Thursday, April 23rd 
Alejandro López Nieto (Free University Berlin) 
Evenodd delay equations with monotone feedback: Period maps and "Sturm" meanders

The periodic solutions of delay equations with monotone feedback and evenodd symmetry can be completely characterized in terms of a planar ordinary differential equation.
During the talk I will explore the period maps associated to such ODEs and show how the global dynamics of the original delay equation can be recovered via a "Sturm" meander,
a construction that originated in partial differential equations.

Thursday, April 30th 
Alejandro López Nieto (Free University Berlin) 
Evenodd delay equations with monotone feedback: Dynamics of global attractors

After recalling the contents presented in the previous session we will dive into the main application of our results: a precise description of some global attractors in delay differential equations.
We will relate our examples to previously known cases and, additionally, we will present a number of new structures.

Thursday, May 7th 
Hauke Sprink (Free University Berlin) 
HL Bianchi models: the boundary cases 
In this talk I will give a short introduction to parameter dependent Bianchi models within HoravaLifshitz (HL) gravity, and discuss the dynamical behaviour for the boundary values of such parameter range.

Thursday, May 14 
Babette de Wolff (Free University Berlin) 
Pyragas control of systems with a negative, unstable multiplier 
We consider the Pyragas control scheme to stabilise periodic orbits of ODEs.
In previous talks, we saw that the Pyragas control scheme preserves the geometric multiplicity of the Floquet multiplier 1.
In this talk, we explore further what restrictions and possibilities this fact gives for actually controlling periodic orbits.
We show that periodic orbits with a positive, unstable multiplier cannot be stabilised via a control gain that is scalar multiple of the identity.
However, such constraints disappear for periodic solutions with a negative, unstable multiplier.
We show that (in a 3D system), we can stabilise a periodic solution with an unstable, negative multiplier, given that the modulus of this multiplier is smaller than e^2.

Thursday, May 28 
Babette de Wolff (Free University Berlin) and she will tell us about 
Stabilisation of discrete waves using equivariant Pyragas control 
We consider the stabilisation of a discrete wave using an equivariant Pyragas control scheme. In the presence of equivariance, the stability of the discrete waves is not (only) determined by the eigenvalues of the monodromy operator, but by the eigenvalues of the reduced monodromy operator. In this talk, we will study the (definition of) the reduced monodromy operator, and how relates to the 'classical' monodromy operator. We then investigate how we can use this knowledge in the context of feedback stabilisation of discrete waves.

Thursday, June 4 
Dennis Chemnitz (Free University Berlin) 
Introduction to Conley index theory 
In the first part of the talk I will continue my talk from last semester by giving an introduction to the basics of singular (co)homology. This will include (co)homology with coefficients, relative (co)homology and long exact sequences of pairs and triples. In the second half I will introduce and motivate the basic definition of Conley index theory. Furthermore the most important existence, welldefinedness and continuation results will be stated.

Thursday, June 11 
Dennis Chemnitz (Free University Berlin) 
Conley index theory and connection matrices 
This talk will continue where last week's talk on the basics of Conley index theory left off. Some methods from algebraic topology will be introduced. Specifically long exact sequences of pairs and triples will be introduced and it will be shown how to use them to calculate Conley index theory and prove existence of heteroclinic connections in certain settings.

Thursday, June 18 
Dennis Chemnitz (Free University Berlin) 
Conley index theory and connection matrices 
This talk will continue where last week's talk on the basics of Conley index theory left off. Some methods from algebraic topology will be introduced. Specifically long exact sequences of pairs and triples will be introduced and it will be shown how to use them to calculate Conley index theory and prove existence of heteroclinic connections in certain settings.

Thursday, June 25 
Phillipo Lappicy (Universidade de São Paulo (ICMC) 
HoravaLifshitz gravity: Bifurcation and Chaos 
The nature of generic spacelike singularities in general relativity is linked to first principles, notably Lorentzian causal structure, general covariance, and scale invariance. To bring a new perspective on the contributions of these principles regarding the chaotic aspects of generic singularities, we consider the initial singularity in spatially homogeneous Bianchi type VIII and IX models in HoravaLifshitz gravity, where relativistic first principles are replaced with anisotropic scalings of Lifshitz type.
For these models we show that general relativity is a critical case that corresponds to a bifurcation where chaos becomes generic. To describe the chaotic features of generic singularities for HoravaLifshitz models near the general relativistic critical case, we introduce symbolic dynamics within Cantor sets and iterated function systems.

Thursday, July 2 
JiaYuan Dai (National Center for Theoretical Sciences) 
Dynamics of two predators competing for a renewable prey 
We are interested in finding (stable) periodic solutions of an ODE system with three equations that describe the dynamics of two predators competing for the same prey. The prey population grows logistically in the absence of predation, and the predators feed on the prey with Holling's typeII response (or MichaelisMenten kinetics). Following the literature in the past four decades, we briefly explain a characterization of parameter regions that may support coexistence states, the existence of a limit cycle when one of predators is absent, and local bifurcations from such a limit cycle. Then we provide our attempts on constructing a Poincaré section and proving nontrivial invariant subsets. This is ongoing research with Dr. P. Lappicy, V. Nicola, and Dr. S. Hannes.
The seminar starts at 14.15

Thursday, July 9 
Tilman Glorius (Free University Berlin) 
Period Mismatch in a Pyragas Controlled System 
We attempt an initial investigation into a Pyragas controlled system with an incorrect delay, i.e.
the delay chosen in the control scheme does not match the period of the target orbit.

Nicola Vassena (Free University Berlin) 
Networks, bifurcation analysis and convex polytopes 
The aim of this talk is to point at connections between the three areas of the title. In particular,
our interest is in equilibria bifurcations for dynamical systems arising from (metabolic) networks.
We concentrate on simple and double zero eigenvalue bifurcations: saddlenode and Takens
Bogdanov, respectively. I will show how, under certain assumptions, this translates into finding
roots of certain multilinear homogeneous polynomials. Depending on time, and on the curiosity
of the audience, I will explore different but intimately related approaches to the problem: an
approach from scratch, an approach via a perturbation argument, an approach via Newton polytopes.
This is an ongoing joint work with Bernold Fiedler.

July 16, at 15:15 
Nicola Vassena (Freie Universität Berlin) 
Networks, bifurcation analysis and convex polytopes 
The aim of this talk is to point at connections between the three areas of the title. In particular, our interest is in equilibria bifurcations for dynamical systems arising from (metabolic) networks.
We concentrate on simple and double zero eigenvalue bifurcations: saddlenode and TakensBogdanov, respectively. I will show how, under certain assumptions, this translates into finding roots of certain multilinear homogeneous polynomials. Depending on time, and on the curiosity of the audience, I will explore different but intimately related approaches to the problem: an approach from scratch, an approach via a perturbation argument, an approach via Newton polytopes.
This is an ongoing joint work with Bernold Fiedler.
