In part 1 of this talk I will give a short introduction to the operator semigroup approach for stochastic partial differential equations driven by Gaussian noise. I will introduce the relevant mathematical background from functional analysis, such as Hilbert-Schmidt and trace-class operators, and from the theory and infinite dimensional stochastic analysis, such as Gaussian measures on Hilbert spaces and stochastic integration in infinite dimensions. To focus on the main issues that arise in this theory, I will only consider equations with additive noise in Hilbert spaces but the concepts introduced can be generalized, with more technical effort, to equations with multiplicative noise and to Banach spaces. I will discuss the stochastic wave equation in more detail as a particular example. In part 2, I will describe a space-time approximation of the linear stochastic wave equation driven by additive nose. For the space-discretization I will introduce the relevant deterministic finite element theory, and for the time-discretization, the relevant theory for rational approximations of the exponential function. Both convergence in the mean-square sense and in the sense of weak convergence of probability measures will be discussed together with sketches of proofs.
In this talk, we will show different cubic polynomial systems in the plane, which simultaneously have limit cycles and invariant curves. Examples of Kolmogorov systems and Kukles systems are given. We show the existence of a cubic system having at least one limit cycle bounded by two invariant parabolas. For this system, we will also obtain the necessary conditions for the critical point in the interior of the bounded region to be a center.