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A theoretical famework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equili Drium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a g: ven equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that ihese theorems are stated in terms of the given problem and not in terms of the corresponding vector field.
An existence and uniqueness theory is developed for general nonlinear and nonautonomous differential-algebraic equations (DAEs) by exploiting their underlying differential-geometric structure. A DAE is called regular if there is a unique nonautonomous vector field such that the solutions of the DAE and the solutions of the vector field are in one-to-one correspondence. Sufficient conditions for regularity of a DAE are derived in terms of constrained manifolds. Based on this differential-geometric characterization, existence and uniqueness results are stated for regular DAEs. Furthermore, our not ons are compared with techniques frequently used in the literature such as index and solvability. The results are illustrated in detail by means of a simple circuit example.
The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.
Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this paper we give a geometric interpretation of higherindex DAEs and indicate problems arising in connection with such DAEs by means of several examples.
A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too .
In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).
"Considerons une particule mobile se mouvant aleatoirement sur la droite (ou sur un segment de droite). Supposons qu'il existe une probabilite F(x,y;s,t) bien definie pour que la particule se trouvant a l'instant s dans la position x se trouve a l'instant t (> s) a gauche de y, probabilite independante du mouvement anterieur de la particule...." Mit diesen Worten beginnt eines der berühmtesten mathematischen Manuskripte des letzten Jahrhunderts. Es stammt vom Soldaten Wolfgang Döblin, Sohn des deutschen Schriftstellers Alfred Döblin, und trägt den Titel "Sur l'equation de Kolmogoroff". Seine Veröffentlichung verbindet sich mit einer unglaublichen Geschichte. Wolfgang Döblin, stationiert mit seiner Einheit in den Ardennen im Winter 1939/1940, arbeitete an diesem Manuskript. Er entschloss sich, es als versiegeltes Manuskript an die Academie des Sciences in Paris zu schicken. Aber er kehrte nie aus diesem Krieg zurück. Sein Manuskript blieb 60 Jahre unter Verschluss im Archiv, und wurde erst im Jahre 2000 geöffnet. Wie weit Döblin damit seiner Zeit voraus war, wurde erkannt, nachdem es von Bernard Bru und Marc Yor ausgewertet worden war. Im ersten Satz umschreibt W. Döblin gleichzeitig das Programm des Manuskripts: "Wir betrachten ein bewegliches Teilchen, das sich zufällig auf der Geraden (oder einem Teil davon) bewegt." Er widmet sich damit der Aufgabe, die Fundamente eines Gebiets zu legen, das wir heute als stochastische Analysis bezeichnen.
Many methods have been proposed for the simulation of constrained mechanical systems. The most obvious of these have mild instabilities and drift problems. Consequently, stabilization techniques have been proposed A popular stabilization method is Baumgarte's technique, but the choice of parameters to make it robust has been unclear in practice. Some of the simulation methods that have been proposed and used in computations are reviewed here, from a stability point of view. This involves concepts of differential-algebraic equation (DAE) and ordinary differential equation (ODE) invariants. An explanation of the difficulties that may be encountered using Baumgarte's method is given, and a discussion of why a further quest for better parameter values for this method will always remain frustrating is presented. It is then shown how Baumgarte's method can be improved. An efficient stabilization technique is proposed, which may employ explicit ODE solvers in case of nonstiff or highly oscillatory problems and which relates to coordinate projection methods. Examples of a two-link planar robotic arm and a squeezing mechanism illustrate the effectiveness of this new stabilization method.
A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.
We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.