In this article we investigate a pitchfork bifurcation of the local attractor of a simple capsizing model proposed by Thompson. Although this is a very simple system it has a very complicate dynamic. We try to reveal some properties of this dynamic with modern numerical methods. For this reason we approximate stable and unstable manifolds which connect the steady states to obtain a complete understanding of the topology in the phase space. We also consider approximations of the Lyapunov Exponents (resp. Floquet Exponents) which indicates the pitchfork bifurcation.

Für zufällige dynamische Systeme mit stetiger Zeit existieren zwei wichtige Klassen von Generatoren: Zum einen stationäre zufällige ifferentialgleichungen, i.e. gewöhnliche Differentialgleichungen, die von einem stationärer zufälligen Vektorfeld getrieben werden, und zum anderen stochastische Stratonovichdifferentialgleichungen mit weißem Rauschen. Während die erste Klasse sich gut in den ergodentheoretischen Rahmen der Theorie der zufälligen dynamischen Systeme einfügt, widersetzte sich die zweite Klasse lange Zeit der dynamischen Untersuchung aufgrund des "Konflikts zwischen Ergodentheorie und stochastischer Analysis". In dieser Arbeit wird gezeigt, daß beide Klassen von zufälligen dynamischen Systemen nicht wesentlich verschieden sind, genauer: Zu jeder stochastischen Stratonovichdifferentialgleichung mit weißem Rauschen (unter den üblichen Regularitätsforderungen an die Vektorfelder, die die Existenz von Flüssen garantieren) existiert eine stationäre zufällige Differentialgleichung derart, daß die erzeugten zufälligen dynamischen Systeme konjugiert sind. Als Anwendung wird eine Version des lokalen Linearisierungssatzes von Hartman/Grobman für stochastische Stratonovichdifferentialgleichungen bewiesen. / For continuous time random dynamical systems there exist two important classes of generators: on the one hand stationary random differential quations, i.e. ordinary differential equations driven by a stationary random vector field, and on the other hand stochastic Stratonovich differential equations with white noise. While the first class fits well into the framework of the theory of random dynamical systems, the second class resisted for a long time the dynamical investigation due to the "conflict between ergodic theory and stochastic analysis". The main result of this thesis is that both classes of random dynamical systems are not essentially distinct, more precisely: For each stochastic Stratonovich differential equation with white noise (under usual regularity assumptions) there exists a stationary random differential equation such that the corresponding random dynamical systems are conjugate. As an application a version of the local Hartman/Grobman theorem for stochastic differential equations is proved.

In this article we investigate a pitchfork bifurcation of the local attractor of a simple capsizing model proposed by Thompson. Although this is a very simple system it has a very complicate dynamic. We try to reveal some properties of this dynamic with modern numerical methods. For this reason we approximate stable and unstable manifolds which connect the steady states to obtain a complete understanding of the topology in the phase space. We also consider approximations of the Lyapunov Exponents (resp. Floquet Exponents) which indicates the pitchfork bifurcation.