A study on phase synchronization in large populations of nonlinear dynamical systems is presented in this thesis. Using the well-known Rossler system as a prototypical model, phase synchronization in one oscillator with periodic external forcing and in two-coupled nonidentical oscillators was explored at first. The study was further extended to consider three-dimensional lattices and globally coupled populations of nonidentical oscillators, in which the mathematical formulation that represents phase synchronization in the generalized N-coupled Rossler system was derived and several computer programs that perform numerical simulations were developed. The results show the effects of coupling dimension, coupling strength, population size, and system parameter on phase synchronization of the various Rossler systems, which may be applicable to studying phase synchronization in other nonlinear dynamical systems as well.

Inverse problems, foundational in applied sciences, involve deducing system inputs from specific output observations. These problems find applications in diverse domains such as aerospace engineering, weather prediction, and oceanography. However, their solution often requires complex numerical simulations and substantial computational resources. Modern machine learning based approaches have emerged as an alternative and flexible methodology for solving these types of problems, however their generalization power often comes at the cost of working with large descriptive datasets, a requirement that many applications cannot afford. This thesis proposes and explores the novel Deep Multi-resolution Operator Network (DMON), inspired by the recently developed DeepONet architecture. The DMON model is designed to solve inverse problems related to chaotic non-linear systems with low-resolution data through intelligently utilizing high-resolution data from a similar system. Performance of the DMON model and the proposed selection mechanisms are evaluated on two chaotic systems, a double pendulum and turbulent flow around a cylinder, with improvements observed under idealized scenarios whereby high and low-resolution inputs are manually paired, along with minor improvements when this pairing is conducted through the proposed the latent space comparison selection mechanism. / Master of Science / In everyday life, we often encounter the challenge of determining the cause behind something we observe. For instance, meteorologists infer weather patterns based on limited atmospheric data, while doctors use X-rays and CT scans to reconstruct images representing the insides of our bodies. Solving these so called ``inverse problems'' can be difficult, particularly when the process is chaotic such as the weather, whereby small changes result in much larger ones over time. In this thesis, we propose a novel method using artificial intelligence and high-resolution simulation data to aid in solving these types of problems. Our proposed method is designed to work well even when we only have access to a small amount of information, or the information available isn't very detailed. Because of this there are potential applications of the proposed method across a wide range of fields, particularly those where acquiring detailed information is difficult, expensive, or impossible.

Yes / Fractional-order chaotic systems have more complex dynamics than integer-order chaotic systems. Thus, investigating fractional chaotic systems for the creation of image cryptosystems has been popular recently. In this article, a fractional-order memristor has been developed, tested, numerically analyzed, electronically realized, and digitally implemented. Consequently, a novel simple three-dimensional (3D) fractional-order memristive chaotic system with a single unstable equilibrium point is proposed based on this memristor. This fractional-order memristor is connected in parallel with a parallel capacitor and inductor for constructing the novel fractional-order memristive chaotic system. The system’s nonlinear dynamic characteristics have been studied both analytically and numerically. To demonstrate the chaos behavior in this new system, various methods such as equilibrium points, phase portraits of chaotic attractor, bifurcation diagrams, and Lyapunov exponent are investigated. Furthermore, the proposed fractional-order memristive chaotic system was implemented using a microcontroller (Arduino Due) to demonstrate its digital applicability in real-world applications. Then, in the application field of these systems, based on the chaotic behavior of the memristive model, an encryption approach is applied for grayscale original image encryption. To increase the encryption algorithm pirate anti-attack robustness, every pixel value is included in the secret key. The state variable’s initial conditions, the parameters, and the fractional-order derivative values of the memristive chaotic system are used for contracting the keyspace of that applied cryptosystem. In order to prove the security strength of the employed encryption approach, the cryptanalysis metric tests are shown in detail through histogram analysis, keyspace analysis, key sensitivity, correlation coefficients, entropy analysis, time efficiency analysis, and comparisons with the same fieldwork. Finally, images with different sizes have been encrypted and decrypted, in order to verify the capability of the employed encryption approach for encrypting different sizes of images. The common cryptanalysis metrics values are obtained as keyspace = 2648, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

Creating chaotic advection is the most efficient strategy to achieve mixing in a microscale or in a very viscous fluid, and it has many important applications in microfluidic devices, material processing and so on. In this paper, we present a quantitative long-term theory of resonant mixing in 3-D near-integrable flows. We use the flow in the annulus between two coaxial elliptic counter-rotating cylinders as a demonstrative model. We illustrate that such resonance phenomena as resonance and separatrix crossings accelerate mixing by causing the jumps of adiabatic invariants. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resulting mixing can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss what must be done to accommodate the effects of the boundaries of the chaotic domain. / Mechanical Engineering

This thesis investigates the relationship between semantic change and theory change in science. The study focuses on thirty years of developments in chaos theory. Because of measurement problems associated with certain nonlinear phenomena, the observability of chaotic systems is severely limited. In such cases, ongoing processes of language change may play a greater role in shaping scientific theories than in cases in which the phenomena are more easily observed. This study is interdisciplinary, drawing on theories from linguistics, philosophy, philology, and sociology. Several mechanisms of semantic change are explored in order to discover their possible influence on theory formation. Developments in chaos theory are described in terms of George Lakoff's radial model of conceptual categories. This model describes concepts in terms of (1) a central cluster which acts as a prototypical example, and (2) various non-central extensions from that center. I argue that in an emerging discipline, non-central extensions are made depending on the interests of the community. As Andrew Pickering observed, communities on the research front select a research direction that will intersect with the interests of more established research communities. This thesis explores several examples of historical developments in chaos research showing how conceptual change in science can be described in terms of Lakoff' s radial category model and Pickering's interest model. / Master of Science

A pendulum with its supporting point vibrating in both the x and the y direction is analyzed. Numerical simulation by computer is used to analyze the motion of the pendulum. Chaotic motion of the system is observed. Threshold values for chaos are obtained by simulation. The Lyapunov exponent and the fast Fourier transform ( FFT ) are used as the criteria to determine if the system is chaotic. Two predictive theoretical criteria, the Melnikov criterion and a period-doubling criterion, are then applied to the system. The results obtained by simulation and by theoretical criteria are shown to be in good agreement. A brute-force approach is used to supplement the results. It is found that the motion of this simple driven pendulum will have very complicated behavior. Multiple attractors can be shown to coexist. / Master of Science

Yes / In this paper, a new fractional order chaotic system without equilibrium is proposed, analyti-cally and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigation were used to describe the system dynamical behaviors including, the system equilibria, the chaotic attractors, the bifurcation diagrams and the Lyapunov expo-nents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attrac-tors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive con-trol theory has been developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state varia-bles for the master and slave. Consequently, the update laws of the slave parameters are ob-tained, where the slave parameters are assumed to be uncertain and estimate corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results are obtained by MATLAB and the Arduino Due boards respectively, where a good consistent between the simulation results and the ex-perimental results. indicating that the new fractional order chaotic system is capable of being employed in real-world applications.

Yes / This article presents a novel four-dimensional autonomous fractional-order chaotic system (FOCS) with multi-nonlinearity terms. Several dynamics, such as the chaotic attractors, equilibrium points, fractal dimension, Lyapunov exponent, and bifurcation diagrams of this new FOCS, are studied analytically and numerically. Adaptive control laws are derived based on Lyapunov theory to achieve chaos synchronization between two identical new FOCSs with an uncertain parameter. For these two identical FOCSs, one represents the master and the other is the slave. The uncertain parameter in the slave side was estimated corresponding to the equivalent master parameter. Next, this FOCS and its synchronization were realized by a feasible electronic circuit and tested using Multisim software. In addition, a microcontroller (Arduino Due) was used to implement the sug-gested system and the developed synchronization technique to demonstrate its digital applicability in real-world applications. Furthermore, based on the developed synchronization mechanism, a secure communication scheme was constructed. Finally, the security analysis metric tests were investigated through histograms and spectrograms analysis to confirm the security strength of the employed communication system. Numerical simulations demonstrate the validity and possibility of using this new FOCS in high-level security communication systems. Furthermore, the secure communication system is highly resistant to pirate attacks. A good agreement between simulation and experimental results is obtained, showing that the new FOCS can be used in real-world applications.

Cette thèse traite du domaine de la cryptographie basée sur des dynamiques chaotiques hybrides.Afin de robustifier la transmission sécurisée de données à l'égard de l'attaque à texte-claire connue, ce travail a été particulièrement axée sur deux directions, l'approche statistique, et l'approche automatique.Les principales contributions de ce travail sont organisées dans ces deux directions.Le choix de la variable mesurée et son influence sur l'émetteur d'un message secret et la possibilité de récupérer la dynamique à la réception.Ceci a été étudié dans le contexte des systèmes chaotiques discrets et continus.L'indépendance statistique des variables d'état des systèmes chaotiques est étudié en relation avec la non-corrélation spatiale de ces états.Ainsi une méthode pour cacher le message secret en fonction de l'évolution de l'émetteur chaotique, et ceci avant son inclusion dans cette dynamique, est proposée.La faisabilité d'un système retardée hybride qui est utilisée pour la transmission sécurisée des données est analysée dans une mise en œuvre analogique.Des simulations et les analyses des résultats obtenus sont faits, afin de prouver l'efficacité des études et des méthodes proposées.La thèse est organisée comme suit: le Chapitre I reprend les notions théoriques et les algorithmes utilisés pour atteindre l'objectif de ce travail.Le chapitre II est consacré à l'étude des exposants de Lyapunov.Les systèmes chaotiques utilisés dans le présent document sont ensuite décrits.Le chapitre III présente une étude de certaines propriétés structurales des systèmes du chapitre II.L'étude se concentre sur le calcul des indices d'observabilité et la détermination des hypersurfaces de la singularité d'observabilité.Le chapitre IV analyse l'indépendance statistique dans le contexte des systèmes chaotiques considérés:la taille de la distance d'échantillonnage (combien d'itérations ou de manière équivalente, combien de temps) pour assurer l'indépendance statistique entre les variables extraites des systèmes chaotiques.Un test original pour l'indépendance statistique (le test Badea-Vlad) a été utilisée; la procédure est applicable à tous les types de variables aléatoires continues, même reparties selon une loi de probabilité inconnue au besoin ici.Le chapitre V illustre le point de vue physique. Le temps transitoire correspond au temps passé par le système chaotique dans le bassin d'attraction avant de rejoindre l'attracteur étrange.De même il est important de savoir après combien de temps les points localisés dans une certaine région de l'attracteur étrange devient non-corrélés.Dans le chapitre VI, sachant l'identifiabilité des paramètres des systèmes chaotiques décrits par des équations polynomiales, une amélioration des inclusions du message dans ce type de cryptographie, est proposé.Le message clair est chiffré en utilisant une substitution classique avec boîtes de transposition, avant son inclusion dans l'émetteur chaotique.Les résultats de l'algorithme proposé sont évalués sur le texte et sur l'image.Le chapitre VII pose quelques questions, et essaie de trouver quelques-unes des réponses à ces questions, dans le cadre du schéma hybride.Comme par exemple, est-il possible de récupérer le message secret en utilisant un observateur, lorsque la dynamique qui lui inclut est retardée?La réponse est positive, et cela est montrée dans le cas d'une transmission intégrale de la sortie du système.Il est important de mentionner que ce travail est pluridisciplinaire, allant de la théorie du contrôle aux statistiques en passant par les domaines de l'électronique, de la mathématique et de l'informatique. / This Thesis deals with the domain of cryptography based on hybrid chaotic dynamics.In order to increase the robustness of the security in data transmission with respect to known text attack, this work was particularly focused on two directions: the statistical approach and the automation control.The main contributions of this work are organized in the mentioned two directions.The choice of the measured variable and its influence on the transmitter of plain messages, alongside the possibility to recover the dynamics at the reception.These are studied in the context of discrete and continuous-time chaotic systems.Statistical independence of the state variables of chaotic systems is investigated in relation with the spatial non-correlation of the states.A method of hiding the secret message, depending on the evolution of the chaotic transmitter and prior to its inclusion in this dynamics is proposed.The feasibility of a delayed time hybrid scheme that is used for secure data transmission is shown in an analog implementation.Simulations and analysis of the obtained results are done in order to prove the efficiency of the proposed studies and methods.The Thesis is organized as follows: Chapter I resumes theoretical notions and algorithms used to achieve the goal of this work.Chapter II is dedicated to the study of the Lyapunov exponents. The chaotic systems used in this report are described.Chapter III presents a study of some structural properties of the chaotic systems from Chapter II.The investigation is focused on the calculation of the observability indexes and the determination of the manifolds of observability singularity.Chapter IV analyses the statistical independence in the context of the considered chaotic systems:how large should be the sampling distance (how many iterations or, equivalently, time) to ensure statical independence between variables extracted from the chaotic systems.An original test for statistical independence (the Badea-Vlad test) was used; the procedure is applicable to all kind of continuous random variables, even of unknown probability law as needed here.Chapter V illustrates the physical point of view.The transient time corresponds to the time spent by the chaotic system in the basin of attraction before rejoining the strange attractor.It is also important to know after how long the points localized in a certain region of the strange attractor become uncorrelated.In Chapter VI, knowing the identifiability of the parameters of chaotic systems described by polynomial equations, an improvement of the inclusion of messages in this type of enciphering is proposed.The plain-message is enciphered using classical substitution and transposition boxes, prior to its inclusion in the chaotic transmitter.The results of the proposed algorithm are evaluated on text and image.Chapter VII rises some questions, and tries to find some answers to these questions, in the context of hybrid dynamical schemes.As for example if it is possible to recover the secret message by using an observer, when the dynamics that includes it is time-delayed.The answer is positive and this is shown in the case of a full transmission of the output of the system.It is important to mention that this work is multidisciplinary, starting from control theory and going to the statistical methods through the fields of electronics, mathematics and computing.

We investigate chaotic ionization of highly excited hydrogen atom in crossed electric and magnetic fields (Rydberg atom) and intra-molecular relaxation in planar carbonyl sulfide (OCS) molecule. The underlying theoretical framework of our studies is dynamical systems theory and periodic orbit theory. These theories offer formulae to compute expectation values of observables in chaotic systems with best accuracy available in given circumstances, however they require to have a good control and reliable numerical tools to compute unstable periodic orbits. We have developed such methods of computation and partitioning of the phase space of hydrogen atom in crossed at right angles electric and magnetic fields, represented by a two degree of freedom (dof) Hamiltonian system. We discuss extensions to a 3-dof setting by developing the methodology to compute unstable invariant tori, and applying it to the planar OCS, represented by a 3-dof Hamiltonian. We find such tori important in explaining anomalous relaxation rates in chemical reactions. Their potential application in Transition State Theory is discussed.