Thursday, April 21st 
JeanPhilippe Lessard (McGill University, Montreal) 
Computerassisted proofs for stable and unstable manifolds via the parameterization method

In this talk, we review the fundamentals of the parameterization method, which provides a powerful way to obtain high order approximations of invariant manifolds in differential equations. The idea is first to introduce an invariance equation whose solution corresponds to the wanted parameterization, second to compute a numerical approximation of a solution of the invariance equation via power series and third to use the tools of rigorous numerics to validate the numerics. We present applications in the context of finite and infinite dimensional dynamical systems, namely ODEs, PDEs and delay equations. More specifically, we demonstrated how to use the rigorous control of the invariant manifolds to prove existence of connecting orbits.

Thursday, April 28th 
Organisational 
During the session, we will organize the talks for the rest of the semester.

Thursday, June 2nd 
Fan Cheng (Freie Universität Berlin) 
Monotonicity of eigenvalues of a geometric operator along the Ricci flow

In this talk, we will consider the eigenvalue of a geometric operator related to Perelman's W functional, compute its variational formula along the Ricci flow, and obtain a condition for monotonicity.

Thursday, June 16th 
JiaYuan Dai (National Chung Hsing University) 
Dynamics of two predators competing for a renewable prey

We consider an ODE system that describes the dynamics of two predators competing for the same prey. Our aim in this part is to study how periodic orbits are possibly triggered by perturbing an elliptic Hopf bifurcation without parameters. For this aim, we prove that the ODE system admits an elliptic Hopf bifurcation and discuss an attempt by using the averaging method. This is ongoing research with Dr. P. Lappicy, Dr. N. Vassena, and Dr. H. Stuke.

Thursday, June 23rd 
Hauke Sprink (Freie Universität Berlin) 
Stable manifolds for heteroclinic chains in Hořava cosmologies

In this talk we give a perspective on results by Liebscher et al who construct stable manifolds for a certain class of heteroclinic chains in the wellknown Bianchi model in general relativity. The aim of our work is to obtain similar results in the parameter dependent Bianchi models within HL cosmology.

Thursday, June 30th 
Nicola Vassena (Freie Universität Berlin) 
Hunting purely imaginary eigenvalues in chemical networks

I will discuss a few sufficient criteria to detect a Jacobian with a pair of purelyimaginary complexconjugated eigenvalues in general reaction networks. Such spectral condition is necessary for Hopf bifurcation, and may constitute an indication of the ability of the network to sustain oscillations.

July 7th 
Tilman Glorius (Freie Universität 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.

Danchen He (Freie Universität Berlin) 
The dynamics of a class of epidemic models with control interventions

This time I will mainly show the dynamic behavior of SVEIQRS model. This system exhibits a backward bifurcation behavior, proving that the wellaccepted condition R0 < 1 is an essential requirement for eliminating a disease that may not be sufficient. Later we will find the critical value Rc where the stable and unstable endemic curves coincide. Only when R0< Rc is the diseasefree equilibrium the only stationary solution. Finally, we will compare the representative value with the subsystem, showing how quarantine and imperfect vaccination affect the system.




