
عنوان و چكيده

Martin Rasmussen, Imperial College London, England

Spectral Theory for Nonautonomous Differential Equations
This talk is an introduction to the spectral theory for nonautonomous differential equations. I will explain four different types of spectra: the Lyapunov spectrum, the SackerSell spectrum, the Morse spectrum and the Bohl spectrum. While both the Lyapunov and the Bohl spectrum are spectral concepts taking individual solutions into account, the SackerSell and the Morse spectrum work with nontrivial decompositions of the space into finitely many components. Differences and similarities between the different types of spectra will be highlighted, and applications will be discussed.
Parts of this talk are joint work with Fritz Colonius (University of Augsburg), Thai Son Doan (Vietnam Academy of Science and Technology), Peter Kloeden (Huazhong University of Science and Technology), and Ken Palmer (National Taiwan University).

Hil Meijer, Twente University, Netherlands


Numerical and theoretical analysis of connecting orbits of iterated maps
The dynamics of smooth iterated maps has been well studied over the past decades. Both local and global bifurcations have been characterized. For local codimension one bifurcations, i.e. saddlenode, perioddoubling and NeimarkSacker and all local codimension 2 cases a substantial theoretical picure has been assembled. Also tools for
numerical bifurcation analysis have been developed, e.g. MatcontM or Auto. Global bifurcations lead to even more complex dynamics such as chaotic attractors. These global bifurcations involve connecting orbits and their tangencies. In this talk I will discuss numerical algorithms for the computation of 1Dinvariant manifolds and connecting orbits of maps.
For the numerical approximation of connecting orbits we work in the setting of numerical continuation. Here we track oneparameter families of zeros of a suitable function defining the connecting orbit. The approximation consists of an orbit piece with boundary conditions such that the first and final point lie in the unstable and stable manifold of a saddle, respectively. We also continue the invariant subspaces. This leads to an effective algorithm to find oneparameter families of
homoclinic and heteroclinic orbits if initial data is available. For an accurate approximation, the initial data is highdimensional. It is cumbersome and errorprone to obtain this by hand. We have developed algorithms to automatically obtain such initial data using onedimensional invariant manifolds.
I will demonstrate two applications of the algorithms in numerical bifurcation analysis. The first is a generalized H\'enon map (GHM) motivated by a global codim 2 bifurcation; a neutralsaddle homoclinic tangency. The theoretical analysis of this bifurcation leads to the GHM.
The numerical analysis of the GHM then shows the boxinabox phenomenon, where the global bifurcation reoccurs on an ever finer scale in the parameter space.
The second model is taken from adaptive control. This map has a parameter set with a quasiperiodic saddlenode bifurcation. Along this set, there is an intricate structure of connecting orbits. We demonstrate how our tools yield a much finer insight into the bifurcation structure. Our numerical study points at a few open questions.

Hans Crauel, Goethe University, Germany

Random Morse decompositions
The essential ingredient of Morse decompositions is a
decomposition of the state space of a dynamical system
into an attractor, a repeller, and a transient part,
consisting of those points which come from the repeller
and go to the attractor as time proceeds.
For deterministic systems a repeller can be obtained
as an attractor for time running backwards.
For random dynamical systems (RDS) there are different notions of attraction, namely pullback versus forward attraction, which both refer to convergence almost everywhere. And then there is also weak attraction, which requires convergence in probability only.
Morse decompositions for RDS thus may depend on the notion of attraction. In particular, the simple deterministic approach to obtain repellers by reversing time does not work for RDS. It has been shown some years ago already that random Morse decompositions exist with respect to weak attractors and repellers, respectively. After a brief introduction into the corresponding results we conclude by presenting a simple example showing that in general neither a pullback nor a forward random Morse decomposition has to exist. Only weak Morse decompositions exist always.

Fraydoun Rezakhanlou, University of California, USA

Periodic Orbits for Stationary Hamiltonian Systems
According to PoincareBirkhoff Theorem, a periodic twist map of a cylinder has at least two fixed points. V.I. Arnold realized that the correct generalization to higher dimensions
concerned the Hamiltonian flows and symplectic maps.
Arnold's conjecture in the case of a torus gives a lower bound on the number of periodic orbits of a Hamiltonian system associated with a periodic Hamiltonian function.
This conjecture was established by Conley and Zehnder in 1984. A parallel generalization of the classical PoincareBirkhoff Theorem is to investigate whether it holds in the stochastic setting. In this talk, I discuss a variant of the PoincareBirkhoff and ConleyZehnder Theorems for Hamiltonian systems associated with stationary
and ergodic Hamiltonian functions.

Henrik Shahgholian, The Royal Institute of Technology, Stokholm, Sweden

Regularity of solutions to “Broken” PDEs
I shall discuss recent developments of PDEs where there
is a qualitative change of the equation across a level set.
Central themes of the discussion concern regularity of solutions as well as that of the level set, where PDE breaks.

Hossein Movasati, IMPAInstitue de Matematica Pura e Aplicada , Brazil 
Title: Differential Equations and Arithmetic
Differential equations are mainly studied from the point of view of dynamics. In this talk I am going to discuss many differential equations which are mainly used in Arithmetic Algebraic Geometry and Number Theory. These includes,
various types of GaussManin connections and PicardFuchs linear differential equations, differential equations solved by modular and quasimodular forms,
algebraic solutions of planar differential equations and algebraic limit cycles etc.

Reza Pakzad, Pittsburgh University , USA

$h$principle for the MongeAmp\`ere equation and prestrained elasticity
We will discuss an hprinciple for the MongeAmp\`ere equation in two dimensions. In particular, through methods of convex integration, we will establish the existence of nonconvex C^1 solutions to the equation $\mathcal{D}et D^2 u = f$ when $f \ge c > 0$, where $\mathcal{D}et D^2$ stands for the very weak Hessian operator. We will also investigate the properties of $C^{1,\alpha}$ solutions for $0<\alpha<1$. We will finally explore connections with some variational problems in prestrained elasticity of plates.

Hayk Mikayelyan, Nottingham Ningbo University , China 
***
CONSTRAINED OPTIMAL REARRANGEMENT PROBLEM LEADING TO A NEW TYPE OBSTACLE PROBLEM
We consider a new type of obstacle problem in the cylindrical domain, where the obstacle is imposed on the integral of the function with respect to the axis direction. We prove existence and regularity results and show that the comparison principle does not hold for the minimizers.
This problem is derived from a classical optimal rearrangement problem in a cylindrical domain, under the constraint that the force function does not depend on the variable of the cylindrical axis.
***

Tomasz Kaczynski, Universit\'e de Sherbrooke, Canada 
Towards a formal tie between combinatorial and classical vector field dynamics
Forman’s discrete Morse theory is an analogy of the classical Morse theory with, so far, only informal ties. Our goal is to establish a formal tie on the level of induced dynamics. Following the Forman’s 1998 paper on “Combinatorial vector ﬁelds and dynamical systems”, we start with a possibly nongradient combinatorial vector ﬁeld. We construct a ﬂowlike upper semicontinuous acyclicvalued mapping whose dynamics is equivalent to the dynamics of the Forman’s combinatorial vector ﬁeld, in the sense that isolated invariant sets and index pairs are in onetoone correspondence.








