Thomas Wanner
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Finding attracting sets using combinatorial multivector fields Open
We discuss the identification of attracting sets using combinatorial multivector fields (CMVF) from Conley-Morse-Forman theory. A CMVF is a dynamical system induced by the action of a continuous dynamical system on a phase space discretiza…
Conley Index for Multivalued Maps on Finite Topological Spaces Open
We develop Conley’s theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we establi…
Cyclic symmetry induced pitchfork bifurcations in the diblock copolymer model Open
The Ohta-Kawasaki model for diblock copolymers exhibits a rich equilibrium bifurcation structure. Even on one-dimensional base domains the bifurcation set is characterized by high levels of multi-stability and numerous secondary bifurcatio…
Conley index for multivalued maps on finite topological spaces Open
We develop Conley's theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we establi…
Cyclic symmetry induced pitchfork bifurcations in the diblock copolymer model Open
The Ohta-Kawasaki model for diblock copolymers exhibits a rich equilibrium bifurcation structure. Even on one-dimensional base domains the bifurcation set is characterized by high levels of multi-stability and numerous secondary bifurcatio…
Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators Open
Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suita…
Connection matrices in combinatorial topological dynamics Open
Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of …
Equilibrium validation in models for pattern formation based on Sobolev embeddings Open
In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the…
A Computer-Assisted Study of Red Coral Population Dynamics Open
We consider a 13-dimensional age-structured discrete red coral population model varying with respect to a fitness parameter. Our numerical results give a bifurcation diagram of both equilibria and stable invariant curves of orbits. We obse…
Creating Semiflows on Simplicial Complexes from Combinatorial Vector\n Fields Open
Combinatorial vector fields on simplicial complexes as introduced by Robin\nForman have found numerous and varied applications in recent years. Yet, their\nrelationship to classical dynamical systems has been less clear. In recent work\nit…
Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces Open
We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in \cite{Mr2017}. The generalization consists in dropping the restrictive assumption in \cite{Mr2017} that every multivector has a uniq…
Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model Open
We present a computer-assisted proof of heteroclinic connections in the\none-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a\nfourth-order parabolic partial differential equation subject to homogeneous\nNeumann bounda…
Rigorous continuation of bifurcation points in the diblock copolymer equation Open
We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through ${\mathbb{Z}}_2$ symmetries in the equ…
Computer-assisted equilibrium validation for the diblock copolymer model Open
The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy…
Towards a
formal tie between combinatorial and classical vector field dynamics Open
Forman's combinatorial vector fields on simplicial complexesare a discrete analogue of classical flows generated by dynamicalsystems. Over the last decade, many notions from dynamical systemstheory have found analogues in this combinatoria…
Degenerate Nucleation in the Cahn--Hilliard--Cook Model Open
Phase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn--Hilliard equation is a dete…