Control algorithms which exploit the unique properties of chaos can vastly improve the design and performance of many practical and useful systems. The program Perfect Moment is built around such an algorithm. Given two points in the system's state space, it autonomously maps the space, chooses a set of trajectory segments from the maps, uses them to construct a composite path between the points, then causes the system to follow that path. This program is illustrated with two practical examples: the driven single pendulum and its electronic analog, the phase-locked loop. Strange attractor bridges, which alter the reachability of different state space points, can be used to increase the capture range of the circuit.

This paper presents techniques that actively exploit chaotic behavior to accomplish otherwise-impossible control tasks. The state space is mapped by numerical integration at different system parameter values and trajectory segments from several of these maps are automatically combined into a path between the desired system states. A fine-grained search and high computational accuracy are required to locate appropriate trajectory segments, piece them together and cause the system to follow this composite path. The sensitivity of a chaotic system's state-space topology to the parameters of its equations and of its trajectories to the initial conditions make this approach rewarding in spite of its computational demands.

The Digital Orrery has been used to perform an integration of the motion of the outer planets for 845 million years. This integration indicates that the long-term motion of the planet Pluto is chaotic. Nearby trajectories diverge exponentially with an e-folding time of only about 20 million years.

When two or more chaotic systems are coupled, they may exhibit synchronized chaotic oscillations. The synchronization of chaos is usually understood as the regime of chaotic oscillations in which the corresponding variables or coupled systems are equal to each other. This kind of synchronized chaos is most frequently observed in systems specifically designed to be able to produce this behaviour. In this thesis, one particular type of synchronization, called impulsive synchronization, is investigated and applied to low dimensional chaotic, hyperchaotic and spatiotemporal chaotic systems. This synchronization technique requires driving one chaotic system, called response system, by samples of the state variables of the other chaotic system, called drive system, at discrete moments. Equi-Lagrange stability and equi-attractivity in the large property of the synchronization error become our major concerns when discussing the dynamics of synchronization to guarantee the convergence of the error dynamics to zero. Sufficient conditions for equi-Lagrange stability and equi-attractivity in the large are obtained for the different types of chaos-generating systems used. The issue of robustness of synchronized chaotic oscillations with respect to parameter variations and time delay, is also addressed and investigated when dealing with impulsive synchronization of low dimensional chaotic and hyperchaotic systems. Due to the fact that it is impossible to design two identical chaotic systems and that transmission and sampling delays in impulsive synchronization are inevitable, robustness becomes a fundamental issue in the models considered. Therefore it is established, in this thesis, that under relatively large parameter perturbations and bounded delay, impulsive synchronization still shows very desired behaviour. In fact, criteria for robustness of this particular type of synchronization are derived for both cases, especially in the case of time delay, where sufficient conditions for the synchronization error to be equi-attractivity in the large, are derived and an upper bound on the delay terms is also obtained in terms of the other parameters of the systems involved. The theoretical results, described above, regarding impulsive synchronization, are reconfirmed numerically. This is done by analyzing the Lyapunov exponents of the error dynamics and by showing the simulations of the different models discussed in each case. The application of the theory of synchronization, in general, and impulsive synchronization, in particular, to communication security, is also presented in this thesis. A new impulsive cryptosystem, called induced-message cryptosystem, is proposed and its properties are investigated. It was established that this cryptosystem does not require the transmission of the encrypted signal but instead the impulses will carry the information needed for synchronization and for retrieving the message signal. Thus the security of transmission is increased and the time-frame congestion problem, discussed in the literature, is also solved. Several other impulsive cryptosystems are also proposed to accommodate more solutions to several security issues and to illustrate the different properties of impulsive synchronization. Finally, extending the applications of impulsive synchronization to employ spatiotemporal chaotic systems, generated by partial differential equations, is addressed. Several possible models implementing this approach are suggested in this thesis and few questions are raised towards possible future research work in this area.

<p>This thesis reviews the experimental investigation of a non-smooth dynamical system consisting of two pendula; a large pendulum attached to a frame with an impact wall, and a small pendulum, which shares its axis of rotation with the large pendulum and can impact against the large pendulum. The system is forced with a sinusoidal horizontal motion, and due to the nonlinearities present in pendula as well as the discontinuous forcing from impacts, exhibits a wide range of behavior. Periodic, quasi-periodic, and chaotic responses all are possible, hysteresis is present, and grazing bifurcations allow for spontaneous change of behavior and the appearance of chaotic responses without following a traditional route to chaos. This thesis follows from existing non-linear dynamics research on forced pendula, impacting systems (such as a bouncing ball) and doubly impacting systems (ball bouncing on top of a bouncing ball).</p> / Thesis

Diss.--Philosophische Fakultät--Universität Erlangen-Nürnberg, 2002. / Bibliogr. p. 189-193.

Diss.--Philosophie--Philosophischen Fakultät 1 der Universität Zürich, 1998. / Bibliogr. p. 266-270.

Thèse de doctorat : Automatique et traitement du signal : INPL : 2006. / Titre provenant de l'écran-titre. Bibliogr.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI Company.

Understanding how students learn, that is, how they recognize, process, and internalize new information, is vital to any teacher’s success. Although many theories exist in this field, I have selected three strong theories to initiate a discussion that I see as suggestive of a new, cohesive theory that represents a synthesis of all three. For the purposes of this report, I have selected the theories of constructivism and social constructivism from Piaget and Vygotsky, Bronfenbrenner’s Ecological Systems theory, and Chaos theory as the basis for my proposed model. In the report, these three theories are deconstructed, and various components of each are then synthesized to suggest a comprehensive model. It is my intent that my proposed model be helpful to teachers in designing and tailoring instruction for their students. By understanding the relationships and inter-relationships of the child to the various systems that affect him/her, the teacher can better engage all students toward a successful outcome. / text