## Abstract

This paper presents a novel four-dimensional chaotic system that is unique in that it exhibits a variety of striking nonlinear dynamical phenomena despite the absence of equilibrium. Computationally, the system is found to have multiple coexisting and hidden attractors and exhibits quasi-periodic, intermittent chaotic, and transient chaotic phenomena as the parameter dynamics change. Within appropriate parameter ranges, the system exhibits rare phenomena such as multiplicity, a controlled number of spike discharges without external excitation, as well as paranoid enhancement triggered by amplitude control, burst oscillations, and a controlled number of multivortex phenomena. These findings have important applications in areas such as neuron-like and voltage-control engineering. Notably, the integration of multiple important nonlinear dynamical behaviors into a system with a hidden attractor, whose realistic validity is achieved through analog simulations and hardware circuits, has rarely been reported in the literature in the past. In addition, the successful application of the chaotic system to image encryption with high confidentiality to meet the requirements of secure communication brings great potential for innovation in the field of nonlinear dynamics and chaotic secure communication, as well as new possibilities for engineering applications in related fields.

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## 1 Introduction

In recent years, people have a deep and wide understanding of the natural law of chaos phenomenon and the performance of chaos in natural science, and there are many domestic and foreign experts and scholars are doing in-depth research in the field of chaos theory, and the researchers are getting more and more extensive and in-depth research on the chaos theory makes the chaos theory get a rapid development [1,2,3]. After entering the twenty-first century, the connection between chaos theory and engineering technology is very close, mainly including chaotic secure communication, chaotic bioengineering, neural network, celestial mechanics, medical research, and so on [3,5,6,7,8,8].

The introduction of nonlinear functions is essential to produce richer chaotic behaviors in dynamic systems. In many references, most scholars have introduced a variety of nonlinear terms to generate rich nonlinear dynamical behaviors. In a previous work [9], Chen et al. improved the symbolic model based on neural networks. In the reference [10], Jacques Kengne et al. proposed and investigated a simple autonomic chaotic circuit with an inverse hyperbolic sinusoidal function antiparallel to the semiconductor pairs. In the reference [11], Wang et al. implemented a four-dimensional hyperchaotic system containing exponential functions with multiplication terms using DSP techniques, and Rocha [12] analyzed Chua's circuits with nonlinearities based on the exponential hyperbolic properties of semiconductor devices. Among these nonlinear functions, the sinusoidal function is one of the algebraically simplest, the most explicit upper and lower bounds, and the physically realizable nonlinear function. It is extremely important to make good use of the nonlinearities to construct a chaotic system with suitable nonlinearities that can be realized in a circuit to produce rich dynamical phenomena. In chaos science, the hidden attractor is a special kind of attractor, which is different from a self-excited attractor in that its attraction basin does not contain the neighborhood of the equilibrium point, and there exist two types of periodic oscillations and chaotic oscillations. People have done a lot of work on the study of nonlinear systems with hidden attractors in recent years [13,14,15], due to the existence of hidden oscillations in nonlinear systems, will make the phenomenon of an unstable state of life and industrial production, so it is also necessary to analyze the dynamics of dynamical systems with hidden attractors. Similar to intermittent chaos and transient chaos but different is the more widely used and interesting quasi-periodic phenomena, usually quasi-periodic behaviors are found in some important fields and nonlinear dynamical systems, such as high-dimensional coupled chaotic systems [16, 17], periodically forced or delayed oscillators [18], spatial quasi-periodic gravitational capillaries of finite depth [19], satellite orbital capacity [20], and the financial interactive markets [21], biochemical cells and systems [22], and others. In this paper. We find the quasi-periodic behavior of the newly constructed systems, and an additional and effective 0-1 test method can be used to further confirm the quasi-periodic paths revealed in this paper [23]. The study of multistability and super-multistability in chaotic systems has become an important research direction in nonlinear dynamics. The coexistence of multiple attractors implies that the system can provide multiple normal operation modes, i.e. when the system is disturbed by noise or the external environment, it can switch to different stabilized states to keep the system in normal operation. Multi-stability in chaotic systems can also be used for encrypted signals or pseudo-random number generators in the field of information engineering. Therefore, it is of theoretical significance and practical engineering value to study the construction of systems with multistable or super-multistable states [24,25,26].

Chaotic systems with four or more dimensions have multiple positive Lyapunov exponents, which means that small perturbations of the system state will diverge exponentially in multiple directions, and the state trajectories are more complex, increasing the unpredictability of the system, and its multi-dimensional complexity, which is superior to the traditional chaotic systems in terms of processing speed, error tolerance, and anti-interference [27,28,29]. In some related technical applications, peripheral devices are often required to modulate the amplitude and position of signals [30, 31]. The unique nature of offset-free control, which can effectively change the offset boost of a chaotic system, can effectively change a bipolar signal into a unipolar signal by changing only one constant to achieve the desired level of boost, and the paranoia enhancement property of a chaotic system can play a key role among them [32]. Chaotic signals with complex dynamic properties have potential applications in fields such as secure communications, image encryption, and bionics. Nonlinear dynamical systems, which are at the core of computational neuroscience research, have similar dynamical properties to the nonlinear phenomena that occur in dimensionless mathematical models of physical circuits [33, 34]. Therefore, it is necessary to analyze and exploit such phenomena generated by circuits. In 2016, Lv et al. designed a four-variable neuron model to describe the effect of electromagnetic induction on neuronal activity and multiple firing patterns could be observed by changing the initial state [35]. In 2022, Zhan et al. explained the dynamical mechanism of the transitions between the firing rhythms of mathematical models of neurons under electromagnetic induction and demonstrated the dynamics of spike burst and spike mode [36]. In the following year, Song et al. investigated and designed a voltage-controlled memristor, and elaborated the discharge mechanism in a nonlinear circuit by analyzing the stability of the equilibrium point [37]. Scholars have conducted a lot of research on neuronal models, however, the reference discussing the multiple discharge behaviors of four-dimensional autonomy is still relatively scarce [38,39,40,41].

Given the above research hotspots and questions raised, a newly constructed four dimensional chaotic system is investigated in this paper. By introducing sinusoidal and exponential functions as nonlinear activation functions and incorporating the classical three-dimensional Jerk autonomy model [30], the system is successfully implemented as a simulation circuit. Within appropriate parameter ranges, the new system is also able to control the number of spike-discharge clusters and the number of multiple vortices, which is a relatively rare phenomenon in four dimensional chaotic systems. Although the system does not have synaptic units and memory transistors as in the neuronal model, dynamical behaviors similar to those of neuronal model discharges are observed, which is important for understanding complex dynamics in neuronal models and state transitions in nonlinear dynamics. In terms of engineering applications, the new system also demonstrates a system with deviation enhancement phenomena and successfully applies a chaotic system to image encryption. Special emphasis is placed on the unpredictability and high sensitivity of the resulting sequence, which provides potential applications in the field of data transmission and information security. This further ensures the feasibility of the complex behavior exhibited by the proposed system in practical applications.

The chapters of this paper are organized as follows: Chapter 2 introduces the construction of the mathematical model and the analysis of the properties. Chapter 3 analyzes the complex dynamical phenomena of the system. Chapter 4 provides an in-depth analysis of the unique dynamical behavior of the system. In Chapters 5 and 6, simulations and hardware circuit experiments demonstrate the physical realizability and high confidentiality of the resulting chaotic sequences. Chapter 7 concludes the paper.

## 2 New four-dimensional chaotic systems with hidden attractors

The proposed new four-dimensional chaotic system model with high complexity and richer nonlinear dynamical phenomena is Eq. (1):

In Fig. 1, the state analysis of the system(1) is carried out. The numerical simulation software used is Matlab and the differential equation solver (Ode45) is used for the approximate solution of the differential equations, the sampling time and simulation duration are 0.01 and 1000, fixed parameters *a* = 0.1, *b* = 0.8, *c* = 0.2, *d* = 0.5, *h* = 1 and *g* = 1 and the initial values of (*x*_{0}, *y*_{0}, *z*_{0}, *w*_{0}) = (0. 1, 0.1, 1, 1), the Lyapunov exponents are presented as LE_{1} = 0.1285, LE_{2} = 0.0020, LE_{3} = −0.1122, LE_{4} = −0.8182, respectively, which exhibit the phenomenon of (+, 0, −, −), thus indicating that the system is in a chaotic state. The state evolution of the system, as well as the specific Lyapunov exponent spectra and bifurcation diagrams of the different parameters, will be verified in the subsequent contents.The study of equilibrium points is essential for the stability of a chaotic system. If the basin of attraction in a system intersects any open neighborhood of a stable equilibrium point, then the attractor of that system is called a self-excited attractor. If a system has an infinite number of equilibrium points, and all of them are stable, or there are no equilibrium points, then the resulting attractor is called a hidden attractor. The hidden attractor has completely different dynamics from the self-excited attractor, so it has important research significance [14]. For making the right-hand side of Eq. (1) equal to zero, as shown in Eq. (2), it is obvious that there is no equilibrium point of the system when *g* is a non-zero parameter. Therefore, the attractor of the system presented in Eq. (1) is hidden, and its attractors in each plane are shown in Fig. 2.

Based on the calculation from Eq. (3), it can be concluded that ∇*V* < 0. Therefore, the dissipativity condition is satisfied, indicating that the system is dissipative. This means that the system converges exponentially with d*V*/d*t* = e^{−0.8t}, and the system's trajectory ultimately reduces to a particular set of zero-volume limits at an exponential rate as *t* → ∞, finally settling on an attractor.

The system proposed in this paper has a simple structure and is easy to realize through physical experiments. Meanwhile, the dynamic display of the system state with parameter variations is investigated, where *x*, *y*, *z* and *w* are the system state variables. To better illustrate the sensitivity of the system to parameter variations, Fig. 3 gives the results with fixed parameters *a* = 0.1, *b* = 0.8, *c* = 0.2, *d* = 0.5, *h* = 1 and *g* = 1 and the initial values of (*x*_{0}, *y*_{0}, *z*_{0}, *w*_{0}) = (0. 1, 0.1, 1, 1), and it can be seen from Fig. 3a, c that, as the value of *k* increases, when *k* ∈ (1.2, 5.76), there exists a Lyapunov exponent greater than zero and a Lyapunov exponent equal to zero, and the system is in a chaotic oscillation state most of the time. When *k* ∈ (8.45, 8.475), the system (1) produces periodic oscillatory state. Observing Fig. 3b, d, the system is in a chaotic state when the fixed parameters *a* = 0.1, *b* = 0.8, *c* = 0.2, *d* = 0.5, *h* = 1, *k* = 1, and *g* ∈ (0, 3.64) ∪ (4.06, 10), and the system (1) moves from chaotic to periodic oscillations for *g* ∈ (3.64, 3.78) ∪ (3.9, 4.06), and the polytropic transitions are characterized by the phenomenon of intermittent chaos, which will be discussed in detail in later sections. The bifurcation diagram in Fig. 3d shows that as the value of the parameter *g* increases, the duration of the chaotic state increases and the stabilization point shifts back. Increasing the parameter *g* reduces the probability of the chaotic attractor transforming into a non-chaotic attractor.

A thorough understanding of the influence of the parameters on the overall dynamics of the system is necessary for a system with good characteristics, and the correct choice of the parameters of the system is necessary to control them to produce accurate dynamics. In Fig. 4 the overall dynamical stability diagram for different parameters, where the Lyapunov exponential bands are activated to better determine the type of behavior that the system exhibits as a function of parameter changes. From the transformation of the colors involved, it can be observed that when the parameters *k* and *g* of the system take the same values, the chaotic dynamics behavior becomes richer as it goes from dark blue to light blue to bright red. When the parameter selected value corresponding to the color tends to zero, the closer to the blue color means that the system attractor is located in the periodic orbit, when the system has only large swaths of blue color in the neuron model represents the resting state, and when the parameter selected value corresponding to the color gradually becomes yellow or even brighter red color, it means that the system enters into the chaotic spiky state, which in the neuron model represents the discharge [35]. It can be seen that as the parameters *k* and *g* change, the attractor moves in the diagonal direction and the type of attractor changes, and then the attractor can exhibit richer and more complex dynamical behaviors than a system that translates in a single direction. These Lyapunov stability diagrams are essential for parameter selection, better control of the system, and practical studies of circuits, systems, and signal processing (e.g., chaotic behavior for encryption), and are an advantage for selecting parameter ranges for applications including chaos-based encryption [40].

## 3 Abundant dynamics of the system

### 3.1 Transformational analysis of the dynamical behavior of the system with parameter evolution

In this section, the coexistence of multiple attractor states is observed in the novel four-dimensional chaotic system, as depicted in Fig. 5. The steady state of the system is analyzed under specific parameters. As shown in the Fig. 5a, c, and e, the trajectory of the chaotic system originates from any point within the chaotic region of the phase space and converges gradually after several iterations. Due to the introduction of the fourth-dimensional variable *w*, the chaotic system will gradually stretch the shape of the phase diagram upward and decrease over time, and the state variables will then converge stably to the point attractor until it converges to a fixed point. In addition, we also observe that different parameters exhibit the coexistence of chaos-period and period-fixed point attractors. Therefore, it can be concluded that the chaotic system gradually tends to stabilize after experiencing transient chaos as the number of iterations increases. The attractors corresponding to different system states are shown in Fig. 5b, d, and f.

As shown in Fig. 5, a transition from chaos to quasi-periodic behavior is observed within a narrow range of chaotic parameters. This range exhibits a small regular pattern of activity that is distinct from periodic behavior. The quasi-periodic attractor has a topology similar to that of chaos, but without a positive Lyapunov exponent, this quasi-periodic behavior is non-chaotic and can be regarded as the "edge of chaos" [16]. It has been shown that biological neurons operating in the vicinity of the "chaotic edge" will show strong adaptability and high learning efficiency, which can stimulate their neuromorphic intelligent responses [22]. When the system take *h* = 6.3, 6.5, 6.7, 8, respectively, with the increase of the parameters, the characteristic quasi-periodic beat pattern frequency slows down the oscillation frequency with the parameters, and in Fig. 5g, a partial period and a small portion of the weak chaos phenomenon are observed. Consequently, the system's timing diagram exhibits an irregular trajectory, indicating a high degree of complexity in the quasi-periodic attractor, as depicted in Fig. 5.

To further demonstrate the existence of quasi-periodic behavior in the system, the 0-1 test algorithm is employed to verify the image of the linear motion trajectory. This method differs from the calculation of Lyapunov exponent and does not necessitate phase space reconstruction. The computation output indicates that the value of *K* is not close to 1, supporting the presence of quasi-periodic variation. This analysis provides additional evidence for the system's complex dynamics.

In this case, *p*(*n*) and *s*(*n*) are used as the transverse and vertical axes for trajectory plotting, the regular ring is generated to represent the quasi-periodic phenomenon, and the Brownian motion is used to represent the chaotic phenomena [23], shown in Fig. 6. The specific process is as follows

In order to analyze the diffusion behavior of *p*(*n*) and *s*(*n*), the displacement mean square error *M*(*n*) can be calculated. The calculation of *M*(*n*) is performed as follows:

The convergence of *p*(*n*) and *s*(*n*), is measured by the convergence of *M*(*n*). If the discrete time series is ordered, then *M*(*n*) is a bounded quantity. If the time series is chaotic, then *M*(*n*) grows linearly with *n*. Finally, calculate the *K* of *M*(*n*) with n linear growth rates, and *M*(*n*) with *n* linear regression coefficients, it’s asynchronous growth rate is defined as

### 3.2 Sensitive behavior of the system with initial values

An important feature of chaotic systems is that they are extremely sensitive to initial conditions. In this section, fixed the parameters *a* = 0.1, *b* = 0.8, *c* = 0.2, *d* = 0.5, and *h* = 1, while the parameter *g* is chosen to be 0.99 and 1.06, respectively.This choice allows us to study the effect of different initial values on the chaotic system. Figure 7a shows the time-domain waveforms of the state variable *x.* There are two neighboring initial value points (0.1, 0.1, 1, 1) (red) and (0.1 + 1 × 10^{–8}, 0.1 + 1 × 10^{–8}, 1 + 1 × 10^{–8}, 1 + 1 × 10^{–8}) (blue). From the upper part of Fig. 7a, it can be seen that the two time-domain waveforms start to undergo a separation change near *t* = 113 s, while in the lower part of the figure, the separation occurs only at *t* = 160.5* s*. It can be seen that the new chaotic system is more sensitive to changes in the state with initial conditions of (0.1, 0.1, 1, 1), and the system produces significant timing errors even when the parameter changes are small. In order to verify whether our chosen initial values are suitable for the newly proposed four-dimensional chaotic system, we proceeded to compare the bifurcation diagrams of (± 0.1, ± 0.1, ± 1, ± 1) in Fig. 7b. The dynamic evolution of these values and their complexity are presented. In order to systematically determine a superior initial value, the energy distribution in the Fourier transform domain is analyzed using the *SE* and *C*_{0} complexity algorithms. The spectral entropy value is obtained through the application of the Shannon entropy algorithm. The *SE* algorithm is specif ied as follows:

**Step 1**: For the pseudo-random sequence of length *M*, use Eq. (8) to remove the part of its direct current, so that its spectrum can more accurately represent the energy size of the signal, and its discrete Fourier transform to get the Eq. (9)

where, \(\overline{y} = \frac{1}{M}\sum_{m = 0}^{M - 1} {y(m)}\)

where, *n* = 0, 1, 2,⋯, *M*-1.

**Step 2**: For the discretized *Y*(*n*) sequence, the power spectrum *p*(*n*) is obtained at a specific frequency. The total power of the sequence, denoted as *p*_{tot}, can be defined. At this stage, the probability of the power spectrum as *p*_{n}

where, *n* = 0, 1, 2,⋯, *M*/2–1

where, \(\sum_{n = 0}^{M/2 - 1} {p_n } = 1\).

**Step 3**: Combining the above Eqs. (10), (11), (12), combined with the concept of Shannon entropy, the spectral entropy of the signal *se* is obtained as Eq. (13), to facilitate the comparative analysis, the spectral entropy is normalized, yielding the normalized spectral entropy.

The *C*_{0} algorithm needs to divide the sequence into regular and irregular parts, and what is wanted is the proportion of irregular parts in the whole chaotic sequence. The more irregular parts in the whole sequence, the closer its corresponding time domain signal is to a random sequence and the more complex it is. The specific calculation steps are as follows:

**Step1**: Similar to the previous Eq. (9), the non-regular part of Eq. (9) is removed after performing the discrete FFT transform on the sequence. It is assumed that the square mean of {*Y*(*n*), *n* = 0, 1, 2,⋯, *M*-1} is obtained.

**Step2**: A parameter *q* is added to Eq. (15), leaving the fraction that exceeds *q* times that mean square value, and setting the value of the remaining fraction to zero, i.e.

**Step3**:Make Fourier inverse transformation to Eq. (16) to get Eq. (17), and finally define the *C*_{0} complexity as Eq. (18)

where *m* = 0, 1 ,⋯, *M*-1

Figure 8a, c shows the numerical simulations obtained by the *SE* spectral entropy complexity algorithm and Fig. 8b, d shows the numerical simulations of the *C*_{0} algorithm. It can be observed that the highest peak complexity of the system is at 0.2111 when the initial value is taken as (0.1, 0.1, *z*, *w*), while the peak complexity of the system with the initial value taken as (−0.1, −0.1, *z*, *w*) is at 0.1513, and none of its highest complexity is as high as that of (0.1, 0.1, *z*, *w*). Moreover, when (*x*, *y*, *z*, *w*) = (0.1, 0.1, 1, 1), the system is in the region of high complexity color, as shown by the location of the spikes in the figure, which is more complex than that of (*x*, *y*, *z*, *w*) = ( −0.1, −0.1, −1, −1), and therefore, it is more suitable for the initial value of the novel chaotic system by choosing (*x*, *y*, *z*, *w*) = (0.1, 0.1, 1, 1) for confidential communication.

### 3.3 Intermittent *chaos*

Intermittent chaos visualizes a system whose state of motion alternates between chaotic and periodic. In the case of a chaotic system with intermittent chaos, the time series consists of chaotic interruptions that are interrupted by irregularities and periodic motions that are regular. As shown in Fig. 9, the red and green colors indicate chaotic oscillations, and the yellow and blue colors indicate periodic oscillations. The two successive attractors have different durations between them and they are randomly distributed in the taken time intervals. System (1) can produce intermittent chaos under fixed parameter conditions, which shows that the system has a high degree of unpredictability.

### 3.4 Transient *chaos*

When a chaotic attractor collides with an unstable periodic orbit on the boundary, a boundary crisis occurs and the attractor transforms into an unattractive set, producing a transient chaotic state. In addition to the transition from chaos to stable point and stable period, the proposed new system also demonstrates the phenomenon from chaos to period. In Fig. 3a. When the parameter *k* ∈ (1.48, 3.11), the LE_{1} of the system is greater than zero and the LE_{2} is approximately equal to zero; while when *k* ∈ (5.7, 5.75), the system exhibits two Lyapunov exponents equal to zero, which indicates that the system is transformed from a chaotic state to a periodic state. After a brief chaotic state, the system enters the periodic state again, and its attractor jumps are shown in Fig. 10, with different colors representing the state jumps of the system in different time intervals.

It is worth mentioning that the transition from transient chaos to intermittent chaos is observed when values of *g* are taken as 4.058, 4.059, and 4.06, respectively, with chaotic oscillations in red and periodic fluctuations in blue. It can be seen in Fig. 11 that by changing only the 0.001 parameter gap, the system enters into a state of intermittent chaos.

Fix the parameters *a* = 0.1, *b* = 4, *d* = 0.5, *k* = 1, and *g* = 1, and change the parameter *c* to observe the transient chaos of the system. In Fig. 12 red color represents the chaotic oscillatory state and the blue color represents the periodic oscillatory state. When *c* = 0.15, in the range of *t* ∈ (0* s*, 1178* s*), the system exhibits a chaotic state increasing with time, but in the blue part of the periodic region, the system exhibits a smooth period. Whereas, when *c* = 0.59, the system exhibits an increasing sequence of periods after entering the chaotic state in the range of *t* ∈ (0* s*, 1118* s*). Although the system exhibits transient chaos under both parameters and enters the periodic state with a difference of only 60 s, the high sensitivity of the chaotic system makes the attractor motion and the change of the chaotic sequence during these 60 s non-negligible. We can utilize this feature to selectively extend the chaotic oscillation time to increase the secrecy and unpredictability of the system. Similarly, in certain engineering applications, the avoidance of chaotic behavior and the utilization of regular periods can be employed to achieve stable currents [20].

## 4 System-specific dynamical phenomena

### 4.1 Multistability

Multistability is important for confidential applications of chaotic systems, and multistability with hidden attractors is more secure [26]. Figure 13a, b show the periodic-chaotic coexistence attractor diagram of the system at *k* = 7.35 for different initial values, where the blue line represents the 1-periodic attractor with initial values (*x*_{0}, *y*_{0}, *z*_{0}, *w*_{0}) = (0.1, 0.1, 1, 1), the red line represents the chaotic attractor with initial values (*x*_{1}, *y*_{1}, *z*_{1}, *w*_{1}) = ( −0.1, −0.1, −1, −1), and the green line represents the 1-periodic attractor with initial values (*x*_{2}, *y*_{2}, *z*_{2}, *w*_{2}) = (0.1, 0.1, −40, −40), and the yellow line indicates the 2-periodic attractor with initial values (*x*_{3}, *y*_{3}, *z*_{3}, *w*_{3}) = (1, 1, 1, 1). In Fig. 13c, d, on the other hand, the chaotic-chaotic attractor coexistence is shown for *k* = 1. In order to more intuitively show that the system exists in multiple states at the same time, the attractor trajectories are made to be slightly shifted in the plane. It can be seen that the topology and the trajectories of the attractors of the system change at the same time in different initial value states. It can be concluded that the topology of the system is flexible and variable at different initial values, and its corresponding chaotic properties are also more complex and variable [42].

The basin of attraction can provide more detailed information about the multistability of the dynamical system. Thus, Fig. 14 shows that when fixing *x*(0) = 0.1, *y*(0) = 0.1, three states occur. The blue and banded green regions dominates the initial phase, and yellow dominate the initial plane in the intermediate regions. Observe that the chaotic bands show up in small regions, which means that the model is sensitive to the initial conditions in this case and produces extreme events. It should be noted that extreme events are associated with many situations such as tsunamis, earthquakes, tornadoes, market crashes, and human brain convulsions [24, 27]. Different types of attractors can be obtained for the chaotic system under different colors corresponding to different initial conditions, which further confirms the multistability property of the system. Since the mapped infinite number of coexisting attractors have different shapes and are located at different positions, the mapping exhibits a heterogeneous extreme multistability.

### 4.2 Offse-boosting

Offset augmentation control can realize the interconversion of chaotic signals between bipolar and unipolar without changing the basic dynamics of the system, but only by changing an additional control constant. The considered four-dimensional chaotic system is characterized by offset enhancement. The state variable *w* appears once in the third equation of the system (1), so the amplitude of the state variable *w* can be easily adjusted by adding the control parameter *m*_{w} to the system (1). This is shown as follows:

As can be seen from Fig. 15, the mean value of the chaotic signal gradually increases as *m* = 0 increases to *m* = 0.6, i.e., when the offset increment controller *m* takes a positive value, the attractor is shifted in the positive direction of the *w*-axis and vice versa. It is worth noting that this modification does not affect the dynamics of the system(1) but provides a controllable capability for the system. This marvelous feature can be utilized in several related technological applications where the desired level of lift can be achieved by using only one constant [31, 32].

### 4.3 Controlled spiking cluster discharge mode

In biology, it is important to study the pattern of potentials and the type of firing of neurons. When a neuron is not stimulated, there is no voltage difference between the two sides of the cell membrane and it is resting. When a neuron is stimulated, the cell membrane opens ion channels and releases ions, resulting in a potential difference between the two sides of the cell membrane, which generates an action potential. Due to the difference in ion concentration inside and outside the cell, to maintain the balance, the ions will enter the cell through new ion channels, causing the potential difference to eventually disappear and the cell to be resting again [34]. In this paper proposed chaotic system, an action similar to the neuronal action potential is observed, as shown in Fig. 16.

Over time, the behavior of periodic cluster discharge and chaotic cluster discharge can be observed as shown in Figs. 17 and 18. In the periodic cluster discharge mode, the discharge frequency and the shape of the spikes of the neurons vary with the applied current. By selecting appropriate parameters, the neuron model can produce a variety of controllable patterns of spikes and action potentials. When the parameter *k* is varied to 0.01 and the parameter *g* is adjusted to 0.86, 0.879, and 4.5, respectively, the system can produce periodic oscillations with 2, 4, and 8 spikes.

In particular, there are interesting coexisting discharge phenomena in the system, i.e., system (1) has a coexisting discharge behavior of cluster discharge modes and mixed-mode oscillations. Coexisting mixed-mode oscillatory discharge modes are a class of complex dynamical phenomena with alternating large and small amplitudes, and the models capable of generating the mixed-mode oscillations are usually systems of nonlinear differential equations incorporating multiple timescales and have been found in the fields of electronic circuits, chemical reactions, and biomathematics, etc. [35] Fig. 19 displays a discharge that showcases triple and bicuspidal high peaks, as well as the lower half of a low-peak discharge. This discharge is identified by large-amplitude spikes that are sporadically spaced out, and small oscillatory intervals that are chaotic. In the time-series plot, these closed orbitals appear as a periodic sequence of large spikes, amidst complex subthreshold oscillatory modes, all within the vicinity of an unstable resting state.This kind of study can be explored in simulating the information transfer law of neuronal networks, and its in-depth study can help to understand the kinetic mechanism of related diseases (e.g., Parkinson's, epilepsy, depression, etc.) induced by abnormal neuronal discharges, as well as related diseases induced by neuronal-like abnormal discharges to be of great significance [36]. In conclusion, the proposed system presents complex mixed-mode discharges and coexisting-mode discharges due to the complexity of the high-dimensional system, and unlike other studies in which external stimuli were applied, the number of spikes in the present study was controlled by changes in the system parameters.

### 4.4 Burst oscillations and controlled multiple vortex attractors

The oscillations of the system are exchanged between a stationary state and a spiky state, forming a bursting phenomenon as shown in Fig. 20. This oscillation connects the two states, and the system must contain both fast and slow processes. The coupling effect between fast and slow time scales causes the system to exhibit periodic motions, including relatively large amplitude and nearly harmonic small amplitude oscillations, which can be dangerous to the system. However, in botanical studies, the phenomenon of burst oscillations enhances the stability of the system or at least reduces the effect of chaos on plant dynamics, e.g., the motion of the plant enters into a state of relaxation for a given value of the damping ratio, which allows the system to periodically switch from large-amplitude oscillations to small-amplitude oscillations, thus maintaining stability, and in the case of the plant's interaction with the wind, the motion of the plant enters into a state of relaxation, ensuring its stability [7].

The dynamic behavior of the system reveals the transition from a periodic state to a chaotic state, similar to intermittent multi-periodic motion. It is observed that the bursting oscillations exhibit a cusp state in variables *x* and *y*, while variables *z* and *w* exhibit a cluster state. It can be observed that among the four variables of the system, *x*, and *y* are in the cusp state in the bursting oscillations while *z* and *w* are in the cluster state, and we observe that the system can also appear in the phenomena of period bursting oscillations and chaotic bursting oscillations as shown in Fig. 21. The attractor orbits are fundamentally different as shown in Fig. 21a, c, e, and g.

In the process of attractor formation, it was discovered that the new system has the capability to generate controllable multi-vortex scrolls. By manipulating the number of spike discharges while keeping other parameters fixed, the value of *h* was adjusted to 14.8, 18, 24, and 58, respectively. Consequently, the system exhibited multi-vortex scrolls consisting of 5, 10, 15, and 20 vortices, as depicted in Fig. 22. The multiple vortex scrolls in controllable attractors are characterized by simple structure, sensitivity to the initial value, and complex dynamical behavior, which can be applied to UAV communication technology to enhance the anti-jamming and anti-interception of the data link [1].

## 5 Experimental verififications

### 5.1 Multisim circuit simulation implementation

The analog circuit of the designed system is shown in Fig. 23. In this section, the overall circuit with the ability to generate nonlinear functions is constructed using the software Multisim, where the nonlinear function generator is generated using Nonlinear Dependent Source with an output of 0.5 sin(*w*) × *y* + 0.1*e*^{y} + 0.3*xy*. The usability of these nonlinear functions in generating electrical activity is confirmed by the successful execution of numerical simulation results on the designed analog circuit.

where, in addition to the resistor *R*_{16} = 12.5kΩ, *R*_{1}-*R*_{23} are all 10 kΩ, the capacitors take the values of *C*_{1} = *C*_{2} = *C*_{3} = *C*_{4} = 10nF, *C*_{5} = *C*_{6} = 100nF.

The operational amplifier is LF353D, and the operating voltage is ± 15 V. The results of the circuit experiments based on the designed circuit are shown in Fig. 24, and the experimental waveforms are intercepted by a digital oscilloscope. The magnitude of the values of *R*_{2}/*R*_{1} and *R*_{12}/*R*_{11} can be adjusted to control the number of spikes and vortices in the chaotic system. It is important to note that the results demonstrate that, as the resistance value changes, the new multivolume chaotic circuit can generate an attractor that aligns with the theoretical phase plane Fig. 2. This verifies the circuit's ability to realize the system.

### 5.2 Hardware implementation of FPGA

Analog hardware implementation can bridge the gap between theoretical exploration and engineering applications and is particularly suitable for hardware implementation of large-scale systems. FPGA is a reliable option for implementing chaotic systems and has high reliability compared to analog circuits with discrete components, unaffected by the aging of the devices and temperature variations. Moreover, it has many internal logic units, memories, and DSP function modules that can be reprogrammed in real time to change system parameters and operation modes. Therefore, it is ideal to use FPGA to generate chaotic signals, and the flowchart shown in Fig. 25.

The paper uses an FPGA chip with 10 K logic cells and a central frequency of 50 MHz. Euler's algorithm with a step size of 0.001 is used to compute the four-dimensional chaotic system (1), as in Eqs. (22), (23). The operation flow in the FPGA as shown in Fig. 26 is consistent with Multisim simulation results, and the multiplier and input/output pins are streamlined to reduce power consumption. The hardware circuit model is significant for building the system and advancing related fields, improving communication security.

## 6 The combination of new chaotic system and DNA algorithm is utilized in image encryption.

Four-dimensional or higher-order systems exhibit more complex dynamics than low-dimensional chaotic systems. Decryption methods such as phase space reconstruction, regression mapping and nonlinear prediction are difficult to decode ultra chaotic encrypted signals. This chapter proposes a color image encryption algorithm that combines the advantages of novel chaotic systems, logistic chaotic systems, and DNA-encoded algorithmic decoding. The encryption results of this system are characterized by large key capacity, high noise resistance, high cropping resistance, and high attack resistance. Experiments show that the new chaotic system combined with the DNA coding algorithm has excellent performance in image encryption, and the practical application value of the chaotic system has been improved.

### 6.1 Algorithms for DNA coding and decoding

DNA is deoxyribonucleic acid. In biogenetics, DNA consists of four bases: A (adenine), T (thymine), C (cytosine), and G (guanine). According to DNA's bottom complementary pairing principle, A and T and C and G can complement each other. Similar to the complementary relationship of bases, 0 and 1 in binary also have complementary relationships, i.e., 01 and 10 and 00 and 11 are complementary. Therefore, four bases A, T, C and G can be used to encode 00, 11, 10, and 01, respectively. According to the Watson–Crick complementarity rule, only 8 out of 24 possible encodings fulfill the law. The following Table 1 lists the 8 encodings that satisfy the complementarity rule. Each pixel value in a grayscale image can be represented as an 8-bit binary number, so each pixel value can be represented using a DNA sequence of length 4. For example, the pixel value of the image is 177, which is defined as 10110001 in binary and CTAG according to DNA coding rule 1. DNA arithmetic is performed according to the rule that every two binary values correspond to one DNA base. Since there are eight eligible DNA coding methods, each of which has its algorithm, each standard algorithm corresponds to 8 different DNA algorithms.

### 6.2 Encryption algorithm ideas and steps

The chaotic sequences generated using the new four-dimensional chaotic system determine the DNA encoding decoding and algorithmic rules. The algorithmic key is associated with the original image, ensuring that each image corresponds to a unique key and improving the system's resistance to attack. The digital image is divided into three two-dimensional matrices, which reduces the need for time and space resources. Each 2D matrix uses chaotic sequences for DNA encoding and operations, enhancing the algorithm's complexity and security. The three channels are merged after row and column permutations to obtain color images. The three chaotic sequences obtained by iterative logistic mapping are used for DNA operation, row substitution, and column substitution, improving the ciphertext image's disordering effect and anti-cropping characteristics. The flow of the encryption algorithm is shown in Fig. 27.

### 6.3 Simulation results and performance analysis

#### 6.3.1 Encryption results and histogram analysis

The color digital image and grayscale image are selected as the encrypted images, and the pixel sizes are 512 × 512 'Lena', 1024 × 1024 'Blood vessels', and 2048 × 1152 'Mars' [43], and the encryption effect is shown in Fig. 28. The result indicates that the histogram distribution of the image encrypted by the algorithm of this paper becomes more uniform, which proves that the information has been successfully hidden. The attacker can not get valuable information about the original image from the pixel distribution of the image. The encrypted image can be wholly deciphered only by using the correct key. The decryption process can completely recover the original image, and the observation shows that the decrypted image has no difference from the original image.

#### 6.3.2 Keyspace and sensitivity analysis

When the key space exceeds 2^100, the security and reliability of the encryption system are generally ensured. In this encryption system, the key comprises the initial values (*X*_{0}, *Y*_{0}, *Z*_{0}, *W*_{0}) of the hyper-chaotic system and the parameters *µ*, as well as the initial values (*x*_{0}, *x*_{01}, *x*_{02}) of the Logistic chaotic system. By altering the decimal places of the key to obtain the decrypted image, and subsequently decrypting the image, the key sensitivities are determined to be 10-1^{6}, 10-1^{6}, 10-1^{6}, 10-1^{6}, 10-1^{5}, 10-1^{6}, 10-1^{6}, and 10-1^{6}. Therefore, the size of the computed key space is 10^{16} × 10^{16} × 10^{16} × 10^{16} × 10^{15} × 10^{16} × 10^{16} × 10^{16} = 10^{127}, which is far more than the above requirement and is sufficient to effectively resist image corruption that may be caused by key attacks.

Figure 29b illustrates the decrypted image under the influence of an erroneous key resulting from a minute alteration of one of the keys *X*_{0} used during the encryption of the 'Lena' image, changing from 0.4937 to 0.4937000000000001 during decryption. Similarly, Fig. 29c presents the decrypted image under the impact of an erroneous key for 'Lena,' where the key *Z*_{0} shifts from 0.4176 during encryption to 0.4175199999999999 during decryption. Despite the exceedingly small range of key alterations in both cases, being only 10^(−16), the original image cannot be discerned from the decrypted images. This underscores the algorithm's remarkably strong key sensitivity. Furthermore, the algorithm's extensive key quantity establishes its resilience against exhaustive key-based attacks.

#### 6.3.3 Information entropy and neighboring location data correlation

Information entropy is a measure of how organized a system is. The greater the uncertainty, the greater the entropy value, which means the greater the amount of information needed to obtain. In the image, information entropy indicates the degree of confusion of the image pixel values. Defined explicitly as *p*(*x*_{i}) represents the proportion of pixels with gray value *x*_{i} in the image, and 2^{N} represents the gray level of the picture. When the image's gray level is *M*, the maximum information entropy is *H*_{max} = log_{2}*M*.Therefore, when *M* = 256 = 2^{8}, *H*_{max} = 8. Table 3 compares the information entropy data of the original image and the ciphertext image, which are calculated according to Eqs. (24), (25).

As can be seen from Table 2, the information entropy of the ciphertext image in the R, G, and B channels is very close to the theoretical information entropy maximum value of 8, which indicates that the confusion degree of the ciphertext image is close to the theoretical limit. Hence, this encryption algorithm can effectively resist the attack based on the information entropy of the image. In a general image, each pixel has a high correlation with its neighboring pixels, and the lower the correlation between the adjacent pixels of the ciphertext image, the stronger the ability of the encryption algorithm to resist attacks. To test the resistance of the encryption algorithm, 5000 pairs of neighboring pixels in horizontal, vertical, and diagonal directions of the encrypted image are randomly selected, and the correlation between them is calculated.

The following can be observed through the three-channel directional correlation coefficient dot plots in Fig. 30 the data values in the horizontal neighboring positions of all three channels of the plaintext image have a strong correlation, and the distribution of the data points is linear, while the correlation between the data values in the vertically neighboring positions of the ciphertext image is almost zero, and the distribution of the data points is completely random. The data comparison in Table 3 shows that the correlation of neighboring pixels in the three channels of the encrypted image tends to be close to zero. In contrast, the original image has a high correlation between pixels. This shows that the data values of neighboring pixels of the channels of the ciphertext image in the horizontal, vertical, and diagonal directions have no correlation, which proves the high level of confusion of the image encrypted by this algorithm.

#### 6.3.4 Analysis of attack resistance

Applying chaotic systems to image encryption algorithms must be robust enough to resist attacks from multiple unfavorable factors in real scenarios. The ciphertext image is added with different pretzel noise and shear attack strengths. In this paper, the 'Mars' image was selected for experimental analysis, as depicted in Fig. 31.

The paper uses noise and shear attacks to test the robustness of the algorithm in image encryption. The algorithm shows excellent resistance to different forms of aggression, including pretzel noise and shear attack, while retaining the primary information of the decrypted image.

## 7 Conclusion

In this paper, a novel four-dimensional chaotic system is proposed by introducing sinusoidal and exponential functions as nonlinear activation terms. The system exhibits a variety of nonlinear dynamical behaviors, including coexisting and hidden attractors, quasi-periodic and intermittent chaos, and transient chaos. At specific parameter values, the system also exhibits rare phenomena such as multiplicity, controllable spike-discharge clusters, burst oscillations, and controllable vortex phenomena. The multiple dynamical behaviors of the system are verified by numerical analysis and its practical effectiveness in image encryption is confirmed. The proposed four-dimensional chaotic system demonstrates diverse nonlinear dynamical behaviors, and this innovation provides new possibilities for future scientific research and technological development, as well as new perspectives and methods for solving practical problems.

## Data availability

Data sharing is not applicable to this paper as no datasets were generated or analysed during the current study.

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This work is supported by the Science and Technology Program" of Gansu Province (No.20JR10RA080).

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Zhang, J., Hou, J., Xie, Q. *et al.* Circuit realization and application of a chaotic system with hidden attractor, controlled spike discharge and offset boosting.
*Nonlinear Dyn* **112**, 18551–18579 (2024). https://doi.org/10.1007/s11071-024-09987-w

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DOI: https://doi.org/10.1007/s11071-024-09987-w