*2.2. Related Work*

Several works on estimating chaotic system parameters have been recently published [13,44,45]. Real-world estimation is difficult for parameters of a complex 3D chaotic system. Most gradient-based methods are sensitive to initial conditions, trapping them in local minima. Estimating 3D chaotic parameters using soft computing techniques at a suitable cost function, is one solution, such as global optimization algorithms. Several cases of chaotic system parameter estimation using optimization algorithms have been reported [44,46,47]. The following section summarizes these algorithms.

Li et al. [35] combined the artificial bee colony algorithm (ABC) and differential evolution (DE) to estimate chaotic system parameters. Gao et al. [48] proposed chaos firefly optimization (CFA) for identifying Lorenz chaotic system parameters. Using chaotic search to update the standard Firefly algorithm improved optimization accuracy and speed. Recent pioneering work has combined the cuckoo search (CS) algorithm and orthogonal learning to estimate Lorenz and Chen chaotic system parameters [49]. He et al. also used particle swarm optimization (PSO) to estimate Lorenz system parameters [50]. This technique does not sufficiently explore the solution space. Small populations produce poor results. Li et al. [51] introduced the chaotic ant swarm (CAS) algorithm to determine chaotic system parameters.

Gholipour et al. [52] estimated chaotic system parameters with the artificial bee colony algorithm. Wei and Yu [53] presented a hybrid cuckoo search (HCS) algorithm inspired by differential evolution. The presented HCS offers two novel mutation strategies to fully exploit the neighborhood. Three chaotic systems with and without time delays were simulated and compared to other optimization methods to test HCS. Experimental results showed HCS's superiority in chaotic system parameter estimation due to its high calculation accuracy, fast convergence speed, and strong robustness. In [54], the authors introduced a two-stage estimation technique that combined the guaranteed approach and swarm intelligence.

Zhuang et al. [55] presented a new hybrid Jaya–Powell method for estimating the parameters of a Lorenz chaotic system. The proposed Jaya–Powell algorithm combines the Jaya algorithm, which seeks the relatively global optimum, with the Powell algorithm, which seeks the relatively local optimum, to provide a more precise and efficient estimate. This algorithm's searching technique makes it easier to strike a middle ground between exploration and exploitation throughout the optimization process. The suggested Jaya– Powell algorithm does not need the careful adjustment of appropriate parameters as it does not rely on any algorithm-specific parameters. Compared with seven benchmark

methods, the proposed hybrid Jaya–Powell algorithm provided more precise estimates and converged more quickly.

The work presented in [56] explored how to use several metaheuristic algorithms for the recognition of parameters in a fractional-order financial chaotic system. The algorithms that have been put into place are the ant colony optimizer, grey wolf optimizer, whale optimization algorithm, and artificial bee colony optimizer. As an objective function, mean square error was used to estimate the system's parameters. Zhang et al. [57] offered a novel method of parameter estimation that made use of numerical differentiation to streamline the preparation of observational data. Given the noisy observations on a subset of dependent variables, numerical differentiation may be used to approximately determine the values of the dependent variables and their derivatives. The parameter estimation issue may be simplified by substituting these approximations into the original system. The precision and efficiency of their technology are shown by numerical examples.

Encouraged by recent developments in data assimilation, Carlson et al. [58] built a dynamic learning technique to estimate missing parameters of a chaotic system using just a subset of available data. The authors convincingly proved, under plausible assumptions that this approach converged to the right parameters when the system under issue was the standard three-dimensional Lorenz system. They computationally showed the effectiveness of this technique on the Lorenz system by recovering any correct subset of the three nondimensional parameters of the system, provided that an appropriate subset of the state was observable. Over the last two decades, studies on how to synchronize a Lorenz chaotic system have been more prominent. Model reference adaptive control (MRAC) synchronization scheme design has been the primary focus of the majority of the research. For this problem, C. Peng, and Y. Li [59] suggested two system identification strategies. The observer–Kalman filter identification method was the first method used. The second kind of discretization was the bilinear transform. The new approach significantly improved the accuracy of the discovered parameters, which were therefore already very near to actual values.

Rizk-Allah et al. [60] presented a unique approach to parameter estimation for the chaotic Lorenz system, using a modified form of particle swarm optimization (PSO). The suggested technique, a memory-based particle swarm optimization (MbPSO) algorithm, modeled the parameter estimation of the Lorenz system as a multidimensional issue. To change the population's orientation and improve search efficiency, MbPSO added two additional variables to the classic PSO. The results showed that the suggested algorithm performed much better than the original PSO, when particle memories were linked to those of other particles. The primary goal of the study [61] was to apply a deep learning technique to the problem of estimating the parameters of chaotic systems, such as the Lorenz system. In this research, the authors used the k-means technique to build out the workflow of a deep neural network (DNN)-based approach. The DNN approach works well for difficult, nonlinear problems. Using the proposed approach, 98% of correct training data and 73% of test data were predicted.

The parameter identification for the discrete memristive chaotic map was the primary topic of the research presented by Peng et al. [62], in which a novel intelligent optimization technique called the adaptive differential evolution algorithm was suggested. To handle the hyperchaotic and attractors that coexist in the investigated discrete memristive chaotic maps, the identification objective function had two unique components: time sequences and return maps. It was shown via numerical simulations that the suggested approach outperformed the other six existing algorithms and maintained the ability to correctly identify the original system's properties, even when subjected to noise interference.

Although chaotic system parameter estimation has been studied for decades, it can still be improved. According to the review, past studies focused on: (1) Estimating a single chaotic system parameter and (2) Not addressing the best optimization technique for exploration and exploitation in a unified framework. Most bio-inspired optimization techniques for chaotic system parameter estimation combine two or more algorithms to improve exploration and exploitation. To the best of our knowledge, little attention has been paid to developing a bio-inspired parameter estimation technique for a chaotic system with few training samples.

#### **3. Materials and Methods**

Let . X = *F*(*X*, *X*0, *θ*0) be a continuous nonlinear chaotic system, whereX = (*x*1, *x*2,..., *xN*) <sup>∈</sup> <sup>R</sup><sup>n</sup> is the chaotic system's state vector, . X is *X*'s derivative, the resulting solution is parameterized by the initial value *X*0, and *θ*<sup>0</sup> = (*θ*1,0, *θ*2,0,..., *θd*,0) are the original parameters. If the system's structure is known, the estimated system may be expressed as . X = *F*(*X*, *X*0, *θ* ), where <sup>X</sup> <sup>=</sup> (*x*1, *<sup>x</sup>*2,..., *<sup>x</sup>N*) <sup>∈</sup> <sup>R</sup><sup>n</sup> is the state vector and *<sup>θ</sup>* = *θ* 1, *θ* 2,..., *θ d* is a collection of estimated parameters. Based on *X*, the fitness function is [49,51]:

$$f\left(\tilde{\theta}\_i^{\boldsymbol{\eta}}\right) = \sum\_{i=0}^{\mathcal{W}} \left[ \left( \mathbf{x}\_1(t) - \tilde{\mathbf{x}}\_{i,t}^{\boldsymbol{\eta}}(t) \right)^2 + \dots + \left( \mathbf{X}\_N(t) - \tilde{\mathbf{x}}\_{i,N}^{\boldsymbol{\eta}}(t) \right)^2 \right],\tag{35}$$

where *t* = 0, 1, ... ... *W* and *i* is the *i*th state vector. Estimating chaotic system parameters aim to reduce fitness function by minimizing *θ n <sup>i</sup>* . Dynamic instability makes chaotic systems difficult to estimate. Due to the problem's many variables and various local search optima, typical optimization in the local optima is difficult [63,64].

A chaos communication system comprises of transmitter, receiver, and channel (noise) performance. In the transmitter, the modulation methods utilized to combine the message signal and chaotic carrier are crucial for system security. As a signal must be sent to the receiver, there is a possibility that intruders may receive the signal. Even if intruders do not know the structure or parameters of a chaotic system, they may use signal processing or sophisticated algorithms to extract the message from the transmitter signal. In chaotic masking, the signal is directly added to the chaotic signal; thus, the fluctuation may be recognized by non-linear dynamic forecasting techniques or power spectrum analysis, if the message amplitude/frequency is high enough. Mixing the message should remove any pattern or information from the sent signal. The carrier chaotic signal will be distorted by channel noise before reaching the receiver. Message recovery requires chaotic synchronization at the receiver. Demodulation is an issue in chaotic communication systems. The recommended solution uses a few signal samples instead of large samples that need more calculation. The communication channel is assumed to be free noise, as the emphasis is on estimating the chaotic system's unknown parameters, not channel attacks.

As discussed later, in a quantum model of FOA, each fruit fly represents a particle that has a state depicted by a wave function, instead of position and velocity. The dynamic behavior of the fruit fly is different from that of the fruit fly in standard FOA algorithms; that is, the accurate values of *x* and *v* cannot be simultaneously calculated. Its searching performance is better than the original particle swarm optimization algorithm. The quantum particle swarm optimization algorithm is a global convergence guarantee algorithm. The capabilities of a QFOA algorithm to enhance convergence speed and low optimization accuracy were achieved through: (1) A mutation operator to increase the diversity of particles in a population (the delta potential well concept to speed up the convergence speed); (2) An operator based on evolutionary generations to update a contraction expansion coefficient (objective or fitness function for global optimization); (3) An elitist strategy to remain the strong particles.
