Cette thèse étudie des ondes localisées pour certaines classes d'équations différentielles non linéaires décrivant des systèmes mécaniques excitables. Ces systèmes correspondent à une chaîne infinie de blocs reliés par des ressorts et qui glissent sur un surface en présence d'une force de frottement non linéaire dépendant de la vitesse. Nous analysons à la fois le modèle de Burridge-Knopoff (avec des blocs attachés à des ressorts tirés à une vitesse constante) et une chaîne de blocs libres glissant sur un plan incliné sous l'effet de la gravité. Pour une classe de fonctions de frottement non-monotones, ces deux systèmes présentent une réponse de grande amplitude à des perturbations au-dessus d'un certain seuil, ce qui constitue l'une des principales propriétés des systèmes excitables. Cette réponse provoque la propagation d'ondes solitaires ou des fronts, en fonction du modèle et des paramètres. Nous étudions ces ondes localisées numériquement et théoriquement pour une grande gamme de lois de frottement et des régimes de paramètres, ce qui conduit à l'analyse d'équations différentielles non linéaires avec avance et retard. Les phénomènes d'extinction de propagation et d'apparition d'oscillations sont également étudiés pour les ondes progressives. L'introduction d'une fonction de frottement linéaire par morceaux permet de construire explicitement des ondes localisées sous la forme d'intégrales oscillantes et d'analyser certaines de leurs propriétés telles que la forme et la vitesse d'ondes. Une preuve de l'existence d'ondes solitaires est obtenue pour le modèle de Burridge-Knopoff pour un couplage faible. / This thesis analyses localized travelling waves for some classes of nonlinearlattice differential equations describing excitable mechanical systems. Thesesystems correspond to an infinite chain of blocks connected by springs and sliding on a surface in the presence of a nonlinear velocity-dependent friction force. We investigate both the Burridge-Knopoff model (with blocks attached to springs pulled at constant velocity) and a chain of free blocks sliding on an inclined plane under the effect of gravity. For a class of non-monotonic friction functions, both systems display a large response to perturbations above a threshold, one of the main properties of excitable systems. This response induces the propagation of either solitary waves orfronts, depending on the model and parameter regime. We study these localized waves numerically and theoretically for a broad range of friction laws and parameter regimes, which leads to the analysis of nonlinear advance-delay differential equations. Phenomena of propagation failure and oscillations of the travelling wave profile are also investigated. The introduction of a piecewise linear friction function allows one to construct localized waves explicitly in the form of oscillatory integrals and to analyse some of their properties such as shape and wave speed. An existence proof for solitary waves is obtained for the excitable Burridge-Knopoff model in the weak coupling regime.

Stochastic differential equations(SDEs) play an important role in many branches of engineering and science including economics, finance, chemistry, biology, mechanics etc. SDEs (with m-dimensional Wiener process) arising in many applications do not have explicit solutions, which implies the development of effective numerical methods for such systems. For SDEs, one can classify the numerical methods into three classes: fully implicit methods, semi-implicit methods and explicit methods. In order to solve SDEs, the computation of Newton iteration is necessary for the implicit and semi-implicit methods whereas for the explicit methods we do not need such computation. In this thesis the common theme is to construct explicit numerical methods with strong order 1.0 and 1.5 for solving Itˆo SDEs. The five-stage Milstein(FSM)methods, split-step forward Milstein(SSFM)methods and M-stage split-step strong Taylor(M-SSST) methods are constructed for solving SDEs. The FSM, SSFM and M-SSST methods are fully explicit methods. It is proved that the FSM and SSFM methods are convergent with strong order 1.0, and M-SSST methods are convergent with strong order 1.5.Stiffness is a very important issue for the numerical treatment of SDEs, similar to the case of deterministic ordinary differential equations. Stochastic stiffness can lead someone to use smaller step-size for the numerical simulation of the SDEs. However, such issues can be handled using numerical methods with better stability properties. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the FSM and SSFM methods are larger than the Milstein and three-stage Milstein methods. The M-SSST methods possess large mean square stability region as compared to the order 1.5 strong Itˆo-Taylor method. SDE systems simulated with the FSM, SSFM and M-SSST methods show the computational efficiency of the methods. In this work, we also consider the problem of computing numerical solutions for stochastic delay differential equations(SDDEs) of Itˆo form with a constant lag in the argument. The fully explicit methods, the predictor-corrector Euler(PCE)methods, are constructed for solving SDDEs. It is proved that the PCE methods are convergent with strong order γ = ½ in the mean-square sense. The conditions under which the PCE methods are MS-stable and GMS-stable are less restrictive as compared to the conditions for the Euler method.

In diese Dissertation werden lineare retardierte Differentialgleichungen (DDEs) und deren Lösungsoperatoren untersucht. Wir stellen eine neue Methode vor, mit der die Lösungsoperatoren für autonome und nicht-autonome DDEs bestimmt werden. Die neue Methode basiert auf dem Pfadintegralformalismus, der aus der Quantenmechanik und von der Analyse stochastischer Differentialgleichungen bekannt ist. Es zeigt sich, dass die Lösung eines Delaysystems zum Zeitpunkt t durch die Integration aller möglicher Pfade von der Anfangsbedingung bis zur Zeit t gebildet werden kann. Die Pfade bestehen dabei aus verschiedenen Schritten unterschiedlicher Längen und Gewichte. Für skalare autonome DDEs können analytische Ausdrücke des Lösungsoperators in der Literatur gefunden werden, allerdings existieren keine für nicht-autonome oder höherdimensionale DDEs. Mithilfe der neuen Methode werden wir die Lösungsoperatoren der genannten DDEs aufstellen und zusätzlich auf Delaysysteme mit mehreren Delaytermen erweitern. Dabei bestätigen wir unsere Ergebnisse sowohl analytisch wie auch numerisch. Die gewonnenen Lösungsoperatoren verwenden wir anschließend zur Stabilitätsanalyse periodischer Delaysysteme. Es werden zwei neue Verfahren präsentiert, die mithilfe des Lösungsoperators den transformierten Monodromieoperator des Delaysystems nähern und daraus die Stabilität bestimmen können. Beide neue Verfahren sind spektrale Methoden für autonome sowie nicht-autonome Delaysysteme und haben keine Einschränkungen wie bei der bekannten Chebyshev-Kollokationsmethode oder der Chebyshev-Polynomentwicklung. Die beiden bisherigen Verfahren beschränken sich auf Delaysysteme mit rationalem Verhältnis zwischen Periode und Delay. Außerdem werden wir eine bereits bekannte Methode erweitern und zu einer spektralen Methode für periodische nicht-autonome Delaysysteme entwickeln. Wir bestätigen alle drei neue Verfahren numerisch. Damit werden in dieser Dissertation drei neue spektrale Verfahren zur Stabilitätsanalyse periodischer Delaysysteme vorgestellt. / In this thesis linear delay differential equations (DDEs) and its solutions operators are studied. We present a new method to calculate the solution operators for autonomous and non-autonomous DDEs. The new method is related to the path integral formalism, which is known from quantum mechanics and the analysis of stochastic differential equations. It will be shown that the solution of a time delay system at time t can be constructed by integrating over all paths from the initial condition to time t. The paths consist of several steps with different lengths and weights. Analytic expressions for the solution operator for scalar autonomous DDEs can be found in the literature but no results exist for non-autonomous or high dimensional DDEs. With the help of the new method we can calculate the solution operators for such DDEs and for time delay systems with several delay terms. We verify our results analytically and numerically. We use the obtained solution operators for the stability analysis of periodic time delay systems. Two new methods will be presented to approximate the transformed monodromy operator with the help of the solution operator and to get the stability. Both new methods are spectral methods for autonomous and non-autonomous delay systems and have no limitations like the known Chebyshev collocation method or Chebyshev polynomial expansion. Both previously known methods are limited to time delay systems with a rational relation between period and delay. Furthermore we will extend a known method to a spectral method for non-autonomous time delay systems. We verify all three new methods numerically. Hence, in this thesis three new spectral methods for the stability analysis of periodic time delay systems are presented.

The analysis of time-delayed dynamics on networks may help to understand many systems from physics, biology, and engineering, such as coupled laser arrays, gene-regulatory networks and complex ecosystems. Beside the complexity due to the network structure, the analysis is further complicated by the presence of the delays. Delay systems are in general infinite dimensional and thus can display complex dynamics as oscillations and chaos. The mathematical difficulties related to the delays hinders the analysis of delay networks. Thus, little is known yet about basic relations between network structure and delay dynamics. It has been shown that networks without delays can be studied efficiently with the generalized modeling approach, which analyzes the stability of an assumed steady state by a direct parametrization of the Jacobian matrix. In this thesis, I demonstrate the extension of the generalized modeling approach to delay networks and analyze networks of delay-coupled delay oscillators, with delayed auto-catalytic growth on the nodes and delayed transport between nodes. For degree-homogeneous networks (DHONs), in which each node has the same number of links, the bifurcation lines that border the stable areas can be calculated analytically, where the topology of the network is described only by the eigenvalues of the adjacency matrix. For undirected networks, the stability pattern in the parameter space of growth and transport delay is governed by two periodic sets of tongues of instability, which depend on the largest positive and the smallest negative eigenvalue. The direct relation between the eigenvalue and the bifurcation lines allows us to predict stability patterns for networks with certain topological properties. Thus, bipartite networks display a characteristic periodicity of tongues. In order to analyze the stability of degree-heterogeneous networks (DHENs), I apply a numerical sampling method based on Cauchy\'s Argument Principle. The stability patterns of these networks resembles the pattern of DHONs, which is governed by the two periodic sets. For networks with sufficiently many links, one set disappears, and the stability of DHENs can be approximates by the stability of a fully-connected network with the same average degree. However, random DHENs tend to be more stable than DHONs, and DHENs with a broad degree-distribution tend to be more stable than DHENs with a narrow distribution. Thus, such networks are more likely to give rise to amplitude death, i.e. the stabilization of an unstable steady state through diffusive coupling. The stability pattern of DHENs can be qualitatively different than the pattern in DHONs. However, for small growth delays, close to the critical delay of the single node system, the bifurcation lines of all DHENs with the same average degree coincide. This, is particularly interesting, because there the stability depends on a global property of the network, which suggests a diverging interaction length. In summary, the extension of generalized modeling to time-delay networks reveals basic relations between the delay dynamics and the topology. The generality of our model should allow to apply these results to a large class of real-world systems.

In this thesis a new frequency domain approach for the analysis of time delay systems is presented. After linearization of a nonlinear delay differential equation (DDE) with constant distributed delay around a constant or periodic reference solution the so-called Hill-Floquet method can be used for the analysis of the resulting linear DDE. In addition, systems with fast or slowly time-varying delays, systems with variable transport delays originating from a transport with variable velocity, and the corresponding spatially extended systems are presented, which can be also analyzed with the presented method. The newly introduced Hill-Floquet method is based on the Hill’s infinite determinant method and enables the transformation of a system with periodic coefficients to an autonomous system with constant coefficients. This makes the usage of a variety of existing methods for autonomous systems available for the analysis of periodic systems, which implies that the typical calculation of the monodromy matrix for the time evolution of the solution over the principle period is no longer required. In this thesis, the Chebyshev collocation method is used for the analysis of the autonomous systems. Specifically, in this case the periodic part of the solution is expanded in a Fourier series and the exponential behavior of the solution is approximated by the discrete values of the Fourier coefficients at the Chebyshev nodes, whereas in classical spectral or pseudo-spectral methods for the analysis of linear periodic DDEs the complete solution is expanded in terms of basis functions. In the last part of this thesis, new results for three applications with time delay effects are presented, which were analyzed with the presented methods. On the one hand, the occurrence of diffusion-driven instabilities in reaction-diffusion systems with delay is investigated. It is shown that wave instabilities are possible already for single-species reaction diffusion systems with distributed or time-varying delay. On the other hand, the stability of metal cutting vibrations at machine tools is analyzed. In particular, parallel orthogonal turning processes with multiple discrete delays and turning processes with a time-varying delay due to a spindle speed variation are studied. Finally, the stability of the synchronized solution in networks with heterogeneous coupling delays is studied. In particular, the eigenmode expansion for synchronized periodic orbits is derived, which includes an extension of the classical master stability function to networks with heterogeneous coupling delays. Numerical results are shown for a network of Hodgkin-Huxley neurons with two delays in the coupling.:1. Introduction 2. System definition and equivalent systems 3. Analysis of nonlinear time delay systems 4. Analytical solution of linear time delay systems 5. Frequency domain approach 6. Hill-Floquet method 7. Applications 8. Concluding remarks A Appendix / In dieser Dissertation wird ein neues Verfahren zur Analyse von Systemen mit Totzeiten im Frequenzraum vorgestellt. Nach Linearisierung einer nichtlinearen retardierten Differentialgleichung (DDE) mit konstanter verteilter Totzeit um eine konstante oder periodische Referenzlösung kann die sogenannte Hill-Floquet Methode für die Analyse der resultierende linearen DDE angewendet werden. Darüber hinaus werden Systeme mit schnell oder langsam variierender Totzeit, Systeme mit einer variablen Totzeit, resultierend aus einem Transport mit variabler Geschwindigkeit, und entsprechende räumlich ausgedehnte Systeme vorgestellt, welche ebenfalls mit der vorgestellten Methode analysiert werden können. Die neu eingeführte Hill-Floquet Methode basiert auf der Hillschen unendlichen Determinante und ermöglicht die Transformation eines Systems mit periodischen Koeffizienten auf ein autonomes System mit konstanten Koeffizienten. Dadurch können zur Analyse periodischer Systeme auch eine Vielzahl existierender Methoden für autonome Systeme genutzt werden und die Berechnung der Monodromie-Matrix für die Lösung des Systems über eine Periode entfällt. In dieser Arbeit wird zur Analyse des autonomen Systems die Tschebyscheff-Kollokationsmethode verwendet. Im Speziellen wird bei diesem Verfahren der periodische Teil der Lösung in einer Fourierreihe entwickelt und das exponentielle Verhalten durch die Werte der Fourierkoeffizienten an den Tschebyscheff Knoten approximiert, wohingegen bei klassischen spektralen Verfahren die komplette Lösung in bestimmten Basisfunktionen entwickelt wird. Im Anwendungsteil der Arbeit werden neue Ergebnisse für drei Beispielsysteme präsentiert, welche mit den vorgestellten Methoden analysiert wurden. Es wird gezeigt, dass Welleninstabilitäten schon bei Einkomponenten-Reaktionsdiffusionsgleichungen mit verteilter oder variabler Totzeit auftreten können. In einem zweiten Beispiel werden Schwingungen an Werkzeugmaschinen betrachtet, wobei speziell simultane Drehbearbeitungsprozesse und Prozesse mit Drehzahlvariationen genauer untersucht werden. Am Ende wird die Synchronisation in Netzwerken mit heterogenen Totzeiten in den Kopplungstermen untersucht, wobei die Zerlegung in Netzwerk-Eigenmoden für synchrone periodische Orbits hergeleitet wird und konkrete numerische Ergebnisse für ein Netzwerk aus Hodgkin-Huxley Neuronen gezeigt werden.:1. Introduction 2. System definition and equivalent systems 3. Analysis of nonlinear time delay systems 4. Analytical solution of linear time delay systems 5. Frequency domain approach 6. Hill-Floquet method 7. Applications 8. Concluding remarks A Appendix