A study of fluxons propagating in annular josephson junctions

In this research we looked at how fluxons propagate in an annular Josephson junction containing a microshort. We studied this from a theoretical stance and looked at how a single fluxon based on the sine-Grodon soliton equation propagates in this type of junction. It has been seen from a variety of studies that fluxons have many applications through the use of Josephson junctions. The aim of this thesis was to see whether a fluxon will show new properties whilst coming into contact with a microshort located in the junction. We also explored the different geometries a Josephson junction can have and whether that would show the fluxon to present new phenomena. We will also examine point particle systems. With this in mind we took a keen interest in how the interaction between two of these particles in a double well potential would present itself and whether a relationship would become apparent. Alongside the point particle system we modelled fluxons in a double well potential and comment on the similarities with the point particle system. With the aid of the computer programmes Mathematica and COMSOL Multiphysics we were able to compute these different theoretical models and present the work in a logical order with a progression from a single point particle in a double well potential to a fluxon in a heart-shaped Josephson junction. We have looked at current theories and ideas present in this area of condensed matter physics and have explained these in the subsequent thesis

On the connection between the Nekhoroshev theorem and Arnold Diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschl\'{e} et al. (2000). A resonant normal form is constructed by a computer program and the size of its remainder $||R_{opt}||$ at the optimal order of normalization is calculated as a function of the small parameter $\epsilon$. We find that the diffusion coefficient scales as $D\propto||R_{opt}||^3$, while the size of the optimal remainder scales as $||R_{opt}|| \propto\exp(1/\epsilon^{0.21})$ in the range $10^{-4}\leq\epsilon \leq 10^{-2}$. A comparison is made with the numerical results of Lega et al. (2003) in the same model.Comment: Accepted in Celestial Mechanics and Dynamical Astronom

A preliminary investigation into the effects of nonlinear response modification within coupled oscillators

This thesis provides an account of an investigation into possible dynamic interactions between two coupled nonlinear sub-systems, each possessing opposing nonlinear overhang characteristics in the frequency domain in terms of positive and negative cubic stiffnesses. This system is a two degree-of-freedom Duffing oscillator coupled in series in which certain nonlinear effects can be advantageously neutralised under specific conditions. This theoretical vehicle has been used as a preliminary methodology for understanding the interactive behaviour within typical industrial ultrasonic cutting components. Ultrasonic energy is generated within a piezoelectric exciter, which is inherently nonlinear, and which is coupled to a bar-horn or block-horn to one, or more, material cutting blades, for example. The horn/blade configurations are also nonlinear, and within the whole system there are response features which are strongly reminiscent of positive and negative cubic stiffness effects. The two degree-of-freedom model is analysed and it is shown that a practically useful mitigating effect on the overall nonlinear response of the system can be created under certain conditions when one of the cubic stiffnesses is varied. It has also bfeen shown experimentally that coupling of ultrasonic components with different nonlinear characteristics can strongly influence the performance of the system and that the general behaviour of the hypothetical theoretical model is indeed borne out in practice

Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode

We study four problems in the dynamics of a body moving about a fixed point, providing a non-complex, analytical solution for all of them. For the first two, we will work on the motion first integrals. For the symmetrical heavy body, that is the Lagrange-Poisson case, we compute the second and third Euler angles in explicit and real forms by means of multiple hypergeometric functions (Lauricella, functions). Releasing the weight load but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle completing some previous treatments of the Euler-Poinsot case. Integrating then the relevant differential equation, we reach the finite polar equation of a special trajectory named the {\it herpolhode}. In the last problem we keep the symmetry of the first problem, but without the weight, and take into account a viscous dissipation. The approach of first integrals is no longer practicable in this situation and the Euler equations are faced directly leading to dumped goniometric functions obtained as particular occurrences of Bessel functions of order $-1/2$.Comment: This is a pre-print of an article published in Celestial Mechanics and Dynamical Astronomy. The final authenticated version is available online at: DOI: 10.1007/s10569-018-9837-

On large deformations of elastic circular arcs

Das Thema dieser Arbeit kommt von einem praktischen Greifersystem, dessen vereinfachter wesentlicher Teil aus zwei eingespannt-freien elastischen kreisförmigen Stäben besteht. Die Bewegung dieses Systems wird durch die Deformationen der elastischen kreisförmigen Stäbe unter der am freien Ende angreifenden Kraft erzeugt. Die Bögen können von jeder möglichen Form sein, von der Geraden bis zum gesamten geschlitzten Ring, und ihre Deformationen können beliebig groß sein. Nach Aufstellung eines mathematischen Modells in Form eines Randwertproblems mit drei Parametern (Kraft, Bogengeometrie) für eine Pendelgleichung wurden Vielfachheit und Stabilität der Lösungen der Pendelgleichung und der entsprechenden Stabkonfigurationen untersucht. Dazu wurde eine leistungsfähige Methode, die "Mannigfaltigkeitsmethode", entwickelt, die auf der Diskussion der Phasenkurven der Pendelgleichung basiert. Mit dieser Methode, gekoppelt mit numerischen Rechnungen, wurden die Bifurkationsdiagramme der Pendelgleichung für verschiedene Kombinationen der Parameter erhalten. Die Bifurkationsdiagramme zeigten Vielfachheit und die Änderungstendenz der Lösungen der Pendelgleichung und der Konfigurationen der verformten Bögen an. Die Bifurkationsdiagramme zeigen für unser Modell turningpoint-, pitchfork- und X-Bifurkationen sowie Hysteresis. Die Kraftparameterebene wurde nach der Zahl der Lösungen in Gebiete unterteilt. Die theoretische Untersuchung des Halbringes mittels elliptischer Integrale wurde durchgeführt, und das Resultat zeigt eine genaue Übereinstimmung mit den Ergebnissen der Mannifaltigkeitsmethode. Für die Stabilitätsuntersuchung versagen klassische Methoden, da das Randwertproblem für die Pendelgleichung keine trivialen Lösungen besitzt. Deshalb wurde die Pendelgleichung in eine parabolische partielle Differentialgleichung eingebettet, die Liapunovstabilität der stationären Lösungen definiert die "P-Stabilität" der Konfiguration des elastischen Stabes. Stabilitätsaussagen werden mit Hilfe der Methode der ersten Näherung und unter Benutzung der Bifurkationsfunktion gewonnen. Zum ursprünglichen praktischen Problem "Feder und Greifer" wurden als Ergebnisse Federcharakteristiken, insbesondere Kraft-Verschiebungs-Kennlinien für kreisbogenförmige Federn gefunden. Für Greifer, die aus zwei symmetrisch angeordneten elastischen Kreisbögen bestehen, wurden Zusammenhänge von Öffnungsweite, Haftreibungskoeffzient und Haltekraft gefunden. Die Methoden, die in dieser Arbeit eingeführt werden, besonders die Mannifaltigkeitsmethode und P-Stabilität, können für die Untersuchung von Bifurkation und Stabilität bei allgemeinen gewöhnlichen Dfferentialgleichungen nützlich sein