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Multi-dimensional Flight Trajectory Prediction (MFTP) in Flight Operations Quality Assessment (FOQA) refers to the estimation of flight status at the future time, accurate prediction future flight positions, flight attitude and aero-engine monitoring parameters are its goals. Due to differences between flight trajectories and other kinds trajectories and difficult access to data and complex domain knowledge, MFTP in FOQA is much more challenging than Flight Trajectory Prediction (FTP) in Air Traffic Control (ATC) and other trajectory prediction. In this work, a deep Koopman neural operator-based multi-dimensional flight trajectory prediction framework, called Deep Koopman Neural Operator-Based Multi-Dimensional Flight Trajectories Prediction (FlightKoopman), is first proposed to address this challenge. This framework is based on data-driven Koopman theory, enables to construct a prediction model using only data without any prior knowledge, and approximate operator pattern to capture flight maneuver for downstream tasks. The framework recovers the complete state space of the flight dynamics system with Hankle embedding and reconstructs its phase space, and combines a fully connected neural network to generate the observation function of the state space and the approximation matrix of the Koopman operator to obtain an overall model for predicting the evolution. The paper also reveals a virgin dataset Civil Aviation Flight University of China (CAFUC) that could be used for MFTP tasks or other flight trajectory tasks. CAFUC Datasets and code is available at this repository: https://github.com/CAFUC-JJJ/FlightKoopman. Experiments on the real-world dataset demonstrate that FlightKoopman outperforms other baselines.

Reaction systems are an abstract model of biochemical reactions in the living cell within a framework of finite (though often large) discrete dynamical systems. In this setting, this paper provides an analytical and experimental study of stability. The notion of stability is defined in terms of the way in which small perturbations to the initial state of a system are likely to change the system's eventual behavior. At the stable end of the spectrum, there is likely to be no change; but at the unstable end, small perturbations take the system into a state that is probabilistically the same as a randomly selected state, similar to chaotic behavior in continuous dynamical systems.

This paper investigates properties of synchronization of trigonometric coupled maps. These maps have some advantages, such as an invariant measure, ergodicity and the possibility of Kolmogorov–Sinai (KS) entropy calculation. Moreover, an encryption algorithm based on a synchronized coupled map is suggested and described in detail. Also, some security analyses are presented to illustrate the effectiveness of the proposed scheme. The cryptosystem's speed is analyzed as well. Results of the various types of analysis are encouraging and it can be concluded that the proposed image encryption technique is a suitable choice for practical applications.

Industry clusters have outperformed in economic development in most developing countries. The contributions of industrial clusters have been recognized as promotion of regional business and the alleviation of economic and social costs. It is no doubt globalization is rendering clusters in accelerating the competitiveness of economic activities. In accordance, many ideas and concepts involve in illustrating evolution tendency, stimulating the clusters development, meanwhile, avoiding industrial clusters recession. The term chaos theory is introduced to explain inherent relationship of features within industry clusters. A preferred life cycle approach is proposed for industrial cluster recessive theory analysis. Lyapunov exponents and Wolf model are presented for chaotic identification and examination. A case study of Tianjin, China has verified the model effectiveness. The investigations indicate that the approaches outperform in explaining chaos properties in industrial clusters, which demonstrates industrial clusters evolution, solves empirical issues and generates corresponding strategies.

This study proposes a short-term traffic flow prediction approach based on multiple traffic flow basic parameters, in which the chaos theory and support vector regression are utilized. First, a high-dimensional variable space can be obtained according to the traffic flow fundamental function. Then, a maximum conditional entropy method is proposed to determine the embedding dimension. And multiple time series are reconstructed based on the phase space reconstruction theory using the time delay obtained by mutual information method and the embedding dimension captured by the maximum conditional entropy method. Finally, the reconstructed phase space is used as the input and the support vector regression optimized by the genetic algorithm is utilized to predict the traffic flow. Numerical experiments are performed and the results show that the approach proposed has strong fitting capability and better prediction accuracy.

Studies on fractional-order chaotic systems have increased significantly in the last decade. This paper presents Rucklidge chaotic system’s dynamical analyses and its fractional-order circuit implementations. Component values required for realizing the circuit of the fractional-order system are calculated for different fractional-orders. The feasibility of the attractor is examined by implementing its electronic circuit with a fractional-order module. The module is constructed based on the Diyi-Chen model since it is easier to implement and cost-effective. In electronic circuit implementations, it is observed that the system’s chaotic state disappears as the fractional degree decreases. Numerical and circuit simulation results are consistent well with the hardware experimental results.

Chua's circuit is a nonlinear dynamic circuit that has assumed a paradigmatic role in mathematical, physical and experimental demonstrations of chaos. Yet even today, complexity and chaos are seen as challenging topics reserved for specialists. Is it possible for young people in junior and senior high school to acquire a difficult concept such as chaos? In order to study this issue, we developed a four-step teaching/learning method in which two groups of students (one from junior and one from senior high school) manually built Chua's circuit and used it as a source for extraordinary images and music. All this allowed students to interact with strange attractors, discovering their fascination and beauty. Both groups of students created 3-D models of attractors, modified the control parameters, explored the sound and music of chaos and, during the process, discovered the connection between science and art. Briefly, they learned through engaging activities, which allowed them to acquire physical and mathematical knowledge of chaos through various modalities. The results show that manipulation of the circuit and related artistic activities can provide a new, creative and enjoyable approach to science learning.

The pre-history of chaos in a rationalist context is taken as a point of departure, starting out with ancient China. The related ancient-Greek "unmixing theory" then leads over to two simple formally 2-body Hamiltonian systems exhibiting chaotic behavior. When the two masses involved are unequal, "pseudoattractors" are formed. Deterministic statistical "thermodynamics" with its dissipative behavior arises when the potential is repulsive. Deterministic statistical "cryodynamics" arises when the potential is attractive. The latter class of Newtonian systems is characterized by "antidissipative" behavior. A geometric proof is sketched in the footsteps of Sinai and Bunimovich. Antidissipative behavior is known empirically from Hubble's law which was so far explained in less fundamental terms. Three experimental examples are proposed.

This paper introduces the notion of second minimal $n$-periodic orbits of continuous maps on the interval according to whether $n$ is a successor of the minimal period of the map in the Sharkovski ordering. We pursue the classification of second minimal $7$-orbits in terms of cyclic permutations and digraphs. It is proven that there are nine types of second minimal 7-orbits with accuracy up to inverses. The result is applied to the problem of the distribution of periodic windows within the chaotic regime of the bifurcation diagram of the one-parameter family of unimodal maps. It is revealed that by fixing the maximum number of appearances of periodic windows, there is a universal pattern of distribution. In particular, the first appearance of all the orbits is always a minimal orbit, while the second appearance is a second minimal orbit. It is observed that the second appearance of the 7-orbit is a second minimal 7-orbit with a Type 1 digraph. The reason for the relevance of the Type 1 second minimal orbit is the fact that the topological structure of the unimodal map with a single maximum, is equivalent to the structure of the Type 1 piecewise monotonic endomorphism associated with the second minimal 7-orbit. Yet another important report of this paper is the revelation of universal pattern dynamics with respect to an increased number of appearances.

Some characteristics of mean sensitivity and Banach mean sensitivity using Furstenberg families and inverse limit dynamical systems are obtained. The iterated invariance of mean sensitivity and Banach mean sensitivity are proved. Applying these results, the notion of mean sensitivity and Banach mean sensitivity is extended to uniform spaces. It is proved that a point-transitive dynamical system in a Hausdorff uniform space is either almost (Banach) mean equicontinuous or (Banach) mean sensitive.

Improving running-in quality is an important project of Green Tribology. In actual environment, the running-in quality of mechanical equipment is greatly affected by system parameters. In view of this, running-in experiments were performed at different rotating speeds on a pin-on-disc tribometer. Chaos theory was used to investigate the effect of rotating speeds on the running-in quality. The experimental results showed that the running-in quality could be better evaluated by using the chaotic characteristic parameter “Convergence (CON)”. The greater the CON, the better the running-in quality. With an increase of the rotating speed, CON first gradually increased and then decreased to a certain level, whilst the running-in quality first gradually became better and then became worse. A moderate rotating speed is advantageous to obtain the optimum running-in quality. This contributes to the service life extension and the efficiency improvement, so as to reduce energy consumption.

The simplest megastable chaotic system is built by employing a piecewise-linear damping function which is periodic over the spatial domain. The unforced oscillator generates an infinite number of nested limit cycles with constant distances whose strength of attraction decreases gradually as moving to outer ones. The attractors and the basins of attraction of the proposed system are almost compatible with those of the system with sinusoidal damping. However, the nonzero Lyapunov Exponent of the latter is consistently below that of the former. A comparative bifurcation analysis is carried out for periodically forced systems, showing the chaotic behavior of coexisting attractors in specific values of parameters. Changing the bifurcation parameter results in expansion, contraction, merging, and separation of the coexisting attractors, make it challenging to find the corresponding basins. Three symmetric pairs of attractors are observed; each one consists of two symmetric attractors (with respect to the origin) with almost the same values of the corresponding Lyapunov Exponent.

Summary—Software engineering processes must be understood to be managed effectively. It is important to go beyond purely metaphorical and descriptive methods to achieve such understanding. This paper proposes a framework, based on dynamic feedback control systems, for building potentially quantitative models of software engineering processes. It then uses this framework to explore some important characteristics of quantifiable outputs of such processes, including observability, measurability, independence, (near) continuity, and (near) monotonicity.

General Terms—Software engineering, process modeling.

The nonlinearity of acoustic signals produced by male cicadas and their propagation in the atmosphere using the theory of dynamical systems and partial differential equations are explored in this paper. Previous research using a Volterra equation has shown that the signal data from the vibrations of cicada tymbals and that from the recordings of the acoustic signals about 5 inches away from the cicada exhibit some nonlinear characteristics. The experimental results shown in this paper confirm the nonlinearity of the signals farther from the cicada. A number of nonlinear acoustic signal propagation models are discussed — among them the Burgers' equation which has been implemented and whose results are quite promising.

Metaheuristic techniques are capable of representing optimization frames with their specific theories as well as objective functions owing to their being adjustable and effective in various applications. Through the optimization of deep learning models, metaheuristic algorithms inspired by nature, imitating the behavior of living and non-living beings, have been used for about four decades to solve challenging, complex, and chaotic problems. These algorithms can be categorized as evolution-based, swarm-based, nature-based, human-based, hybrid, or chaos-based. Chaos theory, as a useful approach to understanding neural network optimization, has the basic idea of viewing the neural network optimization as a dynamical system in which the equation schemes are utilized from the space pertaining to learnable parameters, namely optimization trajectory, to itself, which enables the description of the evolution of the system by understanding the training behavior, which is to say the number of iterations over time. The examination of the recent studies reveals the importance of chaos theory, which is sensitive to initial conditions with randomness and dynamical properties that are principally emerging on the complex multimodal landscape. Chaotic optimization, in this regard, accelerates the speed of the algorithm while also enhancing the variety of movement patterns. The significance of hybrid algorithms developed through their applications in different domains concerning real-world phenomena and well-known benchmark problems in the literature is also evident. Metaheuristic optimization algorithms have also been applied to deep learning or deep neural networks (DNNs), a branch of machine learning. In this respect, the basic features of deep learning and DNNs and the extensive use of metaheuristic algorithms are overviewed and explained. Accordingly, the current review aims at providing new insights into the studies that deal with metaheuristic algorithms, hybrid-based metaheuristics, chaos-based metaheuristics as well as deep learning besides presenting recent information on the development of the essence of this branch of science with emerging opportunities, applicability-based optimization aspects and generation of well-informed decisions.

After the stock market crash of October 19, 1987, interest in nonlinear dynamics and chaotic dynamics have increased in the field of financial analysis. The extent that the daily return data from the Shanghai Stock Exchange Index and the Shenzhen Stock Exchange Index exhibit non-random, nonlinear and chaotic characteristics are investigated by employing various tests from chaos theory. The Hurst exponent in R/S analysis rejects the hypothesis that the index return series are random, independent and identically distributed. The BDS test provides evidence for nonlinearity. The estimated correlation dimensions provide evidence for deterministic chaotic behaviors.

In this article, we establish a model to delineate the emergence of "self" in the brain making recourse to the theory of chaos. Self is considered as the subjective experience of a subject. As essential ingredients of subjective experiences, our model includes wakefulness, re-entry, attention, memory, and proto-experiences. The stability as stated by chaos theory can potentially describe the non-linear function of "self" as sensitive to initial conditions and can characterize it as underlying order from apparently random signals. Self-similarity is discussed as a latent menace of a pathological confusion between "self" and "others". Our test hypothesis is that (1) consciousness might have emerged and evolved from a primordial potential or proto-experience in matter, where inert matter is the carrier of subjective experiences; and (2) "self" arises from chaotic dynamics, self-organization and selective mechanisms during ontogenesis, while emerging post-ontogenically as an adaptive pressure driven by both volume and synaptic-neural transmission and influencing the functional connectivity of neural nets (structure).

We have applied the principles of chaos theory to optimizing the manufacturing process. A simulation system was created in which multiple parameters were modified to denote the effect of chaotic changes on the net profit derived.

The companies that are successful in modeling their manufacturing systems in a simplistic manner can look forward to increased profit, better control and simpler manufacturing systems.

Finding and selecting the correct set of parameters to subject to chaotic changes is time consuming. There are countless operating parameters that could be selected and the interaction of those parameters may have varying degrees of success on trying to optimize the manufacturing process. Certain parametric interactions may increase revenue, while other interactions may actually do the inverse.

To study nonlinear dynamic characteristics of the gyroscope rotor system, a dynamic equation of rolling bearing rotor system considered multiple nonlinear factors such as varying compliance and nonlinear bearing force is presented. Using Runge-Kutta method, it is solved for different parameters. The bifurcation diagrams, the Poincaré maps and the frequency spectrums are given. Study results show that rich periodic, non-periodic (quasi-periodic and chaotic) vibrations and period doubling bifurcation exist in this system with the change of working parameters. Instability of the system can be reduced by selecting reasonable parameters. These results are further verified by the maximum Lyapunov exponents.