Dynamical Systems

A topological route to detect chaos in two families of dynamical systems

Speaker: 

Elisa Sovrano

Date: 

Friday, 21 September, 2018 - 11:30

Venue: 

Room FC1.004

The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2].

Equivariant Bifurcation and Ize Conjecture

Speaker: 

Reiner Lauterbach

Date: 

Friday, 14 September, 2018 - 14:30

Venue: 

Room FC1.031

J Ize conjectured that for any absolutely irreducible representation of a compact Lie group G on a finite dimensional real vectorspace there exists an isotropy subgroup which has an odd dimensional fixed point space. If it were true it had immediate consequences in equivariant bifurcation. Lauterbach & Matthews showed that this is not the case. Their findings of three infinite families of finite groups were supplemented by extensive computer analysis showing a very difficult zoo of groups acting on R4. In this talk we will give a complete list of counter examples in R4.

Dynamics on (adaptive) feedforward networks

Speaker: 

Mike Field

Date: 

Thursday, 21 June, 2018 - 11:30

Venue: 

Room FC1.030

This talk is about describing dynamics on primitive network objects and finding conditions that allow a good "reductive" description of network dynamics. We will give a number of examples when feedback is added.  The examples range from surprising synchrony, an example of the "bullwhip" effect and a remarkable layered network mixing synchrony and chaotic dynamics. Some of this work is part of a joint project with Ana Dias and Manuela Aguiar.

On realizing graphs as complete heteroclinic networks

Speaker: 

Alexander Lohse

Date: 

Wednesday, 20 June, 2018 - 14:30

Venue: 

Room FC1.030

We examine the relation between a heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. In particular, we show that there are robust realizations of a large class of transitive directed graphs that are not complete (i.e. not all unstable manifolds of nodes are included) but almost complete (i.e. complete up to a set of zero measure in the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension).

Random Lorentz gas and deterministic walks in random environments

Speaker: 

Romain Aimino

Date: 

Friday, 15 June, 2018 - 14:30

Venue: 

Room M031

Although one could naively expect that random Lorentz gases are easier to investigate than deterministic periodic ones, this seems not to be the case as essentially no results are available in the non periodic case. In this talk, I will present some general ideas towards studying random Lorentz gases and I will show how to apply them for a class of deterministic walks in random environments wit hone-dimensional uniformly expanding local dynamics. This is a joint work with Carlangelo Liverani.

 

Homogeneous coupled cell systems - unexpected symmetries and how to exploit them in bifurcation analysis

Speaker: 

Sören Schwenker

Date: 

Friday, 22 June, 2018 - 14:30

Venue: 

Room FC1.031

Dynamical systems with an underlying network structure are a subject of great interest as they arise frequently in applications and exhibit many staggering phenomena some of which resemble those in equivariant dynamics. We introduce a theory developed by Rink and Sanders that connects a class of network dynamical systems - namely homogeneous coupled cell systems - to equivariant dynamical systems. The symmetries, however, are generalized in the sense that they do not necessarily form a group but more general structures such as monoids or semigroups.

Stability of quasi-simple heteroclinic cycles

Speaker: 

Liliana Garrido da Silva

Date: 

Friday, 4 May, 2018 - 14:30

Venue: 

Room FC1.031

The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose connections are one-dimensional and contained in flow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form.

The Conley index and its applications to the study of surface flows

Speaker: 

Hector Barge

Date: 

Friday, 27 April, 2018 - 11:30

Venue: 

Room FC1.005

In this seminar we shall introduce the Conley index of an isolated invarant set of a flow on a locally compact metric space. The Conley index is a homotopical tool which encapsulates dynamical information near the isolated invariant set. The definition of this invariant involves the use of some external objects, namely isolating blocks (or, more generally, index pairs). We will give a way to compute this index in "intrinsic terms" for flows defined on surfaces. To do this we will deepen into the structure of the unstable manifold of an isolated invariant set.

Estabilização de ciclos heterodimensionais

Speaker: 

Sebastian Perez

Date: 

Friday, 20 April, 2018 - 14:30

Venue: 

Room FC1.031

Os célebres resultados de S.Newhouse ([N]) mostram que a bifurcação de uma tangência homoclínica asociada a uma sela numa superfície gera tangências homoclínicas robustas (isto é, tangências homoclínicas que persistem por pequenas perturbações) associadas a um conjunto hiperbólico especial chamado ferradura espessa. Além disso, a continuação (hiperbólica) da sela inicial está contida nesse conjunto hiperbólico.

Strange attractors near a homoclinic cycle to a bifocus

Speaker: 

Alexandre Rodrigues

Date: 

Friday, 6 April, 2018 - 14:30

Venue: 

Room FC1.031

In this seminar, we explore the chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions.

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