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Series: CDSNS Colloquium

We present a mean field model of electroencephalographic activity in the brain, which is composed of a system of coupled ODEs and PDEs. We show the existence and uniqueness of weak and strong solutions of this model and investigate the regularity of the solutions. We establish biophysically plausible semidynamical system frameworks and show that the semigroups of weak and strong solution operators possess bounded absorbing sets. We show that there exist parameter values for which the semidynamical systems do not possess a global attractor due to the lack of the compactness property. In this case, the internal dynamics of the ODE components of the solutions can create asymptotic spatial discontinuities in the solutions, regardless of the smoothness of the initial values and forcing terms.

Series: CDSNS Colloquium

If h is a homeomorphism on a compact manifold which is chain-recurrent, we will try to understand when the lift of h to an abelian cover is also chain-recurrent. This has consequences on closed geodesics in manifold of negative curvature.

Series: CDSNS Colloquium

For the tipping elements in the Earth’s climate system, the most important issue to address is how stable is the desirable state against random perturbations. Extreme biotic and climatic events pose severe hazards to tropical rainforests. Their local effects are extremely stochastic and difficult to measure. Moreover, the direction and intensity of the response of forest trees to such perturbations are unknown, especially given the lack of efficient dynamical vegetation models to evaluate forest tree cover changes over time. In this study, we consider randomness in the mathematical modelling of forest trees by incorporating uncertainty through a stochastic differential equation. According to field-based evidence, the interactions between fires and droughts are a more direct mechanism that may describe sudden forest degradation in the south-eastern Amazon. In modeling the Amazonian vegetation system, we include symmetric α-stable Lévy perturbations. We report results of stability analysis of the metastable fertile forest state. We conclude that even a very slight threat to the forest state stability represents L´evy noise with large jumps of low intensity, that can be interpreted as a fire occurring in a non-drought year. During years of severe drought, high-intensity fires significantly accelerate the transition between a forest and savanna state.

Series: CDSNS Colloquium

I will consider the isotropic XY quantum chain with a transverse magnetic field acting
on a single site and analyze the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. It has been shown in the early 70’s
that, in the thermodynamic limit, the state of such system obeys a linear time-dependent
Schrodinger equation with a memory term.
I will consider two different regimes, namely when the perturbation has non-zero or
zero average, and I will show that if the magnitute of the potential is small enough then
for large enough frequencies the state approaches a periodic orbit synchronized with the
potential. Moreover I will provide the explicit rate of convergence to the asymptotics.
This is a joint work with G. Genovese.

Series: CDSNS Colloquium

When perturbed with a small periodic forcing, two (or more) coupledconservative oscillators can exhibit instabilities: trajectories thatbecome unstable while accumulating ``unbounded'' energy from thesource. This is known as Arnold diffusion, and has been traditionallyapplied to celestial mechanics, for example to study the stability ofthe solar system or to explain the Kirkwood gaps in the asteroid belt.However, such phenomenon could be extremely useful in energyharvesting systems as well, whose aim is precisely to capture as muchenergy as possible from a source.In this talk we will show a first step towards the application ofArnold diffusion theory in energy harvesting systems. We will consideran energy harvesting system based on two piezoelectric oscillators.When forced to oscillate, for instance when driven by a small periodicvibration, such oscillators create an electrical current which chargesan accumulator (a capacitor or a battery). Unfortunately, suchoscillators are not conservative, as they are not perfectly elastic(they exhibit damping).We will discuss the persistence of normally hyperbolic invariantmanifolds, which play a crucial role in the diffusing mechanisms. Bymeans of the parameterization method, we will compute such manifoldsand their associated stable and unstable manifolds. We will alsodiscuss the Melnikov method to obtain sufficient conditions for theexistence of homoclinic intersections.

Series: CDSNS Colloquium

Consider an affine skew product of the complex plane. \begin{equation}\begin{cases} \omega \mapsto \theta+\omega,\\ z \mapsto =a(\theta \mu)z+c, \end{cases}\end{equation}where $\theta \in \mathbb{T}$, $z\in \mathbb{C}$, $\omega$ is Diophantine, and $\mu$ and $c$ are real parameters. In this talk we show that, under suitable conditions, the affine skew product has an invariant curve that undergoes a fractalization process when $\mu$ goes to a critical value. The main hypothesis needed is the lack of reducibility of the system. A characterization of reducibility of linear skew-products on the complex plane is provided. We also include a linear and topological classification of these systems. Join work with: N\'uria Fagella, \`Angel Jorba and Joan Carles Tatjer

Series: CDSNS Colloquium

We will consider the
Frenkel-Kontorova models and their higher dimensional generalizations
and talk about the corresponding discrete weak KAM theory. The existence
of the discrete weak KAM solutions is related to the additive
eigenvalue problem in
ergodic optimization. In particular, I will show that the discrete weak
KAM solutions converge to the weak KAM solutions of the autonomous
Tonelli Hamilton-Jacobi equations as the time step goes to zero.

Series: CDSNS Colloquium

One dimensional discrete Schrödinger operators arise naturally in modeling
the motion of quantum particles in a disordered medium. The medium is
described by potentials which may naturally be generated by certain ergodic
dynamics. We will begin with two classic models where the potentials are
periodic sequences and i.i.d. random variables (Anderson Model). Then we
will move on to quasi-periodic potentials, of which the randomness is
between periodic and i.i.d models and the phenomena may become more subtle,
e.g. a metal-insulator type of transition may occur. We will show how the
dynamical object, the Lyapunov exponent, plays a key role in the spectral
analysis of these types of operators.

Series: CDSNS Colloquium

The format of this talk is rather non-standard. It is actually a combination of two-three mini-talks. They would include only motivations, formulations, basic ideas of proof if feasible, and open problems. No technicalities. Each topicwould be enough for 2+ lectures. However I know the hard way that in diverse audience, after 1/3 of allocated time 2/3 of people fall asleep or start playing with their tablets. I hope to switch to new topics at approximately right times.The topics will probably be chosen from the list below.“A survival guide for feeble fish”. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water flow. This is related tohomogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov.How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably “nice” mm–spaces. A notion of rho-Laplacian and its stability. Joint with S. Ivanov and Kurylev.One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodicdynamics. What happens outside KAM tori has been remaining a great mystery. The main quantative invariants so far are entropies. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however,on a zero measure set. We were able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations. Furthermore, a slightly modified construction resolves another long–standing problem of theexistence of entropy non-expansive systems. These modified examples do generate positive positive metric entropy is generated in arbitrarily small tubular neighborhood of one trajectory. The technology is based on a metric theory of“dual lens maps” developed by Ivanov and myself, which grew from the “what is inside” topic.How well can we approximate an (unbounded) space by a metric graph whose parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers (the mostdifficult one is for R^2) are given using dynamics and Fourier series.“What is inside?” Imagine a body with some intrinsic structure, which, as usual, can be thought of as a metric. One knows distances between boundary points (say, by sending waves and measuring how long it takes them to reach specific points on the boundary). One may think of medical imaging or geophysics. This topic is related to the one on minimal fillings, the next one. Joint work with S. Ivanov.Ellipticity of surface area in normed space. An array of problems which go back to Busemann. They include minimality of linear subspaces in normed spaces and constructing surfaces with prescribed weighted image under the Gauss map. I will try to give a report of recentin “what is inside?” mini-talk. Joint with S. Ivanov.More stories are left in my left pocket. Possibly: On making decisions under uncertain information, both because we do not know the result of our decisions and we have only probabilistic data.

Series: CDSNS Colloquium

Given a dynamical system (in finite or infinite dimension) it is very natural to look for finite dimensional invariant subspaces on which the dynamics is very simple. Of particular interest are the invariant tori on which the dynamics is conjugated to a linear one. The problem of persistence under perturbations of such objects has been widely studied starting form the 50's, and this gives rise to the celebrated KAM theory. The aim of this talk is to give an overview of the main difficulties and strategies, having in mind the application to PDEs.