Two-variable boosting bifurcation in a hyperchaotic map and its hardware implementation

There are few reports on the nondestructive adjustment of the oscillation amplitude of the chaotic sequence in the discrete map. To study the lossless regulation of the oscillation amplitude of chaotic sequences, this article proposes a new simple two-dimensional (2D) hyperchaotic map with trigonometric functions. It not only exhibits the offset boosting bifurcation and offset boosting coexistence attractors, but also shows the offset boosting of two state variables with respect to arbitrary parameters in the 2D map. The simulation results of bifurcation diagram, maximum Lyapunov exponent and attractor phase diagram show that the map can produce complex dynamical behaviors. In addition, the introduction of new control parameters into the 2D hyperchaotic map can also make the hyperchaotic map exhibit rich multi-stable phenomena. At the same time, the covariation of the initial state and control parameters can result in arbitrary switching and coexistence of attractors in the phase plane. The 2D hyperchaotic map was tested and verified by hardware experiment platform. Moreover, we design a pseudo-random number generator (PRNG) to test the hyperchaotic map. The results show that the pseudo-random numbers generated by the hyperchaotic map have high randomness.


Introduction
Research on chaotic systems, it has received more and more attentions due to its potential application value. A large number of nonlinear systems (continuous system and discrete system) exhibit chaotic behavior [1,2]. Although the dynamic behavior of the chaotic system is determined by the initial state, the long-term dynamics of the chaotic system caused by the complexity of the chaotic system are still unpredictable. Meanwhile, chaotic systems can be divided into hyperchaotic systems and chaotic systems based on the positive and negative values of the Lyapunov exponent. If the continuous-time chaotic system is to produce hyperchaotic, then, the continuous-time system should be at least a 4 dimensional (4D) chaotic system [3]. In contrast, the discrete chaotic map requires only two dimensions to generate hyperchaotic phenomena [4,5]. Bao et al. recently found in their study of discrete modeling of memristors that rich hyperchaotic behavior can be generated when memristors are coupled with chaotic maps compared to classical 2D discrete chaotic maps [6]. Noticeably, the high complexity of hyperchaotic system makes it useful in more applications [7].
In many continuous chaotic systems, researchers have discovered the coexistence of self-excited or hidden attractors, and this phenomenon of coexistence of self-excited or hidden attractors is called "multistability", which has been reported in [8,9]. In particular, the phenomenon of "multi-stability" is also common in many circuit systems [10][11][12][13][14], such as memristive oscillator circuits [15][16][17], memristive neural networks [18,19], Hindmarsh-Ross neuron oscillators [20], etc. One common characteristic of these circuit systems is their high sensitivity to initial conditions. The discrete chaotic map itself is closely related to the continuous-time chaotic system, in which there is the coexistence of hidden and self-excited attractors. Similarly, discrete chaotic maps would theoretically have a similar "multi-stability" phenomenon. Researchers have discovered the coexistence of hidden and self-excited attractors in Henon maps [21], periodic feedback modulation maps and many nonlinear hyperchaotic maps, which also provides an important basis for studying the "multi-stability" of discrete maps [22]. In addition, the coexistence of "multi-stability" attractors in a chaotic system is of great importance. This is because the disturbances to the initial conditions or system parameters of a chaotic system can greatly change the system stability and result in more complex dynamic behaviors [23][24][25]. In particular, in the chaotic map with initial-boosting, the change of the initial value can lead to the coexistence of the initial-switch boosting of the attractor. The change of the initial state in the chaotic map will generate a wealth of multistate phenomenon, which makes the discrete chaotic map worth investigating [26].
For multi-stability chaotic system, the final state of the system is closely related to its initial conditions [27,28]. The initial state-dependent multi-stability is an inherent property of the chaotic system, and the multistate system can be realized in two ways. The first method is to introduce the resistor into the chaotic system. The memristor can be easily coupled with the chaotic system to form a memristive chaotic system. In theory, the memristive chaotic system has an infinite number of equilibrium points, which tend to introduce multiple attractors that depend on the initial state to coexist. The other method is to introduce trigonometric functions into the offset boostable system. Due to the periodicity of these trigonometric functions, this also determines that the offset-boostable system that introduces trigonometric functions will have infinitely many equilibrium points. So, the nonlinear systems can have an infinite number of offset-boostable attractors that coexist [29]. Recently, under the inspiration of the two methods, researchers proposed a more representative method by introducing the memristor with a trigonometric (sine function or cosine function) dielectric constant into the offset-boostable nonlinear chaotic system [30], which helps to implement more complex initial switch boost chaotic systems. Although some research articles on multi-stable systems have been published in recent years, these methods are only applicable to continuous-time chaotic systems, and the literature of lossless boost bifurcation in discrete chaotic maps is still sparse. Therefore, the study of lossless boost bifurcations and extreme multistability in discrete maps is of great interest [31]. By summarizing the above methods, we propose a simple 2D hyperchaotic map in this paper. Compared with the existing chaotic maps, the proposed chaotic map has a simpler mathematical model and the chaotic map has multiple fixed points [32]. The stability of the fixed point depends on the control parameters. Through analysis, it can be found that the proposed map can achieve many self-similar bifurcations and attractors. In particular, the 2D discrete chaotic map proposed in this paper can realize the initial switching boosting of a single state variable, and can also non-destructively control the boosting bifurcation of another state variable by introducing new control parameters. Under the dual action of control parameters and initial state, the attractor can be switched arbitrarily in the phase plane. Introducing new control parameters can realize the control of the attractor state, making the 2D chaotic map exhibit rich dynamic behavior.
The rest of the paper is arranged as follows. Sect. 2 proposes a mathematical model of a 2D hyperchaotic map with trigonometric functions and analyzes the stability of fixed points. In Sect. 3, the initial switching boosted bifurcation and coexistence attractor of a 2D hyperchaotic map are investigated using numerical simulations, and the performance of its initial state controlling chaotic sequences is evaluated. In Sect. 4, a new hyperchaotic map model is obtained by introducing new control parameters, and the parameter-boosting bifurcation based on the control parameters is studied. Moreover, the performance of the parameter-controlled coexistence attractor and chaotic sequence is evaluated. In Sect. 5, the hyperchaotic map is experimentally verified through a digital experimental platform, and a PRNG is designed using the hyperchaotic map to test the randomness of the generated pseudo-random number (PRNs). Finally, the conclusion of this article is given in Sect. 6.

Mathematical model of 2D hyperchaotic map
By introducing the sine trigonometric function into the 2D simple discrete map, a novel 2D simple hyperchaotic map containing the trigonometric function is proposed, which can be expressed as (1).
where n is natural number, x n and y n are the two state variables of the n-th iteration, a and b are the control parameters of the hyperchaotic map. Firstly, the equation of the fixed point of the 2D simple hyperchaotic map can be obtained as (2).
x * = x * + a sin(y * ) Obviously, the 2D simple hyperchaotic map has an infinite number of linear fixed points, and the fixed points can be expressed as (3).
Fixed points in discrete hyperchaotic maps can be discriminated whether the fixed points are stable or not by calculating the eigenvalues of the Jacobian matrix. Therefore, the Jacobian matrix of a 2D hyperchaotic map can be expressed as (4).
According to the Jacobian matrix, the characteristic equation can be obtained as follows (5).
Through the above derivation of the characteristic equation, we can get cos(μπ )=±1 is the fixed point of (x * , y * ), so we can get the two new characteristic equation as.
Obviously, the eigenvalues of these two new characteristic equations can be divided into the following two cases.
According to the above four eigenvalues, it can be seen that the stability of the equilibrium point is related to the value of the system parameters a and b. Clearly, when the parameter μ of the fixed point is odd, the eigenvalues of the characteristic equation at the fixed point are λ 1 and λ 2 . When − 1 4 ≤ ab < 0, |λ 1 |< 1, then the eigenvalue λ 1 is in the unit circle; when − 1 4 ≤ ab < 2, |λ 2 |< 1, then the eigenvalue λ 2 is in the unit circle. Therefore, when − 1 4 ≤ ab < 0, the two eigenvalues at this time are |λ 1 |< 1 and |λ 2 |< 1, and the eigenvalues λ 1 and λ 2 are always inside the unit circle, which makes all the fixed points stable. In addition, when the parameter μ of the fixed point is an even number, the eigenvalues of the eigenequation at the fixed point are λ 3 and λ 4 . When 1 4 ≤ ab, |λ 3 |< 1, then the eigenvalue λ 3 is in the unit circle; when 1 4 ≤ ab, |λ 4 |< 1, then the eigenvalue λ 4 is in the unit circle. therefore, when 1 4 ≤ ab, the two eigenvalues are |λ 3 |< 1 and |λ 4 |< 1. The eigenvalues λ 3 and λ 4 are always inside the unit circle, which makes all the fixed points stable.
In particular, the spectral entropy (SE) diagrams of the parameters a and b are numerically simulated, as shown in Fig. 1. By observing the entropy spectrum of parameters, we find that when the fixed point parameter μ is odd, it positively corresponds to the spectral entropy complexity distribution in the second and fourth quadrants in Fig. 1; whereas when the fixed point parameter μ is even, it positively corresponds to the spectral entropy complexity distribution in the first and third quadrants in Fig. 1. Since the stability of the fixed point is closely related to the parameter μ of the fixed point, and the periodic change of the parameter μ will not affect the stability of the fixed point. It also provides a foundation for the realization of boosting bifurcation of the state variable x based on the initial value in this paper.

Initial value-based initial-boosting bifurcation and boosting coexistence attractor
The phase diagram, iterative sequence diagram, bifurcation diagram and Lyapunov exponent can be used to describe the state of the system well, as well as to study the influence of parameters and initial values on the system.

Parameter-based bifurcation and attractor
To study the complex dynamic behavior of the 2D hyperchaotic map, set the initial value of the chaotic map (x 0 , y 0 ) = (0.1, 0.1) and the system parameter a=1.8. The Lyapunov exponent diagram of the system parameter b and the bifurcation diagram of the state variables x, y are numerically simulated. As shown in Fig. 2, Fig. 2a shows that when the system parameter b=0, the bifurcation diagram will jump periodically. Meanwhile, due to the existence of chaotic crisis, chaotic mapping will generate multiple periodic windows. In particular, when the value range of the control parameter b is (−7.6, −6.35); (−5.5,−5); (−4.3, −3.3) and (3.3, 4.3); (5.55, 6); (6.35, 7). The chaotic map will have two Lyapunov exponents greater than zero, indicating that the chaotic map is in a hyperchaotic state in this interval. By observing the bifurcation diagram and Lyapunov exponent diagram in Fig. 2, it can be found that there are many complex dynamical behaviors in the hyperchaotic map, including periodic, chaotic, hyperchaotic and quasi-periodic. In addition, To study the complex dynamic behavior caused by the change of system parameter a, the initial value of chaotic map is set as (x 0 , y 0 ) = (0.1, 0.1) and the system parameter b = 2. The Lyapunov exponent diagram of the system parameter a and the bifurcation diagram of the state variables x and y in the interval (−8, 8) and (−2.4, 2.4) are numerically simulated. As shown in Fig. 3, in the bifurcation diagram of the state variable x in Fig. 3a, a periodic jump occurs when the system parameter a = 0. Meanwhile, by observing the bifurcation diagram and Lyapunov exponent diagram in Fig. 3, it is shown that there are complex dynamic behaviors such as chaotic, periodic, quasi-periodic and hyperchaotic in the hyperchaotic map, which indicates that the change of system parameter a will lead to complex dynamic behaviors in the chaotic map.
According to the results of the numerical simulation in Fig. 2, by fixing the system parameter b = 2 and changing the value of the system parameter a, the phase diagram and iterative sequence diagram of the coexisting attractor under different, the blue trace in Fig. 4 represents the initial mapping value of (x 0 , y 0 ) = (0.1, 0.1), the red track represents the initial value of the mapping (x 0 , y 0 ) = (−0.1, −0.1). It can be seen from Fig. 4 that in the hyperchaotic map, there will be the coexistence of periodic attractors, the coexistence of chaotic attractors and the coexistence of hyperchaotic attractors. In addition, according to the numerical simulation results in Fig. 3, by fixing the system parameter a = 1.8 and changing the value of the system parameter b, three sets of typical coexisting attractor phase diagrams and their corresponding iterative sequence diagrams can also be obtained. As shown in Fig. 5, the 2D simple hyperchaotic map exhibits abundant dynamic phenomena, including coexisting quasi-periodic attractors, coexisting chaotic attractors and coexisting hyperchaotic attractors. Meanwhile, when the value of the system parameter is b=3.2, the hyperchaotic map presents transient chaos. As shown in Fig. 6, after transient chaos, the chaotic map will eventually be in a quasi-periodic state.
In addition, we evaluated the performance of the attractors in Figs. 4 and 5. These performance indicators mainly include the Lyapunov exponents and spectral entropy (SE). The calculation results of the performance indicators are shown in Table 1. The results show that the 2D hyperchaotic map presented in this paper has complex dynamic properties, and its dynamic behavior is closely related to two system parameters.

Chaos sequence histogram and 0-1 test
A histogram represents a variable in the form of a bar graph, and the area of the bar graph is proportional to the frequency of the variable value in the iterative sequence. To intuitively understand the advantages of 2D hyperchaotic mapping in chaotic encryption, the parameters are set as a = 1.8, b = 2, and the initial conditions are fixed as x 0 = 0.1, y 0 = 0.1. The length of the iterative sequence is 5000, and the distribution map and histogram of the sequence are numerically simulated. As shown in the histogram of the iterative sequence x in Fig. 7b, except for the large proportion of variable values in the interval (2.1, 2.4) and (3.9, 4.2), the distribution of the histogram is generally uniform, which can be well applied to chaotic encryption. Figure 7d shows the histogram of the iterative sequence y, which is symmetrically distributed and corresponds to the bifurcation diagrams in Figs. 2b and 3b, but the histogram distribution is not uniform. If the 2D hyperchaotic map is used for chaotic encryption, the his-   20 40 60 80 100 20 40 60 80 100 togram can be uniformly distributed by equalization, and the chaotic map can be better used for chaotic encryption.
The 0-1 test is a test method for chaotic characteristics proposed by Gottwald and Melbourne. This test method does not require complex phase space reconstruction. Instead, it only needs the input of a set of chaotic sequences, and determines the chaotic characteristics of iterative sequences according to whether the result of the output gradual growth rate K c is close to 0 and 1. According to the 0-1 test method of Gottwald-Melbourne, the dimensionless function of the ( p, q) plane is obtained as (8) and (9).
and where c is a random constant in the interval (0,2π ), and x(n) is the sampled chaotic sequence, N is the sample data, and the sample data taken in this paper is 1000. Based on the dimensionless functions of p(n) and q(n), the resulting mean square displacement is denoted as M(n) Define the asymptotic growth rate of the mean square displacement M(n) as K c as If the asymptotic growth rate K c ≈ 1, it means that the chaotic sequence has chaotic characteristics. If the asymptotic growth rate K c ≈ 0, it means that the chaotic sequence does not have chaotic characteristics. Therefore, the chaotic characteristics of the iterative sequence when the system parameters take different values can be judged by the Gottwald-Melbourne 0-1 test. When the parameter a = 0.62 and b = 2, the mean square displacement M(n) is bounded over time, and the calculated average asymptotic growth rates are K c ≈ −0.0016 and K c ≈ −0.0014, respectively. The chaotic sequence of the average asymptotic growth rate of is approximately 0, so this chaotic sequence does not have chaotic properties (Fig. 8). When the parameter a = 1.8 and b = 2, the mean square displacement M(n) increases linearly with time, and the calculated average asymptotic growth rates are K c ≈ 0.9953 and K c ≈ 0.9976, respectively. The average asymptotic growth rate of the chaotic sequence is approximately less than 1, so the chaotic sequence has chaotic properties (Fig. 9). When the parameter a = 1.8 and b = 3.1, the mean square displacement M(n) increases linearly with time, and the calculated average asymptotic growth rates are K c ≈ 0.9983 and K c ≈ 0.9978, respectively. The average asymptotic growth rate of the chaotic sequence is approximately less than 1, so the chaotic sequence has chaotic properties ( Fig. 10). At the same time, the results of the 0-1 test are summarized in Table 2. The test results under different parameters in Table 2 correspond to the Lyapunov exponent diagrams in Fig. 2 and Fig. 3.

Initial-boosting bifurcation based on initial value
On the basis of a simple 2D hyperchaotic map mathematical model, by changing the initial value of the state variable x, the numerical simulation realizes the boosting bifurcation based on the change of the initial value. Firstly, the system parameter is fixed as a = 1.8, and the initial value of the chaotic map is set as (x 0 , y 0 ) = (0.1 + 2k π , 0.1) (k=±2, ±1, 0) and (x 0 , y 0 ) = (−0.1 + 2k π , −0.1)(k=±2, ±1, 0). By controlling the system parameter b to increase in the interval (−3.1, 3.1), five sets of bifurcation diagrams with different colors are numerically simulated. As shown in Fig. 11, the bifurcation diagrams with different colors  Fig. 11 represent different initial values. From the bifurcation diagram, it can be found that when the initial value of the state variable x takes the positive initial value and the negative initial value, respectively, the bifurcation structure of the bifurcation diagram will change symmetrically. When k tends to infinity, there will be an infinite number of self-replicating initialboosting bifurcations.
In addition, fix system parameters b = 2, and set the initial value of the chaotic map as (x 0 , y 0 ) = (0.1 + 2k π , 0.1) (k=±2, ±1, 0) and (x 0 , y 0 ) = (−0.1 + 2k π , −0.1) (k=±2, ±1, 0). Five groups of initial-boosting bifurcations with different colors are realized by numerical simulation. As shown in Fig. 12, the bifurcations of different colors represent different initial values. Similarly, when the initial value of the state variable x takes the positive initial value and the negative initial value, the bifurcation structure of the bifurcation graph with different initial values will change symmetrically. When the parameter k tends to infinity, there will be an infinite number of self-replicating boosting bifurcations.

Initial-boosting attractor and iterative sequence
According to the results of the initial switching boost bifurcation given in Fig. 11a, when the parameters of the simple two-dimensional hyperchaotic mapping system are set as a = 1.8, b = 2, and the fixed initial value is (x 0 , y 0 ) = (0.1 + 2kπ, 0.1) (k = ±2, ±1, 0), five groups of attractor phase diagrams and iterative sequences with different initial values are simulated, as shown in Fig. 13. The phase diagram of the attractor switches periodically with the periodic change of the initial value of the state variable x. At the same time, when the value of k tends to infinity, an infinite number of self-similar attractors coexist in the phase diagram of the attractor. Figure 13 shows five sets of initial switching boost coexistence attractors and iterative sequences. It can be seen that the offset behavior along the coordinate axis is closely related to the initial value x 0 , and the initial condition x 0 can control the dynamic amplitude of the chaotic iterative sequence.
For the above five groups of initial-switch boosting attractors, we can use some indicators to evaluate the initial-boosting attractors, such as Lyapunov exponent, permutation entropy(PE, when the embedding dimension is 3, it is recorded as P E (3) , and when the embedding dimension is 5, it is recorded as P E (5) ) and spectrum entropy (SE). The calculation results of different indicators obtained through five groups of iteration sequences are shown in Table 3. By comparing the values in Table 3, it can be seen that the data of the indicators obtained with different initial values are almost the same, and these small differences are caused by numerical calculation errors. This shows that the chaotic map has good initial controllability and robustness.

Boosting bifurcation and boosting coexistence attractor based on parameter control
Different values of the parameters mentioned in the literature can make the chaotic map produce different dynamic behaviors. Therefore, inspired by the above literature [4], a new control parameter is introduced based on the hyperchaotic map proposed in this paper to study the influence of the control parameter on the boosting bifurcation.

Introducing new parameters to the hyperchaotic map model and boosting bifurcation behavior
Inspired by the literature, this section considers the introduction of control parameters to realize the boosting bifurcation with respect to the state variable y. A new control parameter is introduced into the second equation of the hyperchaotic map, and the mathematical model is obtained as (14).  In order to study the effect of the new control parameters on the boost bifurcation of the state variable y, the initial value of the chaotic map is fixed as (x 0 , y 0 ) = (0.1, 0.1). By periodically changing the value of the parameter m, the boost bifurcation diagrams of five groups of state variables y are numerically simulated. As shown in Fig. 14, the bifurcation diagrams with different colors in the figure indicate that the parameter m takes different values. According to the boosting bifurcation diagram in Fig. 14, it can be seen that the boosting bifurcation of the state variable y is closely related to the value of the control parameter m, and when the parameter m tends to infinity, there will be infinitely many self-similar bifurcation structures. Thus, through the initial-boosting bifurcation in Sect. 3 and the boosting bifurcation based on control parameters in this section, we achieve the boosting bifurcation of two state variables with respect to arbitrary parameters in 2D hyperchaotic maps, which also provides a new idea for realizing multivariable boost bifurcation of highdimensional chaotic map.

Parameter control boosting attractor and iterative sequence
By introducing new control parameters into the chaotic map to control the switch boosting of the state variable y. In order to study the lossless adjustment of the state variables by the new control parameters, set the system parameters a = 1.8, b = 2. When the control parameter m changes periodically, the phase diagram and iteration sequence of the corresponding attractor will be shifted periodically, so there will be coexistence of multiple self-similar attractors in the phase diagram, as shown in Fig. 15. In particular, by changing the value of the parameter m and the initial value of the state variable x, the attractor can be switched in any direction in the phase plane. As shown in Fig. 16, the attractors with different colors represent the coexisting attractor phase diagrams obtained when the control parameter m and the state variable x take different values. When the value of the control parameter m tends to infinity, the attractor will generate countless selfreplicating attractors along the ordinate. and the iterative sequence of attractors can be controlled within the dynamic range by the parameter m. Therefore, the control parameters can control the dynamic amplitude of the chaotic/hyperchaotic iterative sequence losslessly, so that the chaotic map can be better applied to the engineering application based on the chaotic map. Through the above-mentioned five sets of boosting bifurcations controlled by the change of control parameters, the boosting attractor can be evaluated through the iterative sequence generated by it. Similarly, the parameters control boosting switch attractor can be evaluated by using indicators such as Lyapunov exponent, permutation entropy (PE) and spectrum entropy (SE). The different calculated values obtained through five sets of iterative sequences are shown in Table 4. By comparing the values in Table 4, it can be seen that the index data obtained by different initial value controls are almost the same. This shows that the chaotic map introduced with new control parameters also has good initial controllability, and the initial controllability of the hyperchaotic map is robust.

Interesting phenomenon that depends on the control parameter m
According to the analysis results in Fig. 15, the periodic change of control parameters makes the attractor of the chaotic map perform lossless switching boost in phase space. In particular, when the parameters of the chaotic map are set as a = 1.93 and b = 2.83, the periodic switching of the control parameter m can produce interesting coexistence of attractors. The phase diagram and iterative sequence diagram of five groups of coexisting attractors are simulated. As shown in Fig. 17, it is shown that the change of control parameters can not only achieve the switching boost of attractors in the phase diagram, but also the number of coexisting attractors can be changed. At the same time, it can be seen from Fig. 17b that the control parameters can effectively control the dynamic amplitude of the attractor iterative sequence, which also confirms that the parameter-controlled iterative sequence has good robustness.
In addition, when the control parameter m is an integer and the value of the control parameter changes, the boosting bifurcation and coexistence attractor controlled by this parameter will appear. When the value of the control parameter m is non-integer, there will be many interesting phenomena. Figure 18 shows the phase diagrams of attractors obtained when the control parameter m takes different non-integer values, showing the asymmetric coexistence of attractors. Thus, the 2D hyperchaotic map introduces a new control parameter m, which can not only realize the coexisting boosting attractor, but also realize the asymmetric coexistence of the attractor.

Performance analysis of 2D hyperchaotic map sequences
Compare the dynamic performance of the existing chaotic maps (2015-2022), the performance advantages of the chaotic graphs proposed in this paper are analyzed. The existing reported chaotic maps include 2D sin discrete chaotic map (2D-SDM) [4], memristive tent map (2D-MTM) [5], 3D memristive sin chaotic map (3D-MS) [7], and initial boost coexistence sine map (Map ISBa) [32], chaotic map with fixed points of closed curves (Map CFa) [33], hidden chaotic map (Map NFIa) [34], first quadratic chaotic map (Map NEM1) [35] and sine Logistic modulation map (2D-SLM) [36], etc. In particular, 2D chaotic map contains two Lyapunov exponents, and its largest Lyapunov exponent (LLE) determines the dynamic behavior of the chaotic map. At the same time, the spectrum entropy (SE) of the chaotic map is also an important index to evaluate the performance of the map. The data obtained by comparing the above indicators are shown in Table 6. From Table 6, it can be seen that the hyper-  chaotic map in this paper has a larger LLE than most of the existing chaotic maps, and the 2D hyperchaotic map has a larger SE, except for the sine Logistic modulation map and initial boost coexistence sine map. It is proved that the 2D hyperchaotic map has high complexity and application value, so the 2D hyperchaotic map in this paper has better dynamic performance than the existing chaotic maps.

Hardware implementation of hyperchaotic map
Taking into account the characteristics of ultra-low power consumption and strong controllability of the microcontroller, this article uses the microcontroller model STM32H750XBPro, the STM32H750 series There are multiple memristive chaotic mapping models in 2D-MTM [5] and 3D-MS [7], and only the memristive chaotic mappings with significant features are summarized respectively microcontroller uses the ARM Cortex-M7 core, and the main frequency of the motherboard is up to 480MHz. The selected microcontroller model is sufficient to meet the needs of this experiment. At the same time, the platform also includes a D/A converter model AD5689, which is a 16-bit precision digital-to-analog converter. The function of the D/A converter is to convert the digital signal into an analog signal. First, use C language to compile and download the relevant program of 2D chaotic map to the microcontroller. That is, the two-channel chaotic sequence of the 2D sine chaotic map can be generated synchronously. Figure 19 shows a photo of the hardware experiment platform and displays the chaotic sequence diagram of the 2D chaotic map on the oscilloscope. As shown in Figs. 20 and 21, Fig. 20 shows the initial controlled chaotic sequence, and Fig. 21 shows the parameter controlled chaotic sequence. The results show that the iterative sequence of the two-channel hyperchaotic map randomly oscillates along the x-and y-axes within their respective fixed amplitude ranges. Therefore, the experimental results prove the correctness and feasibility of the hardware implementation of hyperchaotic mapping in this paper, and show that the initial control chaotic sequence and the parameter-controlled chaotic sequence of the chaotic map in the hardware device have better robustness.

Application in Pseudo-Random Number Generator
Due to its important properties such as initial sensitivity and unpredictability, chaotic systems are widely used Fig. 19 A hardware platform based on microcontroller and a dual-channel chaotic sequence controlled by parameters in many academic and industrial fields, such as pseudorandom number generator (PRNG). In particular, a pseudo-random number generator is designed based on the chaotic sequence of the 2D hyperchaotic map because of its complex dynamic characteristics. Firstly, the system for setting 2D hyperchaotic mapping is set as a = 1.8 and b = 2, and its initial state is set as (0.1, 0.1). Assume that the obtained two chaotic sequences are X (n) = {x 0 , x 1 , x 2 , ...} and Y (n) = {y 0 , y 1 , y 2 , ...}, the floating-point numbers of the two chaotic sequences are converted into 25-bit binary streams X b (n) and Y b (n) using IEEE 754 floating-point standard, and the length of the obtained binary streams should not be less than 10 6 . To test PRNs with high randomness generated by PRNG, we tested the performance of pseudo-random bitstreams generated by 2D hyperchaotic maps by  [37]. The NIST SP800-22 random number test suite contains 15 subtests, each of which can produce 120 P-values. Firstly, the interval [0,1] is divided into ten equal intervals. Then we count the number of P-values in each subinterval. Finally, the χ 2 test was performed on the P-value distribution to obtain the P − value T , In particular, the χ 2 test is used to test whether the difference between multiple rates (or constituent ratios) is significant. If the obtained P − value T is greater than 0.0001 [37], the binary sequence is considered to pass the subtest. Table  6 lists the test results of random numbers generated by different PRNGS in the NIST SP800-22 test. It can be seen that all P − value T of the PRNs by the 2D hyperchaotic map are greater than 0.0001. This shows that the proposed hyperchaotic map can generate pseudorandom numbers with high randomness.

Conclusions
The complex dynamics of the initial switching boost of the chaotic map usually depend on the initial state of the state variables. In this paper, a simple 2D chaotic map model is proposed. Due to the infinite number of straight fixed points, the 2D hyperchaotic map can realize the boost bifurcation of a single state variable by changing the initial state. In addition, the introduction of new control parameters in the 2D hyperchaotic map can realize the boost bifurcation of arbitrary state variables, which provides a good foundation for us to realize the multivariable boost bifurcation of the highdimensional chaotic map. In particular, the change of the new control parameters can make the chaotic map show asymmetric coexisting attractors. The chaotic sequence generated by the 2D hyperchaotic map can be lossless and adjusted by periodic changes of the initial state and control parameters. This also ensures the robustness of our proposed simple 2D hyperchaotic map. Further performance evaluation shows that 2D hyperchaotic maps have better complexity than existing hyperchaotic maps. In addition, a hardware experimental platform is designed that can synchronously realize the lossless adjustment of the oscillation amplitude of the chaotic sequence. At the same time, in order to demonstrate the good application prospects of the 2D hyperchaotic map, the proposed chaotic map is used in the design of PRNG. The test results show that 2D hyperchaotic map can produce PRNG with high randomness. Due to its excellent performance, future work will be focusing on its application in image encryption and secure communication.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest
The authors declare that they have no conflict of interest.