Chaos Theory and Applications

Chaos, which is found in many dynamical systems, due to the presence of chaos, systems behave erratically. Due to its erratic behavior, the chaotic behavior of the system needs to be controlled. Severe acute respiratory syndrome Coronavirus 2 (Covid-19), which has spread all over the world as a pandemic. Many dynamical systems have been proposed to understand the spreading behaviour of the disease. This paper investigates the chaos in the outbreak of COVID-19 via an epidemic model. Chaos is observed in the proposed SIR model. The controller is designed based on the fractional-order Routh Hurwitz criteria for fractional-order derivatives. The chaotic behaviour of the model is controlled by feedback control techniques, and the stability of the system is discussed.

In this research work, we investigate a dynamical systems under piecewise equations with fractional order derivative (FOD). The respective derivative is taken in piecewise to investigate the sudden or abrupt changes occurs in the dynamics of the proposed model. The respective model represent an infectious disease due to corona virus. We establish a detail analysis about sensitivity of the model. Further, we develop a numerical scheme to simulate the model against various fractional order. The concerned results are displayed graphically also.

The dynamical modeling of romantic relationships is explained with a differential equation system designed to explain the development of love/hate feeling between two people over time. In this study, it was assumed that the individual's emotion was two-component, intimacy and passion, instead of a single-component feeling of love. As a result of this assumption, the relationship dynamics is represented by a four-dimensional system of equations. The possible results of this new 4D model were compared with the results of the classical 2D model and it was seen that they could give very different outputs from each other. In addition, situations that cannot be explained by classical models such as the end of passion in long-term relationships, relationships that turn from friendship to love, and couples reunited after separation are interpreted.

Adaptive oscillators can learn and encode information in dynamic, plastic states. The pendulum has recently been proposed as the base oscillator of an adaptive system. In a mechanical setup, the horizontally forced pendulum adaptive frequency oscillator seeks a resonance condition by modifying the length of the pendulum's rod. This system stores the external forcing frequency when the external amplitude is small, while it can store the resonance frequency, which is affected by the nonlinearity of the pendulum, when the external amplitude is large. Furthermore, for some frequency ranges, the pendulum adaptive frequency oscillator can exhibit chaotic motion when the amplitudes are large. This adaptive oscillator could be used as a smart vibratory energy harvester device, but this chaotic region could degrade its performance by using supplementary energy to modify the rod length. The pendulum adaptive frequency oscillator’s equations of motions are discussed, and a field-programmable analog array is used as an experimental realization of this system as an electronic circuit. Bifurcation diagrams are shown for both the numerical simulations and experiments, while period-3 motion is shown for the numerical simulations. As little work has been done on the stability of adaptive oscillators, the authors believe that this work is the first demonstration of chaos in an adaptive oscillator.

Opinion dynamics in relative agreement models seen as an extension of bounded confidence ones, involve a new agents’ variable usually called opinion uncertainty and have higher level of complexity than that of bounded confidence models. After revising the meaning of the opinion uncertainty variable we conclude that it has to be interpreted as the agent’s opinion toleration, that changes the type of the variable from the social to the psychological one. Since the convergence rates to the stationary states in dynamics of sociological and psychological variables are in general different, we study the effect of agents’ psychology and social environment interaction on the opinion dynamics, using concord and partial antagonism relative agreement model in small-world and scale-free societies. The model considers agents of two psychological types, concord and partial antagonism, that differs it from other relative agreement models. The analysis of opinion dynamics in particular scenarios was used in this work. Simulation results show the importance of this approach, in particular, the effect of small variations in initial conditions on the final state. We found significant mutual influence of opinion and toleration resulting in a variety of statistically stationary states such as quasi consensus, polarization and fragmentation of society into opinion and toleration groups of different configurations. Consensus was found to be rather rare state in a wide range of model parameters, especially in scale-free societies. The model demonstrates different opinion and toleration dynamics in small-world and scale-free societies

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

An effective design procedure has been introduced for implementing the fractional order integrator structures with a modified low pass filters (LPFs) and its functionality is verified by realizing a fractional-order chaotic system. In these applications, the state variables of the fractional-order Sprott’s Jerk system are emulated by these first order LPFs. Since the discrete device based designs have the hard adjustment features and the circuit complexities; the realizations of these LPFs are carried out with the Field Programmable Analog Arrays (FPAAs), sensitively. Hence, the introduced LPF based method has been applied to the fractional order Sprott’s Jerk systems and these fractional-order systems, which are built by the several nonlinear functions, have been implemented with a programmable analog device. In this context, the minimum fractional-orders of the Sprott’s Jerk systems are calculated by considering the stability of the fractional-order nonlinear systems. After that, these systems are simulated by employing the Grünwald-Letnikov (G-L) fractional derivative method by using a common fractional-order. Thus, the stability analyses of the fractional-order Sprott’s Jerk system are supported by the numerical simulation results. After the numerical simulation stage, the design procedures of the FPAA based implementations of the Sprott’s Jerk systems have been dealt with in detail. Finally, thanks to the introduced first-order LPF method, the hardware realizations of the Sprott’s Jerk systems have been achieved successfully with a single FPAA device.

An analysis of discrete-time predator-prey systems is presented in this paper by determining the minimum amount of prey consumed before predators reproduce, as well as by analyzing the system's stability and bifurcation. In order to investigate the local stability of the interior equilibrium point of the proposed model, discrete dynamics system theory is employed first. Moreover, the characteristic equation is analyzed to determine the Neimark-Sacker bifurcation of the system. The normal form and bifurcation theory are used to investigate the NS bifurcation around the interior equilibrium point. Based on its analysis, the system exhibits Neimark-Sacker bifurcation when positive parameters are present and non-negative conditions are met. Develop a feedback control strategy to discover the region of stability of the chaotic behavior. By utilizing the maximum laypanuou exponent, the effect of initial conditions on developed systems is further explored. Finally, a computer simulation illustrates the results of the analysis.

Medical imaging, the process of visual representation of different organs and tissues of the human body, is employed for monitoring the normal as well as abnormal anatomy and physiology of the body. Imaging which can provide healthcare solutions ensuring a regular measurement of various complex diseases plays a critical role in the diagnosis and management of many complex diseases and medical conditions, and the quality of a medical image, which is not a single factor but a composite of contrast, artifacts, distortion, noise, blur, and so forth, depends on several factors such as the characteristics of the equipment, the imaging method in question as well as the imaging variables chosen by the operator. The medical images (ultrasound image, X-rays, CT scans, MRIs, etc.) may lose significant features and become degraded due to the emergence of noise as a result of which the process of improvement pertaining to medical images has become a thought-provoking area of inquiry with challenges related to detecting the speckle noise in the images and finding the applicable solution in a timely manner. The partial differential equations (PDEs), in this sense, can be used extensively in different aspects with regard to image processing ranging from filtering to restoration, segmentation to edge enhancement and detection, denoising in particular, among the other ones. In this research paper, we present a conformable fractional derivative-based anisotropic diffusion model for removing speckle noise in ultrasound images. The proposed model providing to be efficient in reducing noise by preserving the essential image features like edges, corners and other sharp structures for ultrasound images in comparison to the classical anisotropic diffusion model. Furthermore, we aim at proving the viscosity solution of the fractional diffusion model. The finite difference method is used to discretize the fractional diffusion model and classical diffusion models. The peak signal-to-noise ratio (PSNR) is used for the quality of the smooth images. The comparative experimental results corroborate that the proposed, developed and extended mathematical model is capable of denoising and preserving the significant features in ultrasound towards better accuracy, precision and examination within the framework of biomedical imaging and other related medical, clinical, and image-signal related applied as well as computational processes.

In this work, we consider the dynamics of opinion among three parties: two small groups of agents and one very persuasive agent, the indoctrinator. Each party holds a position different from that of the others. In this situation, the opinion space is required to be a circle, on which the agents express their position regarding three different options. Initially, each group supports a unique position, and the indoctrinator tries to convince them to adopt her or his position. The interaction between the agents is in pairs and is modeled through a system of non-linear difference equations. Agents, in both groups, give a high weight to the opinion of the indoctrinator, while they give the same weight to the opinion of their peers. Through several computational experiments, we investigate the times required by the indoctrinator to convince both groups.

The electroencephalogram is a promising tool used to unravel the mysteries of the brain. However, such signals are often disturbed by ocular artifacts caused by eye movements. In this study, Independent Component Analysis and Wavelet Transform based ocular artifact removal method, which does not need reference signals, is proposed to obtain signals free from ocular artifacts. With our proposed method, firstly, the ocular artifact regions in the time domain of the signal are detected. Then the signal is decomposed into its components by independent component analysis and independent components containing artifacts are detected. Wavelet transform is only applied to these components with artifact. Zeroing is applied to the parts of the wavelet coefficients obtained as a result of the wavelet transform corresponding to the ocular artifact regions in the time domain. Finally, the clean signal is obtained by inverse Wavelet transform and inverse Independent Component Analysis methods, respectively. The proposed algorithm is tested on a real data set. The results are given in comparison with the method in which the zeroing is applied to the classical independent components. According to the results, it is seen that most of the signal is not affected by the zeroing and the neural part of the EEG signals is successfully preserved.

Chaotic systems are systems that show sensitive dependence on initial conditions, and an immeasurably small change in initial value causes an immeasurably large change in the future state of the system. Besides, there is no randomness in chaotic systems and they have an order within themselves. Researchers use chaotic systems in many areas such as mixer systems that can make more homogeneous mixtures, encryption systems that can be used with high security, and artificial neural networks by taking the advantage of the order in this disorder. Differential equations in which chaotic systems are expressed mathematically are solved by numerical solution methods such as Heun, Euler, ODE45, RK4, RK5-Butcher and Dormand-Prince in the literature. In this research, Feed Forward Neural Network (FFNN), Layer Recurrent Neural Network (LRNN) and Cascade Forward Backpropogation Neural Network (CFNN) structures were used to model the Rucklidge chaotic system by making use of the MATLAB R2021A program Neural Network (NN) Toolbox. By comparing the results of different activation functions used in the modeling, the ANN structure that can best model the Rucklidge chaotic system has been determined. The training of the compared Artificial Neural Networks (ANNs) was carried out with the values obtained from the Euler numerical solution method, which can get satisfactory and fast results.

In order to account for the asymmetry along the ring, the numerical analysis of a semiconductor ring laser using the basic two-mode model and a parameter mismatch in the backscattering coefficients is presented in this paper. The operation of the laser is discovered to be affected by changing the conservative backscattering parameter for a fixed value of the dissipative backscattering parameter, and the bidirectional regime with alternating oscillation can be suppressed. A good correspondence between the numerical results of this paper and the experimental results of the literature is obtained.

This paper reports on the synchronization proprieties in bidirectional coupled current modulated vertical cavity surface-emitting lasers (CMVCSELs) based on the combined model of Danckaert et al.. Regular pulse packages and chaotic behaviors are found in CMVCSEL during the numerical results. The suitable coupling strength leading to high quality of synchronization is determined by numerical analysis. The consequence of the parameter mismatch and the duration of the synchronization process are also highlighted.

This study examines discrete-time T system. We begin by listing the topological divisions of the system's fixed points. Then, we analytically demonstrate that a discrete T system sits at the foundation of a Neimark Sacker(NS) bifurcation under specific parametric circumstances. With the use of the explicit Flip-NS bifurcation criterion, we establish the flip-NS bifurcation's reality. Center manifold theory is then used to establish the direction of both bifurcations. We do numerical simulations to validate our theoretical findings. Additionally, we employ the $0-1$ test for chaos to demonstrate whether or not chaos exists in the system. In order to stop the system's chaotic trajectory, we ultimately employ a hybrid control method.

In this paper, we estabilsh a fractional-order Leslie-Gower prey-predator-parasite model with time delay. The analysis of stability and Hopf bifurcation of this model are presented. By analyzing the corresponding characteristic equations of each equilibrium points, we investigate the impact of time delay on the stability of the model. Regarding time delay as a bifurcation parameter, we obtain that Hopf bifurcation can occur as time delay passes the critical values. We also shown that the system is globally asymptotically stable at equilibria E4. Lastly, by using numerical simulations, some examples are given to support our theoretical results.

In the human glucose-insulin regulatory system, diverse metabolic issues can arise, including diabetes type I and type II, hyperinsulinemia, hypoglycemia, etc. Therefore, the analysis and characterization of such a biological system is a must. It is well known that mathematical models are an excellent option to study and predict natural phenomena to some extent. On the other hand, fractional-order calculus provides a generalization of derivatives and integrals to arbitrary orders giving us a framework to add memory properties and an extra degree of freedom to the mathematical models to approximate real-world phenomena with higher accuracy. In this work, we introduce a fractional-order version of a mathematical model of the glucose-insulin regulatory system. Using the fractional-order Caputo derivative, we can investigate different concentration rates among insulin, glucose, and healthy beta cells. Additionally, the model incorporates two time-lags to represent the elapsed time in insulin secretion in response to blood glucose level and the delay in glucose drop due to increased insulin concentration. Analytical results of the equilibrium points and their corresponding stability are given. Numerical results, including phase portraits and bifurcation diagrams, reveal that the fractional-order increases the chaotic regions, leading to potential metabolic problems. Vice versa, the system seems to work correctly when the behavior evolves to periodic windows.

A cascade function is designed by combining two seed maps that resultantly has more parameters, high complexity, randomness, and more unpredictable behavior. In the paper, a cascade fractal function, i.e. cascade-PLMS is proposed by considering the phoenix and lambda fractal functions. The constructed cascade-PLMS exhibits the required fractal features such as fractional dimension, self-similar structure, and covering entire phase space by the data sequence in addition to the chaotic properties. Due to the chaotic behavior, the proposed function is utilized to generate a pseudo-random number sequence in both integer and binary format. This is the result of an extreme scalability feature of a fractal function that can be implemented on a large scale. A sequence generator is designed by performing the linear function operation to the real and imaginary part of a cascade-PLMS, cascade-PLJS separately, and the iteration number at which the cascade-PLJS converges to the fixed point. The performance analysis results show that the given method has a large keyspace, fast key generation speed, high key sensitivity, and strong randomness. Therefore, the scheme can be efficiently used further to design a secure cryptosystem with the ability to withstand various attacks.

Cryptocurrencies are new kinds of electronic currencies based on communication technologies. These currencies have attracted the attention of investors. However, cryptocurrencies are very volatile and unpredictable. For investors, it is very difficult to make investment decisions in cryptocurrency market. Therefore, revealing changes in the dynamics of cryptocurrencies are valuable for investors. Bitcoin is the most popular and representative cryptocurrency in cryptocurrency market. In this study how dynamical properties of Bitcoin changed through time is analyzed with recurrence quantification analysis (RQA). RQA is a pattern recognition-based time series analysis method that reveals dynamics of the time series by calculating some metrics called RQA measures. This method has been successfully applied to nonlinear, nonstationary, short and chaotic time series and do not assume a statistical model. RQA can reveal important properties of time series data such as determinism, laminarity, stability, randomness, regularity and complexity. By using sliding window RQA we show that in 2021 RQA measures for Bitcoin prices collapse and Bitcoin becomes more unpredictable, more random, more unstable, more irregular and less complex. Therefore, dynamics and stability of the Bitcoin prices significantly changed in 2021.

This study investigates the power systems that involve various numbers of busbars. To prevent the disturbances and instabilities in the power systems, power system stabilizers and various control methods have been used. A hyperchaotic blackout has been created by using an existing hyperchaotic system. Hyperchaotic voltage collapse and hyperchaotic disturbance have been injected to the test systems. The situations of the various power systems are illustrated under proposed hyperchaotic blackout and noise. The stability analysis of the power system has been executed according to the dynamic features of hyperchaos.

A new chaotic system with sine function is presented in this research work. The proposed system presents some special features such as two wing attractors, forward and reverse periodic doubling bifurcation, and dc offset boosting property which can be used to diagnosis the multistability behaviour in the dynamical system. The proposed system has three nonlinear terms and one sine term. The basic information such as equilibrium points, stability and Lyapunov exponents are discussed in detail. The dynamic analysis conducted using classic tools such as bifurcation diagram and Lyapunov exponent plot to verify the chaotic nature in the proposed system. The theoretical study and the simulation results show that the proposed system has chaotic behaviour itself.

Offset boosting is an important issue for chaos control due to its broadband property and polarity control. There are two main approaches to realize offset boosting. One is resort to parameter introducing where an offset booster realizes attractor boosting. The other one is by the means of periodic function or absolute value function where a specific initial condition can extract out any self-reproduced or doubled attractor with different offset. The former also provides a unique window for observing multistability and the latter gives the direction for constructing desired multistability.

Despite the fact that chaotic systems do not have very complex circuit structures, interest in chaotic systems has increased considerably in recent years due to their interesting dynamic properties. Thanks to the noise-like properties of chaotic oscillators and the ability to mask information signals, great efforts have been made in recent years to develop chaos-based TRNG structures. In this study, a new chaos-based dual entropy core TRNG with high operating frequency and high bit generation rate was realized using 3D Pehlivan-Wei Chaotic Oscillator (PWCO) structure designed utilizing RK-Butcher numerical algorithm on FPGA and ring oscillator structure. In the FPGA-based TRNG model of the system, 32-bit IQ-Math fixed-point number standard is used. The developed model is coded using VHDL. The designed TRNG unit was synthesized for Virtex-7 XC7VX485T-2FFG1761 chip produced by Xilinx. Then, the statistics of the parameters of FPGA chip resource usage and unit clock speed were examined. The data processing time of the TRNG unit was achieved by using the Xilinx ISE Design Tools 14.2 simulation program, with a high bit production rate of 437.043 Mbit/s. In addition, number sequences obtained from FPGA-based TRNG were subjected to the internationally valid statistical NIST 800-22 Test Suite and all the randomness tests of NIST 800-22 Test Suite were successful.

Neural networks and fractional order calculus are powerful tools for system identification through which there exists the capability of approximating nonlinear functions owing to the use of nonlinear activation functions and of processing diverse inputs and outputs as well as the automatic adaptation of synaptic elements through a specified learning algorithm. Fractional-order calculus, concerning the differentiation and integration of non-integer orders, is reliant on fractional-order thinking which allows better understanding of complex and dynamic systems, enhancing the processing and control of complex, chaotic and heterogeneous elements. One of the most characteristic features of biological systems is their different levels of complexity; thus, chaos theory seems to be one of the most applicable areas of life sciences along with nonlinear dynamic and complex systems of living and non-living environment. Biocomplexity, with multiple scales ranging from molecules to cells and organisms, addresses complex structures and behaviors which emerge from nonlinear interactions of active biological agents. This sort of emergent complexity is concerned with the organization of molecules into cellular machinery by that of cells into tissues as well as that of individuals to communities. Healthy systems sustain complexity in their lifetime and are chaotic, so complexity loss or chaos loss results in diseases. Within the mathematics-informed frameworks, fractional-order calculus based Artificial Neural Networks (ANNs) can be employed for accurate understanding of complex biological processes. This approach aims at achieving optimized solutions through the maximization of the model’s accuracy and minimization of computational burden and exhaustive methods. Relying on a transdifferentiable mathematics-informed framework and multifarious integrative methods concerning computational complexity, this study aims at establishing an accurate and robust model based upon integration of fractional-order derivative and ANN for the diagnosis and prediction purposes for cancer cell whose propensity exhibits various transient and dynamic biological properties. The other aim is concerned with showing the significance of computational complexity for obtaining the fractional-order derivative with the least complexity in order that optimized solution could be achieved. The multifarious scheme of the study, by applying fractional-order calculus to optimization methods, the advantageous aspect concerning model accuracy maximization has been demonstrated through the proposed method’s applicability and predictability aspect in various domains manifested by dynamic and nonlinear nature displaying different levels of chaos and complexity.

In this study, a method of cooling curve analysis (CCA) was developed to obtain the solid fraction of the liquid metal as a function of both position and time. Obtaining the solid fraction depending on the location is valuable to see the spatial improvement of solidification and examine the factors affecting it (gravity, heterogeneity, the effect of the mold, etc.). In previous studies using CCA, which is a low-cost method, the solid fraction has been found only depending on time, even if Fourier thermal analysis (FTA) was used. Thus, this method, called semi-Newtonian FTA (SNFTA), is unique in that it uses CCA to calculate the solid fraction as a function of both position and time. After developing the SNFTA method combining Newton and Fourier heat equations, its accuracy was tested in two ways. The method was applied to pure tin and the thermal diffusivity and latent heat values of the tin were then estimated. It was observed that the predicted diffusivity and latent heat values deviated little from their original values. Therefore, it can be said that the local solid fraction functions obtained by this method were reliable.