Chaos

The fascinating subject of chaos has been the focus of researchers and scientists from many fields over the last 40 years. The topic is multidisciplinary and continuously opens new opportunities for study in both the theoretical and applied points of view. The high sensitivity to initial conditions, the emergent behavior, and the bifurcations that are peculiar features occurring in nonlinear systems are of increasing interest in various fields of research, ranging from engineering to biology to physics. Moreover, the active use of chaos has also recently been adopted in other areas, such as in computer sciences, telecommunications, and network security.

The aim of this Editorial is to remark the great efforts made by the journal Applied Sciences in promoting, among the various topics typical of its scope, the topic of chaos. Since 2015, 90 contributions have been published in the journal regarding chaos. It is interesting to note, as shown in Figure 1, the constantly increasing trend of papers related to chaos during the last years. Moreover, these 90 papers cover a wide spectrum of subjects in which chaos is considered, discovered, or adopted. This classification, obtained on the basis of the main topic addressed with the use of chaos, is reported in the pie chart of Figure 2.

In this Editorial, we selected 21 contributions published in 2021 and 2022 (up to the current date), containing the terms chaos or chaotic in the title, or which are particularly relevant in the field. In the following, the contents of these papers will be outlined by following an inverse chronological order.

In [1], Shen et al. focus on the intrusion detection problem. Cybersecurity is a particularly fascinating problem, today; therefore, the paper provides a timely contribution in this direction. In particular, the idea proposed in this paper is to generate chaotic weights for classes of patterns that are used in the Class-Level Soft Voting Ensemble (CLSVE), leading to an improvement in the detection rate and to a reduction in the false positive rate.

The paper by Cui et al. [2] includes impressive results regarding the classic Lorenz chaotic system. This paper has been selected for this Editorial even if the term chaos is not included in the title because it is strongly related to the history of chaos. Theoretical and numerical results on the synchronization of two delayed Lorenz systems with multiple positive Lyapunov exponents are reported in the contribution. The Matlab and Spice numerical simulations are discussed, and the electronic circuit’s implementation is studied in detail. The paper remarks that the classic Lorenz system continues to give researchers opportunities for novel investigations.

The paper by Dridi et al. [3] proposes an innovative encryption system operating in cipher blockchains based on chaotic maps to generate pseudo-random sequences of numbers. In the architecture proposed to generate the sequences, more logistic maps are used. The authors present with accuracy the encryption/decryption algorithm and validate the presented approach with more experiments and statistical tests and a comparison with other techniques. The speed of the process has been evaluated by using an entropy-based approach.

In the contribution [4], Chu et al. deal with the problem of trajectory planning in 3D for Unmanned Aerial Vehicles (UAVs), a strong non-convex optimization problem. Therefore, a particle-swarm-based optimization algorithm is considered. In order to improve the performance of the optimization, the initial particles must be distributed using a chaotic sequence to obtain an appropriate uniformity. The randomness and the ergodicity of the used chaotic maps lead to improvements in the performance of the algorithm. The paper merges both artificial intelligence concepts with chaos in order to solve a trajectory control problem. The numerous experiments show the suitability of the global optimization algorithm; moreover, the study is related to static obstacles not involving dynamic changes in the environment.

In the paper [5], Litak et al. introduce important considerations on a forced pendulum systems. Taking into account the very simple structure, the authors discover chaos transition in a simple but rigorous way by using the Melnikov analysis. In fact, it allows us to define the basins of attraction of the system. The obtained results encourage the readers to follow the adopted strategy to study other forced dynamical systems. Indeed, until now, the undiscovered and unexplored aspects of investigating complex systems represent an appealing research topic.

The classic Duffing forced oscillator, like the Lorenz system, represents one of the most studied paradigms for chaos and other complex behavior. The paper [6], by Kydyrbekuly et al., has been selected even though the title does not contrain the word chaos since this type of oscillator is a cornerstone in the chaos theory of forced systems. The authors show that the studied system, consisting of a vertical rotor, presents nonlinear jump resonance. An accurate numerical experiment has been reported, and the study includes a detailed analytical section. The paper is also included in the Editorial because jump resonance can be considered as a precursor of chaos in forced dynamical systems.

The core of synchronization, and, therefore, the key of actively using chaos, is the design of the observer system. Taking into account a master–slave coupling configuration, the slave is an appropriate nonlinear observer of the master. The authors of [7] introduce an adaptive observe scheme for synchronizing two chaotic systems. This is based on the interaction between the observer and an adaptive controller. The suitable observer and the adaptive controller work on transformed systems. The convergence of the observer is proven, and an accurate analytical formulation of the problem is presented. The numerical examples are referred to the chaotic Arneodo system. The theoretical aspects are discussed in detail and with appealing practical results.

The analysis of time-series regarding the water level of rivers is the subject of the contribution [8] by Lee et al. Indeed, chaotic, and therefore in some sense deterministic, oscillations in the river level can open the way to control actions. The authors use the return map approach in order to analyse the time-series. Moreover, the study is also addressed to characterize the attractors of the system and to evaluate all the permutations that allow them to characterize the series. The conclusions positively consider the established chaoticity of the time-series. The contribution opens a discussion on how to use the theory of chaos in evaluating models in hydrogeology and to control river levels.

The paper [9] by Ozawa et al. regards the model of waves within a lattice of simple loss-less devices with a nonlinear load at the end. The paper discusses how waves can be modeled with 1D chaotic maps. The dynamics of the system is accurately analysed for different load parameters. In the contribution, a bifurcation analysis is also reported, as well as an accurate estimation of the Lyapunov exponents.

In the paper [10], Jiang et al. report an accurate new problem in the area of chaotic vibrations. The discovery of chaos in damping systems allows researchers to control the behavior of the system. The time-series analysis is performed by using more parameters that confirm the chaotic behavior of the experimental system. Moreover, attention has been given to multi-time-scale chaotic characteristics by using the so-called Improved Variational Model Decomposition (IVMD) method.

The introduction of chaotic models to study internet dynamics is the subject of the paper [11]. It includes a detailed state-of-the-art and an innovative strategy to use Petri networks in modeling Active Queue Management (AQM) schemes. The analysis was performed by using a two-parameter dynamical map. The numerical results confirm the suitability of the proposed modeling strategy.

The image encryption is the main subject of [12]. Liang et al. propose an hyperchaotic system with a single parameter to generate pseudo-random signals. Moreover, an approach based on elliptic curve cryptography is used. In order to validate their approach, the authors present more experiments with encouraging performance indices.

The topic of the contribution [13] by Huang et al. regards the study of coupled maps in order to generate pseudo-random numbers. The intermittency of the system is analysed. This aspect is new and important since it presents an alternative method to find new classes of pseudo-random number generators inside the same system. The authors also propose an accurate bifurcation analysis. The paper is a receptacle of new ideas.

The evaluation of encryption and decryption performance in image processing by using entropy functions has been proposed by Cheng and Li in [14]. The paper reports some important items such as using thermodynamics principles in evaluating nonlinear signal processing performance. The study also includes numerical examples.

A study regarding new items in adaptive control is the core of [15] by Wang et al. A fuzzy adaptive controller in a self-organizing configuration inspired by cerebellar model articulation is widely studied. The global controller has been used to control a chaotic system based on the FHN neuron model in a master–slave configuration. Both theoretical and numerical results are presented. Moreover, the synchronization capabilities of the controller have been emphasized.

The paper [16] by Lin et al. uncovers the role of the master–slave scheme in hyperchaotic Henon maps for image encryption. The system is integrated in a classical platform of image encryption/decryption. The reported results are encouraging. The approach can be used in various fields, not restricted to image processing applications.

The contribution [17] by Yousri et al. is addressed to the identification of fractional-order parameters related to an electrical model. The identification strategy is related to an optimization problem. The authors use the particle swarm optimization algorithm with the randomization process of the swarming procedure implemented by using more chaotic maps. The paper shows the coupling of chaotic systems with artificial intelligence procedures for solving electrical engineering modeling problems.

In [18], Sambas et al. show significant details related to butterfly chaotic systems and their FPGA implementations. The paper covers the main aspects of chaotic circuits. It represents an innovative proposal of new dynamics and their electrical implementation.

The FPGA capability to realize dynamical systems has been focused in [19] by Dridi et al. for designing stream-based secure chaotic generators. The chaos generator is a 3D Chebishev map. The encryption/decryption problem is applied to secure images.

A third-order jerk chaotic system with cubic nonlinearity is studied in [20] by Arshad et al. The system is implemented by using analog electronics, and experimental results are widely discussed.

The results presented in [21] by Nam and Kang report an accurate classification of chaotic signals. Their approach uses convolutional neural networks and shows an interesting link between chaotic signals and distributed neural networks. More examples have been proposed for the validation of the study.

In

Table 1. Correlation matrix of the selected key topics.

	Numerical and	Physics and	Electronic	Artificial	Environmental	Secure	Aeronautics	Control and	Energy and
	Mathematical	Material	Circuits and	Intelligence	and Civil	Communications	Mechanics	Synchronization	Complex
	Theory and	Science	Power		Engineering	and Image	Robotics		Systems
	Applications		Converters			Encryption	Engineering		Engineering
[1]				X		X
[2]		X						X
[3]		X				X
[4]	X	X		X			X
[5]	X	X
[6]							X	X
[7]	X				X			X
[8]	X				X
[9]	X		X						X
[10]	X	X					X
[11]				X					X
[12]	X					X
[13]	X			X		X
[14]	X	X				X
[15]				X				X
[16]	X					X
[17]	X			X
[18]	X							X
[19]			X		X	X
[20]	X		X
[21]	X			X

, the relationships among the main topics individuated in the pie chart of Figure 2 and the selected papers are emphasized. Moreover, it has been detailed for each contribution not only the main topic, but also the other connections each paper has with the others, thus highlighting the strong interdisciplinarity of the selected studies.

The extraordinary interest in chaos is illustrated in this contribution. Chaos is connected to many disciplines involving applied sciences and technology but also other sciences, such as biology. In all these fields, the interest is not only in the analysis of the onset of chaos, but rather its active use, leading chaos theory to be itself an applied science.