The spreading of evolutionary novelties across populations is the central element of biological adaptation. Models of evolutionary spread have ignored, until quite recently, the randomness inherent in the reproduction process. But having excellent genes is not sufficient to be successful in evolution - one also needs luck to avoid accidents or to be at the right place at the right time. Using microbial evolution experiments and simulations, I elucidate the role of chance in evolutionary processes and show that deterministic models indeed fail to predict the dynamics of evolution as it is observed in microbial evolution experiments. I present novel approaches, combining modeling and experiments, that explain the observed patterns of genetic diversity, spatial spread and adaptation at a cellular scale, where self-driven jamming can impede the expansion, and at a global scale, where the dynamics is sped up by long-range dispersal.

We introduce the term random dynamical system and discuss several objects of these systems like random attractors, random invariant manifolds, etc.
We will also discuss how these random dynamical systems are generated by stochastic evolution equations. Especially we will consider a version of these equations driven by a fractional Brownian motion. To show existence and uniqueness of these stochastic evolution equations we will use techniques which are based on fractional derivatives.

Rigorous numerical methods for establishing the existence of connecting orbits in systems of autonomous differential equations are presented.
Previously we studied the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another. Given a suitable approximate connecting orbit and assuming that a certain associated linear operator is invertible, the existence of a true connecting orbit near the approximate orbit provided the approximate orbit is sufficiently "good" was proved.
More recently we studied orbits connecting hyperbolic equilibria in a parametrized autonomous system. Given a suitable approximate connecting orbit and assuming that a certain associated matrix is invertible, the existence of a true connecting orbit near the approximate orbit and for a nearby parameter value is proved provided the approximate orbit is sufficiently "good".

In this talk we consider the parabolic problem involving a discontinuous hysteresis operator from the viewpoint of free boundary problems. Our main objective is to discuss the regularity properties of the solutions and the structure of the free boundaries. The talk is based on recent results obtained in collaboration with Nina Uraltseva.

Many neuronal systems and models display so-called mixed-mode oscillations (MMOs) consisting of small-amplitude oscillations alternating with large-amplitude oscillations. Different mechanisms have been identified which may cause this type of behaviour. In this talk, we will focus on MMOs in a slow-fast dynamical system with one fast and two slow variables, containing a folded-node singularity. The main question we will address is whether and how noise may change the dynamics.We will first outline a general approach to stochastic slow-fast systems which allows (1) to construct small sets in which the sample paths are typically concentrated, and(2) to give precise bounds on the exponentially small probability to observe atypical behaviour.Applying this method to our model system shows the existence of a critical noise intensity beyond which the small-amplitude oscillations become hidden by noise. Furthermore, we will show that in the presence of noise sample paths are likely to jump away from so-called canard solutions earlier than the corresponding deterministic orbits. This early-jump mechanism can drastically change the mixed-mode patterns, even for rather small noise intensities. The methods used to derive the results range from deterministic bifurcation theory and averaging to martingale techniques and estimates on Markov transition kernels.
Joint work with Nils Berglund (Université d’Orléans) and Christian Kuehn (TU Wien).

ODE systems originating from neural modelling are known to produce mixed-mode oscillations (alternation of large and small amplitude loops). Several different geometric mechanisms explaining small amplitude oscillations have been explored by now, e.g., delayed Hopf bifurcation and folded saddle-node type II. In my talk I will present our recently published work on the existence of periodic mixed mode solutions in which the small loops are influenced by both of these mechanisms.

We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by general-equilibrium macroeconomic models with sticky information.

This talk presents a survey on recent developments of finite theory for nonautonomous dynamical systems. The theory is useful for studying coherent structures in transport and mixing of passive and active scalars in fluids. Several diagnostic quantities have been proposed to visualize these structures, either from the geometric views such as finite time Lyapunov exponents or from probabilistic views such as finite time entropy. Further work is necessary to explore the relation between those methods.
(The talk is based on the joint work with Stefan Siegmund.)

The idea of differential calculus of fractional order is as old as the classical concepts of differentiation and integration that came up in the 17th century. However, it took some time until efforts were made to establish an exhaustive theory of fractional calculus - the first book specifically dedicated to that topic only appeared in 1974.
Until recent times, fractional calculus was considered as a mathematical theory without applications, but in the last decades there has been a growth of research activities on the application of fractional calculus to diverse scientific fields ranging from the physics of diffusion and advection phenomena, to control systems and finance and economics.
The talk aims to give an introduction to basic definitions and properties of fractional calculus and present two examples of application: Firstly, anomalous diffusion as one of the first fields where fractional differentiation was used, and secondly, a model of an electric circuit as an illustration of dynamical systems of non-integer order.

We consider a dynamical system generated by a diffeomorphism or a vector field. The shadowing problem is related to the following question: Under which conditions, for any pseudotrajectory of a dynamical system there exists a close trajectory? This notion is important for stability theory and for theoretical motivation of numerical simulations.
It is well known that a dynamical system has shadowing property in a neighborhood of a hyperbolic set. In fact it is informally believed that almost only hyperbolic systems have shadowing. While construction of non-hyperbolic example with shadowing is easy, converting this informal feeling to rigorous statements is more tricky.
I give an overview of main results in this theory and describe perspectives for future research. In particular problems relating shadowing to skew products and stochastic stability.

This talk will present our work on networks of pulse-coupled phase oscillators in the thermodynamic limit, that is, with the network size tending to infinity. Two models are considered in which oscillators are distributed on a separable metric space, according to a finite Borel measure. The first model is an evolution equation for the oscillator phase field, the second model is a continuity equation in the oscillator phase distribution. Both models are examined with respect to the existence and local stability of synchrony, i.e. all oscillators having one common, time-dependent phase. Continuing, all-to-all pulse-coupled networks of phase oscillators with additive white noise are examined, in the limit where pulses tend to Dirac distributions. This is done using a Fokker-Planck equation for the oscillator phase density. Particular interest is devoted to stationary states (i.e. time-independent phase densities), their existence, uniqueness, stability and bifurcation behaviour at changing noise strength.

In this talk, I am going to discuss three recent connected ideas in the theory of stochastic dynamical systems. We start by defining critical transitions using stochastic fast-slow systems and derive early-warning signs in the near-hyperbolic regime. Then we proceed to study noisy canard-induced mixed-mode oscillations in a regime where hyperbolicity is lost. As a last topic, we propose a new numerical method that uses deterministic continuation algorithms to capture dynamics of stochastic systems.