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In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.

Chaos is ubiquitous in physical systems. The associated sensitivity to initial conditions is a significant obstacle in forecasting the weather and other geophysical fluid flows. Data assimilation is the process whereby the uncertainty in initial conditions is reduced by the astute combination of model predictions and real-time data. This chapter reviews recent findings from investigations on the impact of chaos on data assimilation methods: for the Kalman filter and smoother in linear systems, analytic results are derived; for their ensemble-based versions and nonlinear dynamics, numerical results provide insights. The focus is on characterizing the asymptotic statistics of the Bayesian posterior in terms of the dynamical instabilities, differentiating between deterministic and stochastic dynamics. We also present two novel results. Firstly, we study the functioning of the ensemble Kalman filter in the context of a chaotic, coupled, atmosphere-ocean model with a quasi-degenerate spectrum of Lyapunov exponents, showing the importance of having sufficient ensemble members to track all of the near-null modes. Secondly, for the fully non-Gaussian method of the particle filter, numerical experiments are conducted to test whether the curse of dimensionality can be mitigated by discarding observations in the directions of little dynamical growth of uncertainty. The results refute this option, most likely because the particles already embody this information on the chaotic system. The results also suggest that it is the rank of the unstable-neutral subspace of the dynamics, and not that of the observation operator, that determines the required number of particles. We finally discuss how knowledge of the random attractor can play a role in the development of future data assimilation schemes for chaotic multiscale systems with large scale separation.

To achieve accurate and fast fault location in the distribution network, this paper introduces chaos theory into PSO (Particle Swarm Optimization). It proposes a fault location method for distribution networks with distributed generation based on CPSO (Chaotic Particle Swarm Optimization). The IEEE14-node distribution network model with multiple power sources is established. Then, the coding mode, switching function, and fitness function of distribution network fault information is constructed. Finally, the location results of single-point fault, multi-point fault, and information distortion are analyzed, respectively. The results show that the chaotic particle swarm algorithm proposed in this paper can achieve accurate location and outperform the traditional particle swarm algorithm, genetic algorithm, and the recently proposed manta ray foraging algorithm in convergence, accuracy, and algorithm efficiency.

This paper introduces an improved version of Gorilla Troops Optimizer (GTO) based on Chaotic maps, namely chaotic Gorilla Troops Optimizer (CGTO). GTO like other swarm intelligence algorithms suffers from stagnation in the local optima problem during the optimization process and premature convergence. The proposed CGTO is used to tackle these problems and thus boost the performance of the standard GTO. The performance of the proposed CGTO is tested and evaluated to find optimal solutions for global optimization and multilevel thresholding optimization problems. Three chaotic maps are adopted and evaluated. These maps are Circle, Gauss, and Tent chaotic maps. The experimental results revealed that the proposed CGTO is superior compared with other swarm optimization algorithms. Moreover, the results are validated quantitatively and qualitatively for fundus images. The simulation results showed that the proposed can find dominant regions compared with the original GTO.

The training of the feed forward neural network (FFNN) can be formulated as an optimization problem. In this paper, we present a new Harris hawk optimization algorithm (HHO) to minimize the mean square error (MSE). To balance between the global and local search of the traditional HHO algorithm, we invoke the chaotic map into it. The proposed algorithm is named the Chaotic Harris Hawks Optimization (CHHO) algorithm. We apply the CHHO for training the feed-forward neural network (FFNN). To verify the efficiency of the CHHO algorithm, we test it on five classification datasets and compare it against eight meta-heuristics algorithms in the literature. The experimental results show that the proposed CHHO algorithm has the best overall performance and has more outstanding performance than other meta-heuristic algorithms in terms of performance metrics.

The effects of freeplay and multi-segmented nonlinearities in the pitch degree of freedom on the dynamical responses of a two-degree-of-freedom piezoaeroelastic energy harvesting system are investigated. The nonlinear governing equations of the considered piezoaeroelastic energy harvesting system are derived along with the use of the unsteady representation based on the Duhammel formulation to model the aerodynamic loads. The nonlinear piezoaeroelastic response is carried out in the presence of freeplay and multi-segmented nonlinearities before and after the linear onset of flutter. Such nonlinearities can be introduced to piezoaeroelastic energy harvesters for performance enhancement through the possible existence of sudden jumps and chaotic responses due to the grazing bifurcation. It is shown that the existence of discontinuous effects results in the possibility of harvesting energy at lower speeds than the linear onset speed of instability. Additionally, the increase of the strength of the multi-segmented nonlinearities leads to the presence of aperiodic responses with the presence of several bifurcations limiting the system’s dynamics at low pitch angles.

In this research, advanced technology is used to monitoring chaotic time-series signals. The combination of autoregressive with adaptive network-fuzzy algorithms is suggested for chaotic signal prediction. The autoregressive prediction algorithm is recommended for chaotic time-series prediction. This technique is linear, and the modeling prediction accuracy has a limitation. To reduce the root means square (RMS) error of prediction, the order of autoregressive prediction should be increased which is caused to increase the number of parameters and nonlinearity as well. Thus, the combination of autoregressive prediction with an adaptive network-fuzzy algorithm is suggested to reduce the prediction error in chaotic time-series signals. To test the power of the proposed prediction algorithm, the 2nd order proposed method is compared with the 2nd order and 6th order of AR technique, and the RMS error in these three algorithms are 0.0967, 0.4953, and 0.3159, respectively. So far and compared to the classical autoregressive method, the proposed prediction model is efficient for chaotic time-series signals.

In this paper, we introduce an adaptive sliding mode control (SMC) to deal with a finite-time convergence problem of a multiple inputs and multiple outputs (MIMO) system in the presence of uncertainties and/or external disturbances. The proposed controller is constructed by using the super-twisting sliding mode technique to eliminate a negative effect of the chattering phenomenon on the system. Additionally, an adaptive law is designed to guarantee a rapid adaptation and robust stability of the closed-loop system even without knowledge of the upper bound conditions of uncertainties/perturbations. The stability of a control system is investigated by the Lyapunov theorem. A great efficiency of the proposed algorithm is proven through a numerical simulation of a chaotic system.

With the increasing application of video, security problems caused by video leakage also become more prominent. Due to the randomness, ergodicity, determinism, and sensitivity to initial conditions of chaotic systems, it is possible to generate a large number of uncorrelated pseudo-random chaotic sequences, so chaotic encryption has become an important research direction for video security. This chapter improves on the basis of chaotic encryption algorithm based on logistic mapping and proposes an optimized chaotic encryption algorithm. The algorithm introduces tent mapping, which complements the logistic mapping to achieve superiority, thereby obtaining better performance than the two algorithms. On this basis, a prototype system for real-time video transmission was designed and implemented to verify the feasibility of the encryption algorithm.

In the work, a study of the characteristics of natural visibility graphs was carried out. The data for constructing the graphs were model chaotic realizations and realizations of RR intervals in different cardiac diseases. For visibility graphs, adjacency matrices were built, on the ba-sis of which the following characteristics were calculated: density, integral clustering coefficient, integral centrality, average eccentricity, etc. The results showed that differences in quantitative characteristics of graphs for different types of time series are significant. Quantitative characteristics of visibility graphs can be used as features for classify-ing time series using machine learning methods.

The last few years have plunged us at high speed into a new means of communication, namely the Internet, which has set a new trend for the next millennium. So, the rapid growth of online applications reflects the speed with which most countries can develop. An essential aspect of online communication is related to the trust of users and is a very necessary element to ensure the success of an online application. One of the main elements underlying this trust is the remote authentication of the user through its biometric features while of course protecting these features in different storage media. In this paper, we propose a new palmprint/palm-vein recognition framework based on a hand-craft image feature learning method is suggested. Furthermore, to increase the anti-spoof capability of the system, an effective biometric templates protection method based on chaotic systems was proposed. Experimental results have shown that high accuracy can be obtained with a very high level of template protection, which implies that the proposed cancelable biometric system can operate in highly secure applications.

A four-degree-of-freedom mechanical system composed of two weakly coupled modules with external forcing is modelled and numerically investigated. Each module contains internal vibro-impact oscillator. The mathematical model is built based on the compliant impacts with realistic nonlinear stiffness and damping characteristics. The model of resistance of motion of the modules assumes conditions possible in real linear rolling bearings. There are presented exemplary bifurcation properties of the system including periodic and chaotic dynamics of the system.

The problem of the reconstruction of a multivariable function whose values with chaotic errors are given at a finite number of points is considered in the paper. The problems of this kind arise when solving applied problems in various fields of research, including physics, engineering, economics, etc. We propose a new approach for solving this problem with the help of a metric analysis. The paper gives numerical two examples of the solution of the problem of the reconstruction of multivariable function, demonstrating the effectiveness of the proposed scheme. In the first example, the results of estimating the exact value of the function at the points where the values of the function with errors are known, in the second example, the results of reconstructing the physical characteristics of the core of a nuclear reactor are presented.

Vehicle density estimation has an important role in intelligent traffic systems. As of now, most established studies only focused on areas where people mainly travel by four-wheeled vehicles rather than motorbikes. However, in some countries such as Vietnam where motorbikes are the majority, traffic scenarios will pose different issues. Motorbikes are intrinsically more flexible so they can cause cluttered and chaotic visual. As a result, traffic video data captured in such environment is more challenging to existing systems. In this work, we performed an empirical survey on a set of vision-based counting methods covering a wide range of models and techniques. To our knowledge, there has not been many works dedicated to tackle this problem. Based on our experimental results, some of the top performers is ready to be used in real systems due to their robustness.

The dynamic complexity of the living state as evident on viewing cells microscopically defies immediate comprehension. Experiment and theory now allows us to approach some of the problems of this organized complexity (and one that embraces inherent functional purpose) characterizing the phenomenon of life. In an irreducible whole, and in a not so far from equilibrium open system, for the most part it works on a low-duty cycle. We can distinguish many types of time-dependent behaviour: e.g., oscillatory, rhythmic, clock-like timekeeping (and synchronized), deterministically chaotic, and self-similar or fractal), all simultaneously proceeding on many time scales. Self-synchronized continuous cultures of yeast represent, until now, the most characterized example of in vivo elucidation of time structure. The predominantly oscillatory behavior of network components becomes evident, with spontaneously synchronized cellular respiration cycles between discrete periods of increased oxygen consumption (oxidative phase) and decreased oxygen consumption (reductive phase). This temperature-compensated ultradian ‘clock’ provides coordination, linking temporally partitioned functions by direct feedback loops between the energetic and redox state of the cell and its growing ultrastructure. This model system represents a basic framework is proposed as a universal cellular principle whereby ultradian rhythms are the synchronizing signatures that organize the coherence of the living state. The current challenge is to devise ever increasingly powerful, but non-invasive (or minimally perturbing) techniques to investigate the living organism.

The emergence of chaos in a quantum system is of integral importance to the understanding of the quantum-classical transition, and yet many aspects of it are unexplored. This is in part due to the scarcity of experiments that can probe the dynamics of quantum-chaotic systems in a time-resolved manner, because the subtle chaotic dynamics are often masked by other prominent effects such as decoherence.

Proposition of new methods for the study of complex systems and chaos is working on the frontiers of knowledge and hence it calls for philosophical contemplation besides well-set empirical researches. Complex system studies still suffer methodological paucity. Hence, the research goal is proposing a new generic methodology in the domain of chaotic and complex systems’ behavior study. The research in the theorization section discusses the epistemology of the proposed method (philosophic belief, logical justification and mathematical warrant) for pattern formation in chaos and complexity. Later, in the section on how to use the method in practice, the procedures of finding recursions and their fractalization are illustrated for organizational power. Our discussions deepen generic understanding of the connections between chaos/complexity and fractalization and thus the potential disciplines for the implementation of the introduced method could be a large spectrum embracing the humanities to the natural sciences.

Transmission and processing of information in the brain are highly complicated processes that are defined by a lot of non-trivial interconnections of structural elements of neural networks. To shed light on peculiarities of such communication, a lot of mathematical models were introduced and computer simulations were carried out. In this work, a model describing the impact of neural activity on changes in brain extracellular matrix (ECM) molecules concentration was considered. It was assumed that the rate of neural activity is periodically changed. For this case, various regular and chaotic modes in dynamics of ECM-molecules concentration were observed. The role of the amplitude and frequency of the periodically varying neuronal firing rate in transitions between various dynamical modes in the ECM-model were examined. Bifurcational mechanisms for chaotic oscillations appearance were demonstrated.

Since the beginning of the 90ies, chaos has been viewed as a powerful tool for several applications. The complexity of the chaotic pattern is promising for the generation of true random numbers. Because of its unpredictability, chaos also shed the light on secure communications, especially in the optoelectronics field. Several methods of encryption were studied with a particular focus on the chaos being the carrier among which the message is hidden. This method relies on synchronization (or anti-synchronization) of chaos and implies one emitter and one receiver. This technique is well mastered when it comes to fiber optics, with the latest efforts achieving high data rates and low bit-error rates (BER). At the opposite, there are only a few theoretical reports for free-space chaos synchronization and communications, and experiments are never mentioned, though they could be very relevant.

Mathematical modeling and analysis of system behavior are at the core of the development of high-tech system. Modern engineering sciences, to include naval engineering, call for more sophisticated mathematical and statistical methods in order to efficiently understand and resolve the increasing complexity of the tasks at hand. There is a need of a deeper understanding of the mathematical tools most needed to proper support naval research. Indeed, invariably, in the press to “get on with it,” highly complex issues are addressed by engineers using existing tools and knowledge to produce approximate methods and results that have not been rigorously substantiated and are thus subject to unintended consequences, most importantly, breakage/failure in unpredictable circumstances. Nowhere is this situation more chaotic than in the field of information extraction, signal and image processing, acoustics, and nonlinear vibration analysis where a “witches’ brew” of inadequately posed concepts, questionable/unrecognized assumptions, approximations of convenience, and heuristically defined metrics combined to yield what could be reasonably described as bordering on “technological sorcery.”

Disaster response and search and rescue missions are among the most difficult missions in which an autonomous robot can be deployed. These require a robot to autonomously navigate chaotic, unstructured indoor and outdoor environments. However popular state estimation and mapping methods using vision and lidar are severely handicapped by fog, smoke, or other airborne particulates; conditions common in disaster scenarios. This work presents radar-based methods for state estimation and mapping that are not affected by smoke and fog. We demonstrate the performance of these methods are comparable to other popular methods in favorable conditions. We also demonstrate visual and lidar-based methods degrade quickly in fog, while our methods do not.

This chapter will start with a simple theoretical model to introduce chaos in a global manner. Fluctuations observed since the early days of lasers will then be related to chaotic fluctuations and techniques to assess the chaotic nature of the observed fluctuations will be described. As recent experiments in QCLs have shown the basic components of chaotic dynamics, these results will highlight the diversity of such non-linear phenomenon. Parameters having a strong influence on the observed dynamics will be explored, and more precisely the temperature, the carrier-to-photon lifetime and the level of injection current. We will see that QCLs are more prone to chaos at low temperature and that a quasi-continuous bias is more appropriate to generate complex dynamics. In order to explore all the degrees of freedom offered by the external feedback configuration, the current bias will be modulated and this will lead to a synchronization of the chaotic fluctuations with the modulation wave. The evolution of the frequency and phase-shift of the spiking pattern will be reviewed and will give first indications about the use of chaos for communication purposes. The end of the chapter will be dedicated to two other reinjection methods, namely phase-conjugate feedback and cross-polarization reinjection. The latter methods is able to trigger all-optical square-wave patterns whose characteristics will be analyzed, albeit not yet understood with a numerical analysis.

Construction of device for fuzzy chaos signal generation is valuable problem in the field of communication and practical application. Problems of communication can be modeled by different types of differential equations. In this paper is used fuzzy differential equation for modelling the chaotic dynamics of an electronic circuit. Our aim of this study is to design an electronic circuit for generation of chaotic fuzzy signals.

In a cloud platform, task scheduling of IoT applications is tedious because of the non-homogenous features of the IoT. An effective scheduling algorithm for IoT applications is essential for minimizing computation time. This chapter introduces an optimization algorithm for the identification of an optimum solution of task scheduling for IoT applications in a different multiprocessor cloud platform. The task-scheduling algorithm based on the fitness-scaling chaotic genetic ant colony (FSCGAC) algorithm is utilized in this chapter. During the simulation, a series of experiments occur and show competitive performance over the other methods in a significant way.

In this work, we suggest a novel algorithm for construction of large chaotic confusion and diffusion S-boxes for gray scale and color image scrambling. The modification of the value of a pixel by these S-boxes, depend not only on its gray level intensity but also on its position in the original image. In order to raise the impact of avalanche effect and to make our method robust against differential attack, we have developed a strong link between the ciphered pixel and the next original. The simulations carried out by our system on a large number of randomly selected images show the effectiveness of this crypto system against any known attack.

Big data is a concept that is more on the agenda with increasing data sources. One of the most important features of big data is that there is a structure that continues to flow continuously (streaming). It is not that the input of the same data is continuous. Big data is generated by the accumulation of variable data in a non-stop/continuous system. From this point of view, we cannot say that big data is in a regular structure. The chaotic structure should be examined with chaotic methods. The concept of creativity also has a chaotic structure. Neural networks formed by perceptions in the brain are stimulated by some other perceptions. The fact that neurons stimulate different networks with unexpected reactions creates ideas that may seem strange in our minds. If this formation is a new idea for us and others, this is called the creative idea. This unexpected behavior of the brain is related to the proximity of nerve cells and their transmission. The transmitting signals operate in different ways. Formal differences are also effective in signal formation. Neurons that are actively activated outside the neurons that are active in parallel to the perceptions are, in one aspect, similar to big data. This study aims to explain how to create brain creativity by using big data and artificial intelligence through the equivalence of chaotic structures in the informatics world. If we let two models of two different types of interdependencies depend on time and examine the states of change according to each other, the strange attractors to be determined by artificial intelligence will be the creative ideas that will be presented to the insurgency.

This chapter consists of four sections. Section 5.1 reiterates with the Hopf bifurcation theorem. Section 5.2 provides two specific systems that generate chaotic motions by means of numerical simulations. Section 5.3 explains Shilnikov’s theorem. Finally, Sect. 5.4 treats the emergence of chaos in the nonlinear system of the delay-differential equations. To link the chaos theory to economic modelling, at the end of each section, we include simple examples of applications to economics.

Epileptic seizures constitute an important group of neurological disorders in brain that affect many people globally each year. Complexity of EEG signals due to their high-dimensional nature, as well as artifacts in data due to equipment flaws, pose significant challenges to physicians in diagnosing epileptic seizures directly and manually from EEG signals. In this paper, a method is proposed to combine signal processing and machine learning for diagnosing epileptic seizures and tested on the Bonn University database. We used Tunable Q-Factor wavelet transform (TQWT) method to transform signals. Subsequently, various statistical properties, frequency, chaotic and fractional were extracted from the TQWT sub-bands. Subset selection techniques were used in order to reduce the features of the methods used and their results were compared. Finally, a SVM method with different kernels were tested, where the empirical results show the high efficiency of the proposed method in the diagnosis of epileptic seizures.

This work deals with the contributions Machine Learning techniques can bring into the coffee growing conglomerate, committees and other points in the production and marketing chain involved in the dynamics of this commodity. It is well known that the different variables that interact with prices both nationally and internationally have a direct, dramatic affect on the sector under study. In this work, we summarize an extensive review of the coffee price dynamics and the forecast techniques used in this eld. In addition, the internal coffee price in Colombia has been modeled using a long short-term memory (LSTM) recurrent neural network that was chosen as the one of better performance out of three original models. The archetype that evidenced a pertinent superiority of fitness within the parameters specified for this type of model is composed of a linear self-regressive component, plus a multi-layer perceptron-type artificial neural network with twenty (40) LSTM cells neurons in the hidden layer. This epitome captures the chaotic coffee price dynamics. The normalized residuals of the model are uncorrelated and homoscedastic and follow a normal distribution. The results indicate that the current price depends on the prices that occurred in the last four (4) years. This tool can be used to help the coffee growing community to better design alternatives to overcome difficulties with the price of the grain, and this makes it a Logistics solution for them.

The randomness of the traditional measurement matrix in compressed sensing is too strong to be implemented on hardware, and when compressed sensing is used for image encryption, the measurement matrix transmitted as a key will consume time and storage space. Combined with the sensitivity of the chaotic system to the initial value, this paper uses Logistic-Chebyshev chaotic map to obtain random sequences with fewer parameters and construct measurement matrix. To test the measurement performance of the chaotic matrix, compare it with the Gaussian measurement matrix and the Bernoulli measurement matrix in the same compression encryption scheme. Pixel scrambling operation is carried out on the compressed image to complete the final encryption step, and the encrypted image is obtained. The reconstruction algorithm adopts the orthogonal matching tracking method to restore the image. The experimental simulation results show that the chaotic matrix has more advantages than the other two random matrices in image quality, and the encryption and decryption time is shorter.