further simulation shows that coexisting pseudo-double-scroll attractors are observed with d = 1.28, h = 3, where blue: (x₀, y₀) = (0.1 + d, 0.1) = (1.38, 0.1) and red: (x₀, y₀) = (0.1 − d, 0.1) = (−1.18, 0.1), as shown in Figure 3c,d. When the attractor doubles twice in the x-dimension and two different offsets are endowed with different values, the coexisting four attractors are captured by proper initial conditions with d = 2, h = 4, where green: (x₀, y₀) = (0.1 − h − d, 0.1) = (−5.9, 0.1), magenta: (x₀, y₀) = (0.1 − d, 0.1) = (−1.9, 0.1), turquoise: (x₀, y₀) = (0.1 + d, 0.1) = (2.1, 0.1), and gold: (x₀, y₀) = (0.1 + d + h, 0.1) = (6.1, 0.1), as illustrated in Figure 3e,f.

3.2. Symmetric Pseudo-Multi-Scroll Attractor in the y-Dimension

This approach can be applied across multiple dimensions, allowing for the doubling of the attractor in the y-direction as well:

$$\left\{\begin{array}{l}{x}_{n+1}=1-a{x}_n^2+G(y_n)\\ y_{n+1}=y_n+\mathrm{sgn}(y_n\left)\right(b{x}_n-G(y_n))\end{array}\right.$$

(4)

where a is 1.4, b is 0.3, p is 0.38, and the initial condition (x₀, y₀) = (0.1, 0.1). Parameters a, b, and p are system parameters, and x_n and y_n are system variables, and represent the nth-iteration of the system (4). When G(y_n) = |y_n| − p, this study sets the function sgn(y_n) for polarity balance in the y-dimension [23]. Doubled coexisting attractors can be visited by a suitable set of initial conditions. Small offset p makes a pseudo-double-scroll attractor, as shown in Figure 4a. Discrete sequences are plotted in Figure 4b.

Repeated operations process more doubled coexisting attractors, as in the case above, and shown by Equation (5). As shown in Figure 5, four attractors are reproduced in different forms, and offset constants p and q are selected accordingly, where G’(y_n) = |y_n| − q:

$$\left\{\begin{array}{l}{x}_{n+1}=1-a{x}_n^2+G(G^{\prime }(y_n))\\ y_{n+1}=y_n+\mathrm{sgn}(y_n\left)\mathrm{sgn}\right(G^{\prime }(y_n)\left)\right(b{x}_n-G(G^{\prime }(y_n)))\end{array}\right.$$

(5)

where a is 1.4 and b is 0.3. Parameters a, b, and q are system parameters, and x_n and y_n are system variables, and represent the nth-iteration of the system (5). Pseudo-four-scroll attractors with p = 0.38, q = 0.76 are plotted in Figure 5a,b. Coexisting pseudo-double-scroll attractors with p = 0.38, q = 1 are shown in Figure 5c,d, where green: (x₀, y₀) = (0.1, 0.1 + q) = (0.1, 1.1) and red: (x₀, y₀) = (0.1, 0.1 − q) = (0.1, −0.9). Four coexisting attractors with p = 0.5, q = 1 are shown in Figure 5e,f, where green: (x₀, y₀) = (0.1, 0.1 − p − q) = (0.1, −1.4), magenta: (x₀, y₀) = (0.1, 0.1 − p) = (0.1, −0.4), turquoise: (x₀, y₀) = (0.1, 0.1 + p) = (0.1, 0.6), and gold: (x₀, y₀) = (0.1, 0.1 + p + q) = (0.1, 1.6).

3.3. Compound Symmetric Structure of Pseudo-Multi-Scroll Attractor in Phase Space

As attractor doubling operations accumulate, they yield an increasing number of attractors that occupy symmetrical phase space. Now, let the attractor double in both dimensions x and y. Assume the attractor undergoes two doublings in each dimension, as implemented by the equation,

$$\left\{\begin{array}{l}{x}_{n+1}=x_n+\mathrm{sgn}(x_n)(\mathrm{sgn}(F^{\prime }(x_n))(1-a{F}^2(F^{\prime }(x_n))+G(G^{\prime }(y_n))-F(F^{\prime }(x_n))))\\ y_{n+1}=y_n+\mathrm{sgn}(y_n)\mathrm{sgn}(G^{\prime }(y_n))bF(F^{\prime }(x_n))-G(G^{\prime }(y_n))\end{array}\right.$$

(6)

where a is 1.4, b is 0.3. Parameters a, b, d, h, p, and q are system parameters, and x_n and y_n are system variables, and represent the nth-iteration of the system (6). A larger offset clearly separates coexisting attractors, and a smaller offset links coexisting attractors together. The attractor doubles twice in the x-dimension and y-dimension. Different colors represent different phase orbits obtained at different initial values. These are commonly known as the coexisting chaotic attractors. Coexistence modes are different and colors are different. When the substitution operation of the absolute value function is repeated, pseudo-sixteen-scroll attractors with d = 1.28, h = 2.552, p = 0.38, and q = 0.76 are plotted in Figure 6a. Coexisting pseudo-four-scroll attractors in the y-dimension with d = 1.28, h = 2.552, p = 0.5, and q = 1 are shown in Figure 6b, where green: (x₀, y₀) = (0.1, 0.1 − p − q) = (0.1, −1.4), magenta: (x₀, y₀) = (0.1, 0.1 − p) = (0.1, −0.4), turquoise: (x₀, y₀) = (0.1, 0.1 + p) = (0.1, 0.6), and turquoise: (x₀, y₀) = (0.1, 0.1 + p + q) = (0.1, 1.6). Coexisting pseudo-four-scroll attractors in the x-dimension with d = 1.28, h = 2.552, p = 0.38, and q = 0.76 are shown in Figure 6c, where green: (x₀, y₀) = (0.1 − d − h, 0.1) = (−3.732, 0.1), magenta: (x₀, y₀) = (0.1 − d, 0.1) = (−1.18, 0.1), turquoise: (x₀, y₀) = (0.1 + d, 0.1) = (1.38, 0.1), and turquoise: (x₀, y₀) = (0.1 + d + h, 0.1) = (3.932, 0.1). A total of sixteen coexisting attractors are produced by Equation (6), as shown in Figure 6d. By adjusting the offsets in various combinations, we can arrange coexisting attractors differently. The quantity of these attractors increases exponentially, specifically by 2ⁿ, where n represents the number of iterations of the absolute value function.

4. Physical Experiment Based on Microcontroller

Typically, researchers use analog circuits to replicate the chaotic oscillation. However, these analog circuits face issues such as poor confidentiality, vulnerability to interference, and unstable signals. In contrast, CH32-based digital circuits are easier to integrate, more resistant to noise, and offer flexible parameter adjustments for chaotic systems, making them ideal for replicating neuronal activities [24]. The computation and output of the chaotic map are managed by the CH32V307 microcontroller, which utilizes the 32-bit RISC-V instruction set and architecture. This microcontroller can reach clock speeds up to 144 MHz and includes 256 KB of flash memory and 64 KB of SRAM. Its hardware support for floating-point operations significantly boosts calculation speed, making it suitable for chaotic map implementation. Additionally, the CH32V307 features two 12-bit DAC converters that transform chaotic digital signals into analogue voltage outputs, which can be visualized on an oscilloscope.

The equation is integrated into the main program loop, where it begins iterating after initializing variables and setting the iteration count. Each iteration scales the calculated chaotic data to 0–4095 and writes these values into the DAC_DORx register, initiating the DAC conversion process that transforms digital chaotic signals into analogue voltage outputs. These outputs are the initial values for subsequent iterations, continuing until the specified number of iterations is reached. Finally, the analogue voltage signals from both outputs are transmitted to the oscilloscope for monitoring. The physical experiment based on the microcontroller is shown in Figure 7. Attractor doubling one time with a = 1.4, b = 0.3 under (x₀, y₀) = (0.1, 0.1) observed from the oscilloscope is shown in Figure 8, where the operation of attractor doubling is built in the x- or y-dimension. Attractor doubling two times with a = 1.4, b = 0.3 under (x₀, y₀) = (0.1, 0.1) observed from the oscilloscope is shown in Figure 9, and a pseudo-four-scroll attractor and coexisting four attractors are verified in the circuit. Here, we verify the correctness of the above MATLAB numerical simulation and theoretical analysis through the circuit platform.

5. Applications: Mobile Robot Path Planning

5.1. Pseudo-Random Number Generator

The chaotic sequences are represented as X = {x₁, x₂,…x_i,…x_n} and Y = {y₁, y₂,…y_i,…y_n}, which can be effectively utilized as a pseudo-random number generator. These sequences underwent rigorous evaluation using the tests provided by the National Institute of Standards and Technology [27]. The evaluation process involves a comprehensive suite of statistical procedures designed to detect and measure random biases within binary sequences, ensuring the reliability and robustness of the pseudo-random number generator for various applications in secure communications and cryptographic systems. The National Institute of Standards and Technology test suite evaluates binary pseudo-random number generator sequences generated through a series of 15 tests. Frequency Test: Assesses if the number of 0s and 1s in a bit sequence are approximately equal, which is a fundamental property of a random sequence. Block Frequency Test: Similar to the Frequency Test but applied to larger blocks, this test checks for uniform distribution within these blocks. Cumulative Sums (Cusum) Test: Evaluates the randomness by calculating the cumulative sum of the partial sums of the sequence, testing for shifts from randomness. Runs Test: Checks if the lengths of consecutive identical bits (runs) conform to the expected distribution for a random sequence. Longest Run of Ones in a Block Test: Determines if the longest run of 1s within fixed-length blocks fits the expected distribution for randomness. Rank Test: Examines the rank of matrices formed by the bit sequence to ensure it reflects randomness. Discrete Fourier Transform (Spectral) Test: Analyzes the periodicity of the sequence through spectral analysis to detect repetitive patterns. Non-overlapping Template Matching Test: Evaluates the frequency of predefined, non-overlapping templates in the bit sequence. Overlapping Template Matching Test: Similar to the Non-overlapping Template Test but checks for overlapping templates. Maurer Universal Statistical Test: Assesses the compressibility of the bit sequence, where less compressible sequences are considered more random. Approximate Entropy Test: Measures the entropy (complexity) of the sequence to determine its randomness. Random Excursions Test: Evaluates if the deviations of the bit sequence conform to a random walk model. Random Excursions Variant Test: Extends the Random Excursions Test to assess various deviation scenarios. Serial Test: Examines the frequency of occurrence of blocks of a certain length within the bit sequence. Linear Complexity Test: Checks if the linear complexity of the sequence matches that of a random sequence. Each test produces P-values compared to a predetermined significance level (α = 0.01). If the P-values are greater than or equal to α, the null hypothesis is accepted, indicating that the sequence is random [5]. On the other hand, if the P-values are less than α, the null hypothesis is rejected, suggesting that the sequence is non-random. The quantization function P_i is defined as follows:

$$P_i=⌊(X_i+\left|X_{\min }\right|.K)⌋\mathrm{mod}N$$

(7)

where the minimum value of the chaotic solution is represented by X_min, and k = 10¹², N = 128, a = 1.4, b = 0.3 with (x₀, y₀) = (0.1, 0.1) in this work. Here, three cases are selected as examples to show the high performance of the proposed method. Case 1: original Hénon Equation (1); Case 2: one-time attractor doubling in Equation (2); Case 3: two-time attractor doubling in the Equation (3). The results of the National Institute of Standards and Technology statistical tests for the proposed pseudo-random number generator are displayed in

Table 1. NIST statistical test of the proposed pseudo-random number generator with a = 1.4, b = 0.3 under (x₀, y₀) = (0.1, 0.1).

No.	Statistical Test Terms	Original Hénon	One-Time Doubling Hénon	Two-Time Doubling Hénon
01	Frequency	0.589203	0.793912	0.892038
02	Block frequency	0.391830	0.391038	0.510291
03	Cumulative sums	0.471192	0.503819	0.491038
04	Runs	0.581291	0.710812	0.839109
05	Longest run	0.737391	0.820391	0.962815
06	Rank	0.628192	0.658290	0.721021
07	FFT	0.518293	0.503819	0.618291
08	Non-overlapping template	0.819371	0.899103	0.881796
09	Overlapping template	0.881937	0.910938	0.957863
10	Universal	0.137193	0.310291	0.392018
11	Approximate entropy	0.518291	0.738102	0.801827
12	Random excursions	0.338937	0.502948	0.610282
13	Random excursions variant	0.371837	0.409381	0.519202
14	Serial	0.381930	0.373819	0.391082
15	Linear complexity	0.612719	0.729719	0.762819

. We observed that the complexity is enhanced with the increase in operation times. To evaluate the uniformity of the P-values, P-value_T is determined using the following calculation:

P-value_T = igamc(9/2, χ²/2),

(8)

The game function is the complementary incomplete gamma function used to calculate P-values in statistical tests [5]. The satisfactory P-values are denoted by F_i, and the incomplete gamma function igamc(.) is computed as follows:

$${\chi }^2={\displaystyle \sum _{i=1}^{10}\frac{{(F_i-s/10)}^2}{s/10}}$$

(9)

5.2. Mobile Robot Path Planning

Drones, window cleaning robots, and restaurant delivery robots all utilize pathfinding optimization. This method can indeed be applied to robots in various scenarios, enhancing their efficiency and performance. Therefore, in this section, we delve into the optimization of mobile robot path planning [28,29]. This paper introduces the aforementioned discrete chaotic sequences into the traditional squirrel search algorithm [30]. Random sequences were replaced with chaotic sequences, resulting in the construction of the chaos squirrel search algorithm [31]. Comparative tests were conducted on mobile robot path planning problems. As shown in Equation (10), the random sequence in the x-dimension was normalized to a specified interval, the size of which is determined by h. Here, h is set to 1, yielding random numbers within the interval [0, 1].

$$t_i=\frac{x_i-Min(x)}{Max(x)-Min(x)}·h$$

(10)

To enhance population diversity, this random sequence is introduced during the population initialization phase of the traditional squirrel search algorithm. As shown in Figure 10, the operational environment of the mobile robot was modeled using a grid method. Each grid cell was labeled in a sequence from bottom to top and from left to right, with the side length of each grid cell designated as unit length 1. The grid model was represented using a binary matrix as depicted in Equation (11).

$$M=\left[\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& 1& 1& 0& 0\\ 1& 1& 1& 0& 1& 0& 1& 1& 0& 0\\ 1& 1& 1& 0& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 1& 1& 1& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 1& 1& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(11)

Using Equation (12), the grid labels were converted from serial numbers to two-dimensional coordinates:

$$\left\{\begin{array}{l}{x}_i=b\times (\mathrm{mod}(i-1,MM)+1),\\ y_i=b\times (\mathrm{int}((i-1)/MM)+1)\end{array}\right.$$

(12)

Here, x and y represent the grid’s horizontal and vertical coordinates, b denotes the side length of a unit grid, MM indicates the map’s dimensions, int() is the floor function, and mod() is the modulus function.

When solving the mobile robot path planning problem using the chaos squirrel search algorithm, each path is represented as an individual in the population, with each one consisting of two components (x, y). These two components represent a node position in the path. Therefore, with S as the specified starting point and E as the specified ending point, a path with n intermediate nodes obtained through the path planning optimization algorithm can be represented by Equation (13).

$$L=\{S,(x_1,y_1),(x_2,y_2)\dots (x_i,y_i)\dots (x_n,y_n),E\}$$

(13)

The point (x_i, y_i) represents the i-th intermediate node traversed by the path.

The length of each path is used as the fitness function, as shown in Equation (14).

$$fit={\displaystyle \sum _{i=1}^n\sqrt{{(x_i-x_{i-1})}^2+{(y_i-y_{i-1})}^2}}$$

(14)

where (x_i, y_i) represents the current coordinates of the robot, and (x_i-1, y_i-1) denotes the coordinates of its previous position, with n being the number of steps taken by the robot. Passive voice translation:

The basic squirrel search algorithm and chaos squirrel search algorithm with different sequences [30,31], including the original Hénon Equation (1), one-time doubling Hénon Equation (2), two-time doubling Hénon Equation (3), and Logistic map [32], are utilized to solve the mobile robot path planning problem in a 20 × 20 grid environment. The basic parameters for all five algorithms are consistent: a maximum of 200 iterations, a population size of 50, and a safety value of 0.8, with a ratio of 0.3 for explorers and 0.2 for scouts. Each algorithm is independently run 30 times, and convergence curves and path diagrams are plotted based on the mean values, as shown in Figure 11 and Figure 12. In Figure 11, the squirrel search algorithm is denoted by SSA, the Logistic-based squirrel search algorithm is denoted by Logistic, the original Hénon-based squirrel search algorithm is denoted by OH, the one-time attractor doubling-based squirrel search algorithm is denoted by OTDH, and the two-time attractor doubling-based squirrel search algorithm is denoted by TTDH. A comparison of the effectiveness of mobile robot path-planning algorithms is shown in

Table 2. Comparison of the effectiveness of mobile robot path planning algorithms in a 20 × 20 grid environment.

Algorithm	Shortest Path	Turning Points	Time/s	Iterations
Squirrel Search Algorithm	31.1633	13	1.2868	35
Logistic [32]	30.9090	14	0.8892	34
Original Hénon Equation (1)	30.7477	14	0.8351	34
One-time doubling Hénon Equation (2)	29.8503	14	0.7956	32
Two-time doubling Hénon Equation (3)	29.3980	12	0.7751	29

.

The chaotic data are introduced into the squirrel search algorithm to create a new algorithm. As shown in Table 2, With the increment of attractor doubling, the shortest path can be obtained with fewer iterations and shorter duration in this work, but it also shows higher accuracy and better optimization performance. However, this method introduces the non-differentiable signum function, necessitating a continuous equivalent treatment for the calculation of the system’s Lyapunov exponents. Moreover, the incorporation of the absolute value function complicates the feedback dynamics.

6. Conclusions and Discussion

In this paper, symmetric pseudo-multi-scroll attractors are obtained based on offset boosting from attractor doubling. This method effectively expands the phase orbits in discrete maps, by which any desired number of scrolls in an attractor is constructed depending on the operation times of symmetrization. The number of multiple scrolls will increase according to 2ⁿ, where n indicates how many times the absolute value functions have been applied during the iterations. Furthermore, circuit implementation based on the microcontroller is carried out to verify the numerical simulation of the attractor doubling. Finally, the application of a pseudo-random number generator is employed to show its high performance. The chaos squirrel search algorithm method employing the two-time doubling Hénon equation stands out among other algorithms, demonstrating superior optimization performance by achieving the shortest path with higher precision in a shorter duration and with fewer iterations [30].

Based on the findings of this paper, future research directions could focus on optimizing the methods for constructing complex attractors, particularly through enhancements of symmetrization and offset boosting to reduce computational complexity and increase efficiency. Further exploration of other nonlinear functions and mapping methods may provide more diverse multi-scroll attractor structures [4]. Additionally, beyond pseudo-random number generation, there is potential for extending the application of chaotic sequences to fields such as encrypted communication, image processing, and information hiding, leveraging their advantages and developing new application schemes. Through these research avenues, the generation methods and application scope of multi-scroll attractors can be significantly advanced, driving further development in chaotic system theory and practice [31].