Nonlinear dynamical systems are all around us, describing time-dependent processes in all the sciences, engineering, and many situations of daily life. Yet, their mathematical analysis tends to be extremely challenging as we often have to deal with local intricate singularities, global nonlinear effects, or large-scale models. One may not hope to obtain a complete classification theory but motivating models and examples have to be studied in tandem with the development of sufficiently generic mathematical techniques. In fact, most of the research in nonlinear systems employs a combination of tools using analytical pen-and-paper, computational, and formal asymptotic methods. Interestingly, if a system exhibits a multiscale structure, e.g., a separation of time scales, then one has more hope to obtain better analytical results as the scale separation can be exploited, while numerical computations often tend to get more difficult in comparison to single scale dynamics. This balance of methodological difficulty, in conjunction with ample practical examples from a broad range of sciences, characterizes my work in multiscale dynamics.

Many dynamical models, particularly those arising in applications, are incomplete. One often misses small-scale or finite-size intrinsic effects as well as external disturbances and environmental fluctuations. Therefore, using a stochastic dynamics approach provides a good compromise to keep modelling complexity tractable but to still capture important effects and phenomena. There are many different approaches to stochastic systems, and usually one finds that a mixture of techniques is the most powerful strategy to analyze a given problem. Examples of such techniques are direct sample path estimates providing fine scale estimates, probability distribution methods connecting to partial differential equation theory, focusing on stochastic effects (metastability, large deviations, stochastic/coherence resonance, isochronicity, etc), abstract random dynamical systems approaches, and specialized numerical methods for stochastic dynamics. Furthermore, models from applications as well as data-driven approaches naturally involve the challenge to deal with noise. My work in the area of stochastic dynamics is driven by constructing an applicable, yet mathematically generic, theory that focuses on dynamics near instability.

Networks are ubiquitous: The world around us is composed of individual parts, which interact. This is true on all levels - whether in physics, chemistry, biology, mathematics, engineering, epidemiology, technology, politics or economics. Structure and collective behaviour emerges from the properties of the individual parts and from the laws of their interaction. Many breakthrough results have been obtained in the study of networks but dynamics on and of networks is still relatively poorly understood, particularly when considering the gaps between theory and applications. Many dynamical systems posed on finite, but very large, networks are virtually impossible to treat by pen-and-paper methods. Hence, one has to employ mathematical techniques such as multiscale reduction, symmetry arguments, moment closure, or infinite-network limits. The analytical results must then be compared to numerical simulations. These ideas naturally link network dynamics to the other areas of my research, e.g., microscopic stochastic network dynamics often leads to macroscopic local or nonlocal differential equation models.

Based upon a well-grounded understanding of dynamical systems on finite-dimensional state spaces given by ordinary differential equations (ODEs), a next natural step is to also consider partial differential equations (PDEs). These equations arise often very naturally if spatial dependence of physical processes, e.g., diffusion or advective transport, are taken into account. Furthermore, many network dynamics models, such as mean-field or continuum limits, also lead to models beyond ODEs, frequently given by operator equations. The first mathematical task is to derive these new dynamical systems from microscopic first principles, and prove the existence and regularity of solutions. Yet, from the viewpoint of applications, existence of solutions is only a very first step as the models have derived to help us in the understanding of new dynamical phenomena. As an example, one often wants to understand pattern formation such as wave motion or the spatial structure of long-term steady states.

There are evidently many practical limits to purely analytical calculations in dynamical systems. Local nonlinear theory is already challenging, and global extensions are possible if one has additional exploitable structure e.g. multiple spatial scales, symmetry, energy conservation, or multiple time scales. Yet, general global nonlinear phenomena are frequently just accessible via numerical methods. A first step are usually forward simulation algorithms, which efficiently and accurately produce individual trajectories of dynamical systems. In many cases, very robust and well-established simulation methods exist but it can still be extremely difficult to understand the behaviour from just having access to sample paths. My research goal in numerical dynamics is to contribute to tailor-made numerical methods to provide access more directly to critical objects in the study of nonlinear dynamics. This includes the computation of invariant manifolds, improving and extending numerical continuation methods for parametric studies, as well as uncovering patterns numerically.

The biosciences are a source of a wide variety of interesting dynamical problems since complexity of biological systems is staggering, and their immediate impact on our lives (e.g. epidemic spreading, ecological diversity, systemic human diseases) is enormous. In my work, I have focused on problems in ecology, epidemiology, and neuroscience. In all three areas, I have been mostly interested in biological systems exhibiting interesting bifurcations and transitions upon parameter variation. In particular, generic dynamical systems methods are particularly well-suited to classify, comprehend and exploit parameter dependencies in biology as most models come with many, potentially even stochastically varying, parameters. Typical systems with many important parameters occur in the study of epidemic outbreaks, the death of species, or the onset of bursting/spiking behaviour in neurons. The methodology used in my work on biological problems regularly combines different mathematical approaches outlined above as biological systems are often multiscale, stochastic, networked, and/or spatially structured.

Similar to the biosciences (and historically far earlier), physical problems are deeply integrated in the science of nonlinear dynamical systems via providing key motivating examples. In addition, from a theoretical viewpoint, many areas in physics are highly interactive methodologically with a mathematical dynamics perspective e.g. statistical physics, biophysics, quantum physics, econophysics, just to name a few. In my work, I focus on macroscopic problems in physics, i.e., theoretical physics above quantum, atomic or molecular scales. A typical research area, where macroscale effects are crucial is fluid dynamics since the motion of many fluids is well-described by differential equations. In relation to this area, there has been an important recent surge of interest in climate dynamics, and large parts of my recent physics-driven research center around taming mathematical challenges arising in climate science. Mathematical techniques are particularly important in studying climate change as it basically impossible to just run experiments, and even numerical simulations of large climate models tend to be expensive and highly parametrized.