We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the 2-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point.

In this talk we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic waves associated with the Kawahara equation.

Riemannian splines have been studied since the 1990's and in fact one of the leading groups is Fatima Leite's at Coimbra. A simple spline is a curve controlled by a force, whose normal component deviates it from the geodesic path, whereas the tangential component controls the speed. We revisit the theme with ideas from Geometric Mechanics.

First of all, mention that non-smooth systems are driven by applications and they play an intrinsic role in a wide range of technological areas and many fundamental difficulties arise, even for local phenomena. The aim of this paper is to survey a qualitative and geometric analysis on the dynamics of a 3D fold-fold singularity following various texts existing in the literature.

We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian, and the stochastic chains with memory of variable length which have finite state space.

These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process.

Finally we consider the case where the interactions between components are given by a critical directed Erdös-Rényi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings. This is a joint work with E. Löcherbach.

I will discuss the existence of infinitely many complete metrics with constant positive scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $S^m \times R^d$ and $S^m \times H^d$. As a consequence, one obtains infinitely many periodic solutions to the singular Yamabe problem on $S^m \setminus S^k$, for all $0 \leq k \lt (m−2)/2$. I will also show that all Bieberbach groups are periods of bifurcating branches of solutions to the Yamabe problem on $S^m\times R^d$. This is a joint work with R. Bettiol, UPenn.

In this talk we will consider spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant, with initial data on the outgoing initial null hypersurface satisfying a Price law, and we will study the extendibility of the corresponding maximal globally hyperbolic development. This is a joint work with João L. Costa, José Natário and Jorge D. Silva and this talk is the continuation of the one by Jorge Drumond Silva.

This talk serves as an introduction to the problem of the stability of the Cauchy horizon and the future extend​a​bility of the maximal globally hyperbolic development for the Einstein-Maxwell-scalar field system with a cosmological constant.

We will start by reviewing some basic notions of General Relativity, in particular the structure of black holes in spherical symmetry, leading to the formulation of the celebrated strong cosmic censorship conjecture, from a PDE perspective.

We will finish by describing the characteristic initial value problem for the study of the Einstein-Maxwell-scalar field system inside a black hole, with an exponential Price law over the event horizon, which is the object of recent joint work with João L. Costa, Pedro M. Girão and José Natário.

This will be a prelude to the talk by Pedro M. Girão, that will be focused on the last topic.

Let $f$ and $g$ be piecewise $C^r$ maps of the interval, with $r > 1$ and non-flat at the discontinuity sets $C_f$ and $C_g$, respectively, and let $h$ be a topological conjugacy between $f$ and $g$. We note that the maps $f$ and $g$ can be discontinuous and/or have different lateral derivatives (zero, finite or infinite) at the non-flat discontinuity sets $C_f$ and $C_g$, respectively. Let $A$ be a cycle of intervals of $f$, whose $supp A$ is a chaotic topological attractor. We prove that, if $h$ is $C^1$ in a single point with non-zero derivative then the conjugacy is a $C^r$ diffeomorphism.

We consider the quasilinear non-local Benney system

\[ \begin{cases}i u_t + u_{xx} = |u|^2 u + buv \\ \displaystyle v_t + a \left( \int_{\mathbb{R}^+} v^2 \ dx \right) v_x = -b(|u|^2)_x , \quad (x,t) \in \mathbb{R}^+ \times [0,\infty[ . \end{cases} \]

We study the existence and uniqueness of the local strong solutions to the initial boundary value problem, their possible blowup, the existence of global weak solutions and we exhibit bound-state solutions in some special cases.This talk is based on a submitted paper by J. P. Dias and F. Oliveira with the same title.

We present a careful description of the relationship between pullback and uniform attractors, leading to a detailed description of the uniform attractor and providing the understanding of its dynamical structures. That description is used to show continuity (upper and lower semicontinuity) and structural stability (topological and geometrical) of uniform attractors, at least for a non-autonomous perturbation of a semigroup.

A large class of biological and socioeconomic dynamical systems have a hierarchical structure with time and energy values ranging from the molecular level to the macroscopic level. A macroscopic variable which has come to play a role in understanding the relation between the different scales in the hierarchy is evolutionary entropy. This statistical parameter, a function of the interaction at the micro-level, describes the diversity of pathways of energy flows between the elements that compose the micro-level. Evolutionary entropy characterize the robustness or stability of the hierarchy, that is the rate at which macroscopic variables that describe the system return to their steady state condition after a random perturbation.

This talk is concerned with an analysis of the response or sensitivity of evolutionary entropy to perturbations in the microscopic parameters that describe the hierarchy.

The problem we address is an elaboration of studies which originally began in the study of demographic networks. We will appeal to the methods explored in those studies to derive a general expression for the response of entropy to changes in the microscopic variables that describe the interaction that defines the hierarchy.

Joint work with Lloyd Demetrius, Harvard University and Max Planck, Berlin.

The purpose of this talk is to present conditions under which vakonomic and nonholonomic mechanics for linearly constrained mechanical systems coincide. The main result states that, if the vakonomic vector field is tangent to the Whitney sum $\mathcal{D} \oplus (\mathcal{D}^0)^{(1)}$, where $\mathcal{D}$ is the constraint distribution and $(\mathcal{D}^0)^{(1)}$ is the first derived system of the annihilator $\mathcal{D}^0 \subset T^*M$ of $\mathcal{D}$ in the cotangent bundle of the configuration manifold $M$, then the the projection on $\mathcal{D}$ of the vakonomic trajectories with initial condition in this Whitney sum coincide with the nonholonomic trajectories.

Global surfaces of section for three-dimensional flows allows one to study dynamics via the first return map. When such global surfaces of section do not exist, it is still possible to consider the so called systems of transversal sections: a transversal foliation to the vector field in the complement of a finite set of periodic orbits. Such systems may provide interesting information about the underlying dynamics. In this talk, I will introduce the notion of systems of transversal sections for Reeb flows on the connected sums of real projective three spaces. In particular, I will present sufficient conditions so that a set formed by three periodic orbits admits a system of transversal section in its complement.

This result is motivated by the restricted three body problem for energies slightly above the first Lagrange value. This is joint work with N. de Paulo (UFSC) and U. Hryniewicz (UFRJ).

These are results of an ongoing joint work with J. Menacho. They are mostly contained in a paper of the same title appeared in SIAM J. Appl. Math. 75 (2015), no. 2, pp. 745-761. We study the hyperbolic system of equations of the so called Linear Transport Model in a True Moving Bed chromatography device with four ports. By using methods based on a suitable energy-functional we show that all solutions approach exponentially a unique steady-state solution. Then, with the use of Asymptotic Analysis techniques we calculate the limit profiles of these steady-state solutions when the mass transfer coefficient between the liquid and solid phases tends to infinity. Along this singular limit sharp boundary layers appear near some ports. We are able to obtain explicit and simple formulas for these limit profiles.

I will present shortly the results concerning fractional generalization of the Navier-Stokes problem reported in two recent papers [1, 2]. The abstracts of that two manuscripts are copied below.

The following reference list contains the most important references; an excellent paper [4], forming a base for the local in time solvability, by Yoshikazu Giga and Tetsuro Miyakawa, and recent publications [6, 5] by Jiahong Wu containing earlier studies of the fractional Navier-Stokes equation.

Abstract 1

We consider the Navier-Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard $L^2$ a priori estimates and we consider its regular approximations with the fractional power operator $(-P\Delta)^{1+\alpha}$, $\alpha \gt 0$ small, where $P$ is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard $L^2$ a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator $(-P\Delta)^s$ with $s\gt \frac{5}{4}$.

Using Dan Henry's semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small $f$ are shown to be global in time and regular.

Abstract 2

We consider fractional Navier-Stokes equations in a bounded smooth domain $\Omega\in\mathbb{R}^N$, $N\ge 2$. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solutions. Using these generalizations we construct next global solution of the original Navier-Stokes problem. Focusing finally on the 3-D model with zero external force we show that such solutions regularize after a certain time $T=T(\|u_0\|_{X_2})$.

References

1. J. W. Cholewa, T. Dlotko, Fractional Navier-Stokes equations, submitted, January 2016.
2. T. Dlotko, Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., DOI 10.1007/s00245-016-9368-y (2016).
3. T. Dlotko, M. B. Kania, C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations 259 (2015), 531-561.
4. Y. Giga, T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267-281.
5. J. Wu, The quasi-geostrophic equation and its two regularizations, Comm. Partial Differ. Equ. 27 (2002), 1161-1181.
6. J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003), 284-312.

A non-commutative integrable system (NCIS) on a symplectic manifold $(X^{2n}, \Omega)$ is given by a collection of functions $\{f_1, \dots, f_k\}$ where $(k\geq n)$, satisfying the following two assumptions:

1. Involutivity: the first $r=2n-k$ functions Poisson commute with all $k$ functions:

   \[\{f_i,f_j\}=0,\qquad (i=1,\dots,r; j=1,\dots,k).\]
2. Independence: the functions are independent almost everywhere: \[\operatorname{d} f_1\wedge \dots \wedge\operatorname{d}f_k\neq 0 \qquad \text{on a dense open set}.\]

When $k=n$ one recovers the classical notion of a commutative integrable system (CIS). The same way a CIS is related to a Lagrangian fibration, NCIS are related to isotropic fibrations. In this lecture I will explore this relationship and some beautiful connections with Poisson geometry, integral affine geometry and symplectic groupoids, leasing to a classification of regular NCIS.

This talk is based on various ongoing collaborations with Marius Crainic (Utrecht), David Martinez Torres (PUC-Rio), Daniele Sepe (UFF-Rio), Camille Laurent-Gengoux (Metz) and Pol Vanhaeck (Poitiers).

We assume that $W: \mathbb{R}^m\rightarrow \mathbb{R}$ is a nonnegative potential with exactly two nondegenerate zeros $\pm a \in \mathbb{R}^m$ and that there exist two (unique modulo translation) heteroclinic orbits $\overline{u}_- \ne \overline{u}_+$ connecting $-a$ to $a$, that is two minimizers of the Ginzburg-Landau functional $J(\varphi)= \int_ {\mathbb{R}} (W(\varphi)+\frac12|\varphi’|^2) \ ds$:

\[ J(\overline u_\pm) = \min_A J(\varphi) , \quad A\equiv \{\varphi \in W^{1,2}_{loc}(\mathbb{R};\mathbb{R}^m): \lim_{s\rightarrow\pm\infty} \varphi(s)=\pm a\} . \]

M. Schatzman in her remarkable paper Asymmetric heteroclinic double layers considered the PDE \[ \Delta u=W_u(u) , \quad W_u=(D_{u_1}W,\dots, D_{u_m}W)^T \] and, under a nondegeneracy condition on $\overline u_\pm$, proved the existence of a solution $u^S: \mathbb{R}^2\rightarrow\mathbb{R}^m$ that connects $\overline u_-$ to $\overline u_+$ in the sense that

\begin{align*}\lim_{y\rightarrow\pm\infty} u^S(x,y) & = \pm a , \\\lim_{x\rightarrow\pm\infty} u^S(x,y) & = \overline u_\pm(y-\eta_\pm)\end{align*} for some $\eta_\pm \in \mathbb{R}$.

Assuming that $W$ is a $C^\infty$ function, we give an alternative elementary proof of this result. Our approach is based on the idea of regarding $u^S$ as an Heteroclinic map $\mathbb{R} \ni x \rightarrow u^S(x, \cdot) \in A$ that connects the minimizers $\overline u_\pm(\cdot-\eta_\pm)$ of the Effettive Potential $J(\varphi)$.

We shall recall briefly how can be the local phase portraits of the equilibrium points of an analytic differential system in the plane, and we shall put our attention in the centers. First in the kind of integrability of the different types of centers, and after in the focus–center problem, i.e. how to distinguish a center from a focus. This is a difficult problem which is not completely solved. We shall provide some new results using the divergence of the differential system.