Dia: Dimecres 14 de gener de 2026
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Gustaf Söderlind, Lund University, Sweden
Títol: Logarithmic norms with applications to nonlinear dynamics
Resum: Originally defined for matrices in 1958, the logarithmic norm bounds solutions to initial value problems of the form
u̇ = Au + r, u(0)=u0 ,
where r is a forcing function or perturbation and A is a bounded linear operator.
Using differential inequalities one obtains
∥ u(t) ∥ ≤ etM[A] ∥ u0 ∥ + ∫0t e(t-τ)M[A] ∥ r(τ) ∥ dτ ,
where the logarithmic norm M[A] is the best possible constant such that this bound holds for all t≥ 0.
The logarithmic norm has since been extended to cover nonlinear maps and differential operators, unifying and quantifying notions such as definiteness for matrices; monotonicity for nonlinear maps; and ellipticity for differential operators. It plays an essential role in stability analysis, with broad and versatile applications in initial value problems, elliptic boundary value problems and initial-boundary value partial differential equations, as well as their discretizations.
Based on the new book [Sode24], the talk gives a brief introduction to the essential background material, and proceeds to develop the theory for nonlinear maps. Some classical nonlinear systems (such as van der Pol, Lotka--Volterra, the Kermack--McKendrick SIR model, the Oregonator equation, and the Lorenz system) are analyzed with respect to stability, stiffness and splitting. Alternative but related techniques involving the divergence of the vector field are also discussed, illustrating phase volume compression as a necessary condition for stability.
[Sode24] G. Söderlind, Logarithmic Norms, Springer Series in Computational Mathematics 63, (2024)
A càrrec de: Christos Efthymiopoulos, University of Padova
Títol: Manifold models of spiral structure in galaxies
Resum: The unstable invariant manifolds associated with unstable periodic orbits or low-dimensional tori in the corotation region at the end of galactic bars have been used extensively in the modelling of galactic spiral structure in the last 20 years. After a short review of main results on the subject, the seminar will discuss how the conventional wisdom of dynamical systems' theory, including the invariance and stability properties of the manifolds, and the passage from manifolds to Lagrangian coherent structures, can lead to observational consequences as regards the morphology and the mean velocity flow through galactic spirals. We will finally discuss the comparison between manifold models and self-consistent N-body simulations of galaxies with chaotic spirals.
Last updated: Tue Jan 13 12:03:25 2026