He is an expert on dynamical systems with the focus on multiscale analysis,

stochastic bifurcation and network dynamics. And today he will give a talk on

GSBT for fast slow PDEs. Please Christian, the screen is yours.

So thanks to BAS for the very nice introduction and invitation. So let me just quickly sort of

say a few words about the talk. What's the sort of today's plan? So the plan is that I will start a

little bit slower because I think it's quite an interdisciplinary audience I would say. I mean

already the name of the chair where I'm giving this talk and their dynamics control and numerics

is pretty broad. So I thought I would spend really quite a bit of time motivating a little

bit the problem so that everybody has the sort of at least background and understands why these

fast slow systems or these multi-scale systems are important and why we want to lift a certain

theory from ordinary differential equations to partial differential equations because

somehow this is a sort of key topic in sort of I think almost all application areas that you could

think of you know biology, physics, engineering, climate, whatever the same problem appears over

and over again and we are trying to develop a little bit sort of a of a base theory in multi-scale

PDEs for these different application situations. So let me say this is joint work with Maximilian

Engel and Felix Hummel and there's many other people who are currently joining so there are

future projects in preparation and let me start a little bit with the background introducing what

these fast slow systems are and why they are important and at least for all these that everybody

has seen a little bit. Okay so the classical textbook example so second lecture of sort of

undergraduate non-linear dynamics is the van der Poel oscillator. It's a relatively simple set

of two ODE's where in the second equation you have a small parameter which sort of means that

x formally is fast and y is slow. So because you know the tiny epsilon says that y is effectively

evolving slowly compared to x in most parts of phase space not everywhere but in most parts.

So how does this look? Well you could sort of start with certain initial data in green and

integrate them forward and then you see indeed your intuition is correct if you just sort of pick

at random any initial data. First you mostly see movement in x because x is much faster than y.

So q here is a globally unstable equilibrium you can check that this equation has a globally

unstable equilibrium point at the origin and indeed these trajectories almost vertically

fly away from this if you start near there. Oh horizontally fly away of course so yeah in this

case I've drawn it horizontally apologies. So now you could think how do I analyze this behavior?

So the natural thing as an mathematician would be to say well you know epsilon is small let's

let us try to take the limit. So if you take the limit in this van der Poel oscillator what you get

is called the fast subsystem because it's basically only the fast dynamics is evolving and y is

basically acting as a parameter. So you have these horizontal sort of invariant subspaces

and within each subspace you have a dynamical system in x and if you analyze that you can

quite easily check okay there are different regions for example in this sort of you know

horizontal line here you have three equilibria you have one equilibrium in the middle that's

unstable and two that are sort of stable so you have bistability and then if you go up in y in

the parameter at the sort of top you for example only have one equilibrium because you have changed

y and that equilibrium is actually globally stable and you get attracted to it. So you can analyze

this fast motion in this called singular limit quite easily because you somehow have reduced

the dimension by taking this limit because you have can analyze each of these systems for y separately

but somehow this doesn't give the full dynamics yeah so the full dynamics

the full dynamics is different so in the full dynamics you also at some point have to take

into account the slow scale so y will be moving at some point and you do that by scaling time

so you go from the fast time scale t to the slow time scale s so you slow time down by epsilon

this moves effectively the epsilon that was in front of the minus x and the second component

and it moves it up to the first equation. Well by the way I think somebody is not muted I hear some

I'm sorry yeah I as I'm not host because I left before I can't unmute them but if you give me the

host back I'm happy to take care of that. Okay let me quickly see if I can make the host