Presented work on computer-assisted proofs for renormalisation fixed points at the cross-disciplinary School of Mathematics and Physics Research Colloquium

Description of Activity

Abstract of presentation: Dynamical systems are used to model how quantities change over time. They have enormous numbers of applications across a wide variety of topics spanning physics, chemistry, biology, cosmology and celestial mechanics, social systems, and so on. In its simplest form, a dynamical system consists of a function that is applied repeatedly to some initial condition in order to produce a sequence of values that we call an orbit, representing how the quantity changes over time.

Some dynamical systems display a special type of behaviour called a period-doubling cascade, in which we observe cyclic behaviour whose period (i.e., the time taken to repeat) doubles repeatedly as we vary some experimental parameter. What is most striking and surprising is that certain features of these "doubling cascades" turn out to be universal; the same qualitative and even quantitative behaviour is observed across a wide variety of (apparently very different) systems.

An explanation for this universality rests on the existence of a fixed point to a "renormalisation operator" that acts on dynamical systems to simplify them (in the sense of halving the doubled periods).

We show how rigorous computer-assisted techniques can be used to prove the existence of such renormalisation fixed points, to bound the spectrum of the derivative of the operator at the fixed point, which helps to explain universal behaviour, and to bound the eigenfunctions and eigenvalues that determine the universal properties of systems with noise.

Our computations use multi-precision interval arithmetic with rigorous directed rounding modes to bound tightly the coefficients of the relevant power series and their high-order terms, and the corresponding universal constants.

Period	17 May 2022
Held at	School of Mathematics & Physics
Degree of Recognition	Local