Fractional calculus enables the possibility of using real number powers or complex number powers of the differentiation operator. The fundamental connection between fractional calculus and subordination processes is explored and affords a physical interpretation for a fractional trajectory, that being an average over an ensemble of stochastic trajectories. With an ensemble average perspective, the explanation of the behavior of fractional chaotic systems changes dramatically. Before now what has been interpreted as intrinsic friction is actually a form of non-Markovian dissipation that automatically arises from adopting the fractional calculus, is shown to be a manifestation of decorrelations between trajectories. Nonlinear Langevin equation describes the mean field of a finite size complex network at criticality. Critical phenomena and temporal complexity are two very important issues of modern nonlinear dynamics and the link between them found by the author can significantly improve the understanding behavior of dynamical systems at criticality. The subject of temporal complexity addresses the challenging and especially helpful in addressing fundamental physical science issues beyond the limits of reductionism.

Four aspects of the dynamics of continuous-time dynamical systems are studied in this work. The relationship between the Lyapunov exponents of the original system and the Lyapunov exponents of induced Poincare maps is examined. The behavior of these Poincare maps as discriminators of chaos from noise is explored, and the possible Poissonian statistics generated at rarely visited surfaces are studied.

The focus of this thesis is to theoretically and experimentally investigate two new schemes of synchronizing chaotic attractors using chaotically operating diode resonators. The first method, called synchronization using control, is shown for the first time to experimentally synchronize dynamical systems. This method is an economical scheme which can be viably applied to low dimensional dynamical systems. The other, unidirectional coupling, is a straightforward means of synchronization which can be implemented in fast dynamical systems where timing is critical. Techniques developed in this work are of fundamental importance for future problems regarding high dimensional chaotic dynamical systems or arrays of mutually linked chaotically operating elements.

A geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the standard horseshoe, but where the set of points that escape are now reinjected and contribute to richer dynamics. We show it is observed in the unfolding of a three-dimensional vector field possessing an inclination-flip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic topology to show that for suitable parameters the flow contains a strange attractor. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

In this thesis we will discuss connections between Hamiltonian systems with a periodic kick and planar diffeomorphisms. After a brief overview of Hamiltonian theory we will focus, as an example, on derivations of the Hâenon map that can be obtained by considering kicked Hamiltonian systems. We will conclude with examples of Hâenon maps of interest. / by Zalmond C. Barney. / Thesis (M.S.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

Systems with time delays have a broad range of applications not only in control systems but also in many other disciplines such as mathematical biology, financial economics, etc. The time delays cause more complex behaviours of the systems. It requires more sophisticated analysis due to the infinite dimensional structure of the space spaces. In this thesis we investigate stability properties associated with output functions of delay systems. Our primary target is the equivalent Lyapunov characterization of input-tooutput stability (ios). A main approach used in this work is the Lyapuno Krasovskii functional method. The Lyapunov characterization of the so called output-Lagrange stability is technically the backbone of this work, as it induces a Lyapunov description for all the other output stability properties, in particular for ios. In the study, we consider two types of output functions. The first type is defined in between Banach spaces, whereas the second type is defined between Euclidean spaces. The Lyapunov characterization for the first type of output maps provides equivalence between the stability properties and the existence of the Lyapunov-Krasovskii functionals. On the other hand, as a special case of the first type, the second type output renders flexible Lyapunov descriptions that are more efficient in applications. In the special case when the output variables represent the complete collection of the state variables, our Lyapunov work lead to Lyapunov characterizations of iss, complementing the current iss theory with some novel results. We also aim at understanding how output stability are affected by the initial data and the external signals. Since the output variables are in general not a full collection of the state variables, the overshoots and decay properties may be affected in different ways by the initial data of either the state variables or just only the output variables. Accordingly, there are different ways of defining notions on output stability, making them mathematically precisely. After presenting the definitions, we explore the connections of these notions. Understanding the relation among the notions is not only mathematically necessary, it also provides guidelines in system control and design. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection

Realizamos uma montagem experimental da experiência da torneira gotejante, com desenvolvimento dos sistemas de aquisição de dados e de controle do experimento e os respectivos programas de aquisição e analise de dados. O sistema apresentou uma grande variedade de comportamentos, tais como crises, intermitências, bifurcações (tangentes de Hopf), orbitas semi-periódicas e ciclos-limite. Utilizamos técnicas de dinâmica simbólica com a determinação de orbitas periódicas, e representações na forma de grafos das gramaticas simbólicas associadas a diversos atratores. Foram feitas estimativas e representações gráficas de complexidade de conjunto. Foram comparados os resultados obtidos com diversos tipos de bico de torneira. / We built a dripping faucet experiment. We have developed the data acquisition and control systems, as well as the acquisition and data analysis softwares. The dynamical system showed a great variety of non-linear behavior such as crises, intermittencies, Hopf and tangent bifurcations, quasi-periodic orbits and circle limits. We have employed symbolic dynamics tools for periodic orbits extraction and grammar approximations. The grammars associated to some attractors were represented as directed graphs, with approximations and pictoral representations of their set complexity. The results obtained with different types of neeple faucet were compared.

Realizamos uma montagem experimental da experiência da torneira gotejante, com desenvolvimento dos sistemas de aquisição de dados e de controle do experimento e os respectivos programas de aquisição e analise de dados. O sistema apresentou uma grande variedade de comportamentos, tais como crises, intermitências, bifurcações (tangentes de Hopf), orbitas semi-periódicas e ciclos-limite. Utilizamos técnicas de dinâmica simbólica com a determinação de orbitas periódicas, e representações na forma de grafos das gramaticas simbólicas associadas a diversos atratores. Foram feitas estimativas e representações gráficas de complexidade de conjunto. Foram comparados os resultados obtidos com diversos tipos de bico de torneira. / We built a dripping faucet experiment. We have developed the data acquisition and control systems, as well as the acquisition and data analysis softwares. The dynamical system showed a great variety of non-linear behavior such as crises, intermittencies, Hopf and tangent bifurcations, quasi-periodic orbits and circle limits. We have employed symbolic dynamics tools for periodic orbits extraction and grammar approximations. The grammars associated to some attractors were represented as directed graphs, with approximations and pictoral representations of their set complexity. The results obtained with different types of neeple faucet were compared.

Zhao, Jin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 79-83). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Synchronization of Master-slave Systems --- p.1 / Chapter 1.2 --- Output Regulation --- p.2 / Chapter 1.3 --- Typical Nonlinear Systems --- p.4 / Chapter 1.4 --- Organization --- p.4 / Chapter 2 --- Synchronization of Chua's Circuit and Van der Pol Oscillator via Inter- nal Model Approach --- p.6 / Chapter 2.1 --- Introduction --- p.6 / Chapter 2.2 --- Problem Formulation --- p.8 / Chapter 2.3 --- Preliminaries --- p.10 / Chapter 2.4 --- Solvability of the Problem --- p.13 / Chapter 2.4.1 --- The solution of the regulator equations --- p.14 / Chapter 2.4.2 --- Steady-state generator --- p.15 / Chapter 2.4.3 --- Internal model --- p.19 / Chapter 2.4.4 --- Stabilization --- p.20 / Chapter 2.4.5 --- Simulation --- p.22 / Chapter 2.5 --- Conclusions --- p.27 / Chapter 3 --- Robust Output Regulation of Output Feedback Systems with Nonlinear Exosystems --- p.28 / Chapter 3.1 --- Introduction --- p.28 / Chapter 3.2 --- Assumptions and Preliminaries --- p.29 / Chapter 3.3 --- Solvability of the Synchronization Problem --- p.33 / Chapter 3.4 --- Comparing Two Approaches for Output Regulation --- p.42 / Chapter 3.4.1 --- Differences between the two approaches for the output regulation problem --- p.42 / Chapter 3.4.2 --- Solvability of the regulator equations --- p.43 / Chapter 3.4.3 --- Solvability of stabilization --- p.47 / Chapter 3.5 --- Conclusions --- p.49 / Chapter 4 --- Applications of Robust Regional Synchronization via Output Regulation Techniques --- p.50 / Chapter 4.1 --- Problem Formulation --- p.50 / Chapter 4.2 --- Duffing Oscillator Synchronizes with Chua's Circuit --- p.51 / Chapter 4.2.1 --- Transfer the synchronization problem into the stabilization problem --- p.53 / Chapter 4.2.2 --- Boundedness of Chua's circuit --- p.57 / Chapter 4.2.3 --- Stabilization --- p.59 / Chapter 4.2.4 --- Simulation Results --- p.64 / Chapter 4.3 --- The Chaotic SMIB Power System Synchronizes with Van der Pol Oscillator --- p.64 / Chapter 4.3.1 --- Transfer the synchronization problem into the stabilization problem --- p.68 / Chapter 4.3.2 --- Stabilization --- p.71 / Chapter 4.3.3 --- Simulation Results --- p.74 / Chapter 4.4 --- Conclusions --- p.76 / Chapter 5 --- Conclusions --- p.77 / Bibliography --- p.79

The concern of this thesis was to investigate the nonlinear dynamics inherent in oscillating waterfalls.