Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

Complex oscillations including chaotic motions have been identified in off-shore and submerged mooring systems characterized by nonlinear fluid-structure interactions and restoring forces. In this paper, a means of controlling these nonlinear oscillations is addressed. When applied, the controller is able to drive the system to periodic oscillations of arbitrary periodicity. The controller applies a perturbation to the nonlinear system at prescribed time intervals to guide a trajectory towards a stable, periodic oscillatory state. The controller utilizes the pole placement method, a state feedback rule designed to render the system asymptotically stable. An outline of the proposed method is presented and applied to the fluid-structure interaction system and several examples of the controlled system are given. The effects of random noise in the excitation force are also investigated and the subsequent influence on the controller identified. A means of extending the controller design is explored to provide adequate control in the presence of moderate noise levels. Meanwhile, in the presence of over powering noise or system measurements that are not well defined, certain filtering and estimation techniques are investigated for their applicability. In particular, the Iterated Kalman Filter is investigated as a nonlinear state estimator of the nonlinear oscillations in these off-shore compliant structures. It is seen that although the inclusion of the nonlinearities is theoretically problematic, in practice, by applying the estimator in a judicious manner and then implementing the linear controllers outlined above, the system is able to estimate and control the nonlinear systems over a wide area of pseudo-stochastic regimes. / Graduation date: 2006

Nichtlineare multistabile Systeme unter dem Einfluss von Rauschen weisen vielschichtige dynamische Eigenschaften auf. <br /> Ein mittleres Rauschlevel zeitigt ein Springen zwischen den metastabilen Zustaenden. <br /> Dieser “attractor-hopping” Prozess ist gekennzeichnet durch laminare Bewegung in der Naehe von Attraktoren und erratische Bewegung, die sich auf chaotischen Satteln abspielt, welche in die fraktalen Einzugsgebietsgrenzen eingebettet sind. Er hat rauschinduziertes Chaos zur Folge. <br /> Bei der Untersuchung der dissipativen Standardabbildung wurde das Phaenomen der Praeferenz von Attraktoren durch die Wirkung des Rauschens gefunden. Dies bedeutet, dass einige Attraktoren eine groessere Wahrscheinlichkeit erhalten aufzutreten, als dies fuer das rauschfreie System der Fall waere. Bei einer bestimmten Rauschstaerke ist diese Bevorzugung maximal. <br /> Andere Attraktoren werden aufgrund des Rauschens weniger oft angelaufen. Bei einer entsprechend hohen Rauschstaerke werden sie komplett ausgeloescht. <br /> Die Komplexitaet des Sprungprozesses wird fuer das Modell zweier gekoppelter logistischer Abbildungen mit symbolischer Dynamik untersucht. <br /> Bei Variation eines Parameters steigt an einem bestimmten Wert des Parameters die topologische Entropie steil an, die neben der Shannon Entropie als Komplexitaetsmass verwendet wird. Dieser Anstieg wird auf eine neuartige Bifurkation von chaotischen Satteln zurueckgefuehrt, die in einem Verschmelzen zweier Sattel besteht und durch einen “Snap-back”-Repellor vermittelt wird. <br /> Skalierungsgesetze sowohl der Verweilzeit auf einem der zuvor getrennten Teile des Sattels als auch des Wachsens der fraktalen Dimension des entstandenen Sattels beschreiben diese neuartige Bifurkation genauer. <br /> Wenn ein chaotischer Sattel eingebettet in der offenen Umgebung eines Einzugsgebietes eines metastabilen Zustandes liegt, fuehrt das zu einer deutlichen Senkung der Schwelle des rauschinduzierten Tunnelns. <br /> Dies wird anhand der Ikeda-Abbildung, die ein Lasersystem mit einer zeitverzoegerden Interferenz beschreibt, demonstriert. Dieses Resultat wird unter Verwendung der Theorie der Quasipotentiale erzielt. <br /> Sowohl dieser Effekt, die Senkung der Schwelle für rauschinduziertes Tunneln aus einem metastabilen Zustand durch einen chaotischen Sattel, als auch die beiden Skalierungsgesteze sind von experimenteller Relevanz. / Nonlinear multistable systems under the influence of noise exhibit a plethora of interesting dynamical properties. A medium noise level causes hopping between the metastable states. This attractorhopping process is characterized through laminar motion in the vicinity of the attractors and erratic motion taking place on chaotic saddles, which are embedded in the fractal basin boundary. This leads to noise-induced chaos. The investigation of the dissipative standard map showed the phenomenon of preference of attractors through the noise. It means, that some attractors get a larger probability of occurrence than in the noisefree system. For a certain noise level this prefernce achieves a maximum. Other attractors are occur less often. For sufficiently high noise they are completely extinguished. The complexity of the hopping process is examined for a model of two coupled logistic maps employing symbolic dynamics. With the variation of a parameter the topological entropy, which is used together with the Shannon entropy as a measure of complexity, rises sharply at a certain value. This increase is explained by a novel saddle merging bifurcation, which is mediated by a snapback repellor. Scaling laws of the average time spend on one of the formerly disconnected parts and of the fractal dimension of the connected saddle describe this bifurcation in more detail. If a chaotic saddle is embedded in the open neighborhood of the basin of attraction of a metastable state, the required escape energy is lowered. This enhancement of noise-induced escape is demonstrated for the Ikeda map, which models a laser system with time-delayed feedback. The result is gained using the theory of quasipotentials. This effect, as well as the two scaling laws for the saddle merging bifurcation, are of experimental relevance.

In einem klassischen Kontext bedeutet Synchronisierung die Anpassung der Rhythmen von selbst-erregten periodischen Oszillatoren aufgrund ihrer schwachen Wechselwirkung. <br /> Der Begriff der Synchronisierung geht auf den berühmten niederläandischen Wissenschaftler Christiaan Huygens im 17. Jahrhundert zurück, der über seine Beobachtungen mit Pendeluhren berichtete. Wenn zwei solche Uhren auf der selben Unterlage plaziert wurden, schwangen ihre Pendel in perfekter Übereinstimmung. <br /> Mathematisch bedeutet das, daß infolge der Kopplung, die Uhren mit gleichen Frequenzen und engverwandten Phasen zu oszillieren begannen. <br /> Als wahrscheinlich ältester beobachteter nichtlinearer Effekt wurde die Synchronisierung erst nach den Arbeiten von E. V. Appleton und B. Van der Pol gegen 1920 verstanden, die die Synchronisierung in Triodengeneratoren systematisch untersucht haben. Seitdem wurde die Theorie gut entwickelt, und hat viele Anwendungen gefunden. <br /> <br /> Heutzutage weiss man, dass bestimmte, sogar ziemlich einfache, Systeme, ein chaotisches Verhalten ausüben können. Dies bedeutet, dass ihre Rhythmen unregelmäßig sind und nicht durch nur eine einzige Frequenz charakterisiert werden können. <br /> Wie in der Habilitationsarbeit gezeigt wurde, kann man jedoch den Begriff der Phase und damit auch der Synchronisierung auf chaotische Systeme ausweiten. Wegen ihrer sehr schwachen Wechselwirkung treten Beziehungen zwischen den Phasen und den gemittelten Frequenzen auf und führen damit zur Übereinstimmung der immer noch unregelmäßigen Rhythmen. Dieser Effekt, sogenannter Phasensynchronisierung, konnte später in Laborexperimenten anderer wissenschaftlicher Gruppen bestätigt werden. <br /> <br /> Das Verständnis der Synchronisierung unregelmäßiger Oszillatoren erlaubte es uns, wichtige Probleme der Datenanalyse zu untersuchen. <br /> Ein Hauptbeispiel ist das Problem der Identifikation schwacher Wechselwirkungen zwischen Systemen, die nur eine passive Messung erlauben. Diese Situation trifft häufig in lebenden Systemen auf, wo Synchronisierungsphänomene auf jedem Niveau erscheinen - auf der Ebene von Zellen bis hin zu makroskopischen physiologischen Systemen; in normalen Zuständen und auch in Zuständen ernster Pathologie. <br /> Mit unseren Methoden konnten wir eine Anpassung in den Rhythmen von Herz-Kreislauf und Atmungssystem in Menschen feststellen, wobei der Grad ihrer Interaktion mit der Reifung zunimmt. Weiterhin haben wir unsere Algorithmen benutzt, um die Gehirnaktivität von an Parkinson Erkrankten zu analysieren. Die Ergebnisse dieser Kollaboration mit Neurowissenschaftlern zeigen, dass sich verschiedene Gehirnbereiche genau vor Beginn des pathologischen Zitterns synchronisieren. Außerdem gelang es uns, die für das Zittern verantwortliche Gehirnregion zu lokalisieren. / In a classical context, synchronization means adjustment of rhythms of self-sustained periodic oscillators due to their weak interaction. The history of synchronization goes back to the 17th century when the famous Dutch scientist Christiaan Huygens reported on his observation of synchronization of pendulum clocks: when two such clocks were put on a common support, their pendula moved in a perfect agreement. In rigorous terms, it means that due to coupling the clocks started to oscillate with identical frequencies and tightly related phases. Being, probably, the oldest scientifically studied nonlinear effect, synchronization was understood only in 1920-ies when E. V. Appleton and B. Van der Pol systematically - theoretically and experimentally - studied synchronization of triode generators. Since that the theory was well developed and found many applications. <br /> Nowadays it is well-known that certain systems, even rather simple ones, can exhibit chaotic behaviour. It means that their rhythms are irregular, and cannot be characterized only by one frequency. However, as is shown in the Habilitation work, one can extend the notion of phase for systems of this class as well and observe their synchronization, i.e., agreement of their (still irregular!) rhythms: due to very weak interaction there appear relations between the phases and average frequencies. This effect, called phase synchronization, was later confirmed in laboratory experiments of other scientific groups. <br /> Understanding of synchronization of irregular oscillators allowed us to address important problem of data analysis: how to reveal weak interaction between the systems if we cannot influence them, but can only passively observe, measuring some signals. This situation is very often encountered in biology, where synchronization phenomena appear on every level - from cells to macroscopic physiological systems; in normal states as well as in severe pathologies. With our methods we found that cardiovascular and respiratory systems in humans can adjust their rhythms; the strength of their interaction increases with maturation. Next, we used our algorithms to analyse brain activity of Parkinsonian patients. The results of this collaborative work with neuroscientists show that different brain areas synchronize just before the onset of pathological tremor. Morevoever, we succeeded in localization of brain areas responsible for tremor generation.

The first line of defence against wireless attacks in Radio Frequency Identi cation (RFID) systems is authentication of tags and readers. RFID tags are very constrained in terms of power, memory and size of circuit. Therefore, RFID tags are not capable of performing sophisticated cryptographic operations. In this dissertation, we have designed light-weight authentication schemes to securely identify the RFID tags to readers and vice versa. The authentication schemes require simple binary operations and can be readily implemented in resource-constrained Radio Frequency Identi cation (RFID) tags. We provide a formal proof of security based on the di culty of solving the Syndrome Decoding (SD) problem. Authentication veri es the unique identity of an RFID tag making it possible to track a tag across multiple readers. We further protect the identity of RFID tags by a light-weight privacy protecting identifi cation scheme based on the di culty of the Learning Parity with Noise (LPN) complexity assumption. To protect RFID tags authentication against the relay attacks, we have designed a resistance scheme in the analog realm that does not have the practicality issues of existing solutions. Our scheme is based on the chaos-suppression theory and it is robust to inconsistencies, such as noise and parameters mismatch. Furthermore, our solutions are based on asymmetric-key algorithms that better facilitate the distribution of cryptographic keys in large systems. We have provided a secure broadcast encryption protocol to effi ciently distribute cryptographic keys throughout the system with minimal communication overheads. The security of the proposed protocol is formally proven in the adaptive adversary model, which simulates the attacker in the real world.

Coupled shape oscillations and translational motion of an incompressible gas bubble in a liquid container in response to forced vibration is studied numerically. Bond number (Bo) and the ratio of the vibration amplitude to the bubble diameter (A/D) are found to be the governing non-dimensional numbers. Bubble response is studied in both 2D and 3D. Different schemes are used for 2D and 3D simulations. In 2D, the flow solver is coupled to a Volume of Fluid (VOF) algorithm to capture the interface between the two phases while in 3D the interface is captured using a level set algorithm. The oscillation outcome ranges from small amplitude and regular oscillations for small Bo and A/D to large amplitude, nonlinear, and chaotic oscillations for large Bo and A/D. Chaotic behavior occurs due to the coupling between the nonlinear shape oscillations and large amplitude oscillatory translational motion. By further increase of the forcing, the inertia of the liquid results in the formation of a liquid jet which penetrates within the bubble core and pierces the bubble and a toroidal bubble shape is formed. The toroidal bubble shape then goes through large amplitude shape oscillations and smaller bubbles are formed. A summary of the 3D simulations provides a map which shows the bubble oscillation outcome as a function of Bo and A/D. The interaction between two bubbles is studied in 2D as well and the effect of vibration amplitude, frequency and liquid to gas density ratio on the interaction force is investigated.

Coupled shape oscillations and translational motion of an incompressible gas bubble in a liquid container in response to forced vibration is studied numerically. Bond number (Bo) and the ratio of the vibration amplitude to the bubble diameter (A/D) are found to be the governing non-dimensional numbers. Bubble response is studied in both 2D and 3D. Different schemes are used for 2D and 3D simulations. In 2D, the flow solver is coupled to a Volume of Fluid (VOF) algorithm to capture the interface between the two phases while in 3D the interface is captured using a level set algorithm. The oscillation outcome ranges from small amplitude and regular oscillations for small Bo and A/D to large amplitude, nonlinear, and chaotic oscillations for large Bo and A/D. Chaotic behavior occurs due to the coupling between the nonlinear shape oscillations and large amplitude oscillatory translational motion. By further increase of the forcing, the inertia of the liquid results in the formation of a liquid jet which penetrates within the bubble core and pierces the bubble and a toroidal bubble shape is formed. The toroidal bubble shape then goes through large amplitude shape oscillations and smaller bubbles are formed. A summary of the 3D simulations provides a map which shows the bubble oscillation outcome as a function of Bo and A/D. The interaction between two bubbles is studied in 2D as well and the effect of vibration amplitude, frequency and liquid to gas density ratio on the interaction force is investigated.

The first line of defence against wireless attacks in Radio Frequency Identi cation (RFID) systems is authentication of tags and readers. RFID tags are very constrained in terms of power, memory and size of circuit. Therefore, RFID tags are not capable of performing sophisticated cryptographic operations. In this dissertation, we have designed light-weight authentication schemes to securely identify the RFID tags to readers and vice versa. The authentication schemes require simple binary operations and can be readily implemented in resource-constrained Radio Frequency Identi cation (RFID) tags. We provide a formal proof of security based on the di culty of solving the Syndrome Decoding (SD) problem. Authentication veri es the unique identity of an RFID tag making it possible to track a tag across multiple readers. We further protect the identity of RFID tags by a light-weight privacy protecting identifi cation scheme based on the di culty of the Learning Parity with Noise (LPN) complexity assumption. To protect RFID tags authentication against the relay attacks, we have designed a resistance scheme in the analog realm that does not have the practicality issues of existing solutions. Our scheme is based on the chaos-suppression theory and it is robust to inconsistencies, such as noise and parameters mismatch. Furthermore, our solutions are based on asymmetric-key algorithms that better facilitate the distribution of cryptographic keys in large systems. We have provided a secure broadcast encryption protocol to effi ciently distribute cryptographic keys throughout the system with minimal communication overheads. The security of the proposed protocol is formally proven in the adaptive adversary model, which simulates the attacker in the real world.

Chaos theory is well known as one of three revolutions in physical sciences in 20th-century, as one physicist called it: Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurable process; and chaos eliminates the Laplacian fantasy of deterministic predictability". Specially, when chaos synchronization was found in 1991, chaos theory becomes more and more attractive. Chaos has been widely applied to many scientific disciplines: mathematics, programming, microbiology, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics. One of most important engineering applications is secure communication because of the properties of random behaviours and sensitivity to initial conditions of chaos systems. Noise-like dynamical behaviours can be used to mask the original information in symmetric cryptography. Sensitivity to initial conditions and unpredictability make chaotic systems very suitable to construct one-way function in public-key cryptography. In chaos-based secure communication schemes, information signals are masked or modulated (encrypted) by chaotic signals at the transmitter and the resulting encrypted signals are sent to the corresponding receiver across a public channel (unsafe channel). Perfect chaos synchronization is usually expected to recover the original information signals. In other words, the recovery of the information signals requires the receiver's own copy of the chaotic signals which are synchronized with the transmitter ones. Thus, chaos synchronization is the key technique throughout this whole process. Due to the difficulties of generating and synchronizing chaotic systems and the limit of digital computer precision, there exist many challenges in chaos-based secure communication. In this thesis, we try to solve chaos generation and chaos synchronization problems. Starting from designing chaotic and hyperchaotic system by first-order delay differential equation, we present a family of novel cell attractors with multiple positive Lyapunov exponents. Compared with previously reported hyperchaos systems with complex mathematic structure (more than 3 dimensions), our system is relatively simple while its dynamical behaviours are very complicated. We present a systemic parameter control method to adjust the number of positive Lyapunov exponents, which is an index of chaos degree. Furthermore, we develop a delay feedback controller and apply it to Chen system to generate multi-scroll attractors. It can be generalized to Chua system, Lorenz system, Jerk equation, etc. Since chaos synchronization is the critical technique in chaos-based secure communication, we present corresponding impulsive synchronization criteria to guarantee that the receiver can generate the same chaotic signals at the receiver when time delay and uncertainty emerge in the transmission process. Aiming at the weakness of general impulsive synchronization scheme, i.e., there always exists an upper boundary to limit impulsive intervals during the synchronization process, we design a novel synchronization scheme, intermittent impulsive synchronization scheme (IISS). IISS can not only be flexibly applied to the scenario where the control window is restricted but also improve the security of chaos-based secure communication via reducing the control window width and decreasing the redundancy of synchronization signals. Finally, we propose chaos-based public-key cryptography algorithms which can be used to encrypt synchronization signals and guarantee their security across the public channel.

Parallel-plate electrostatic actuators driven by a voltage difference between two electrodes suffer from an operation range limited to 30% of the gap that has significantly restrained their applications in Microelectromechanical systems (MEMS). In this thesis, the travel range of an electrostatic actuator made of a micro-cantilever beam electrode above a fixed electrode is extended quasi-statically to 90% of the capacitor gap by introducing a voltage regulator (controller) circuit designed for low frequency actuation. The developed large-stroke actuator is valuable contribution to applications in optical filters, optical modulators, digital micro-mirrors and micro-probe based memory disk drives. To implement the low-frequency large-stroke actuator, the beam tip velocity is measured by a vibrometer, the corresponding signal is integrated in the regulator circuit to obtain the displacement feedback, which is used to modify the input voltage of the actuator to reach a target location. The voltage regulator reduces the total voltage, and therefore the electrostatic force, once the beam approaches the fixed electrode so that the balance is maintained between the mechanical restoring force and the electrostatic force that enables the actuator to achieve the desired large stroke. A mathematical model is developed for the actuator based on the mode shapes of the cantilever beam using experimentally identified parameters that yields good accuracy in predicting both the open loop and the closed loop responses. The low-frequency actuator also yields superharmonic resonances that are observed here for the first time in electrostatic actuators. The actuator can also be configured either as a bi-stable actuator using a low-frequency controller or as a chaotic resonator using a high-frequency controller. The high-frequency controller yields large and bounded chaotic attractors for a wide range of excitation magnitudes and frequencies making it suitable for sensor applications. Bifurcation diagrams reveal periodic motions, softening behavior, period doubling cascades, one-well and two-well chaos, superharmonic resonances and a reverse period doubling cascade. To verify the observed chaotic oscillations, Lyapunov exponents are calculated and found to be positive. Furthermore, a chaotic resonator with a quadratic controller is designed that not only requires less voltage, but also produces more robust and larger motions. Another metric of chaos, information entropy, is used to verify the chaotic attractors in this case. It is found that the attractors have a common information entropy of 0.732 independent of the excitation amplitude and frequency.