ACDC Seminar by Dirk Doorakkers on the 2nd of September
On the 2nd of September at 16.00, Dirk Doorakkers will give an Amsterdam Dynamics Seminar. We will go for drinks at Bar Boele after the talk.
The talk will take place in the Maryam seminar room (9A-46).
Title: A functional analytic approach to invariant manifold theory for differential equations on Banach space with slowly evolving parameters
Abstract: Fast-slow systems of ordinary differential equations (ODEs) are a well-established topic in the mathematical study of dynamical systems. Such systems involve at least two different variables with a wide timescale separation, and have many applications in biology, physics and engineering; for example in the modeling of ecosystems and neuronal dynamics. Also, many mathematical models for complex biophysical systems incorporate spatially extended systems of differential equations, such as integro-differential equations or partial differential equations (PDEs). However, when such spatially extended systems interact with parameters evolving on a comparatively slow timescale, relatively scant mathematical literature appears to be available so far, as compared to the large body of ODE literature on fast-slow systems. Spatially extended systems interacting with slow parameters are relevant to a lot of applications, for example to models for cortical dynamics in neuroscience that incorporate parameters representing modulatory processes.
In this talk, I discuss several aspects of my PhD research under supervision of Daniele Avitabile and Jan Bouwe van den Berg, in which we develop an invariant manifold theory for differential equations on abstract Banach space with slowly evolving parameters. We use a Lyapunov-Perron approach for our proofs, which contrasts with established literature using geometric approaches for discrete-time (semi)flows. Instead, our approach can be considered ‘top-down’ departing from a given system of differential equations. Our proofs clear up several gaps and inconsistencies in the literature. This way we recover Fenichel-like Theorems for fast-slow systems whereby the fast system may live on a possibly infinite-dimensional Banach space.
Our theory is relevant to fast-slow systems with spatially extended fast variables, and is immediately applicable for example to Neural Field Equations (NFEs). These are integro-differential equations that can be interpreted as coarse-grained models for cortical activityI discuss the translation of the Fenichel-like theory for systems on abstract Banach space as mentioned before, to NFEs with slowly modulated synaptic kernel. The resulting theory for example provides rigorous support for the formal construction of relaxation oscillations in such NFEs.
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