4^{th} IST-IME Meeting
Honoring the 60^{th} birthdays of
Luís Magalhães and Carlos Rocha
September 3-7, 2012
4^{th} IST-IME Meeting
Honoring the 60^{th} birthdays of
Luís Magalhães and Carlos Rocha
September 3-7, 2012
Rigidity of Lagrangian intersections play a fundamental role in symplectic geometry and topology. In this talk, after a general introduction, I will address a natural Lagrangian intersection problem in the context of toric symplectic manifolds: displaceability of torus orbits. The emphasis will be on recent joint work with Leonardo Macarini on their non-displaceability and with Strom Borman and Dusa McDuff on their displaceability.
We will discuss selected topics of current research interest in the theory of dynamical systems, in the rich interplay of hyperbolic dynamics, ergodic theory, and quantitative recurrence. The topics will include qualitative versus quantitative behavior of recurrence, product structure of hyperbolic measures, and Hausdorff dimension of level sets of invariant quantities. We also intend to transmit the unified character of the field.
We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels. As a consequence we show that the global behaviour of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin, proving Liouville-like results on the real axis. If the function is real-analytic it extends holomorphically to a maximal horizontal strip of the complex plane. We show that a meromorphic pdf function on the complex plane must be entire. Positivity is therefore a very strong functional constraint which reveals its power only in a differential context.
Joint work with A. C. Paixão
In this lecture I will discuss some of our recent results relative to the description of the “asymptotic dynamics” (pullback attractors) of non-autonomous partial differential equations.
A sequence of braids is presented for which two associated dynamic-geometric structures converge: the associated pseudo-Anosov maps and the associated hyperbolic 3-manifolds. The two limiting objects, however, have little to do with one another. We discuss some issues related to this puzzling state of affairs.
In the description of the long-time behavior of solutions to nonautonomous differential equations the notion of a pullback attractor plays a similar role as the global attractor in autonomous dynamical systems. We present the theorem on the existence of a pullback attractor if the evolution process is a family of closed operators. The abstract result is formulated in the context of the smoothing properties of the process and for pullback attractors attracting a given universe, i.e. a chosen class of possibly time-dependent families of sets. We also present an application of the result to reaction-diffusion equations.
We present some results, obtained in collaboration with Mário Figueira and Filipe Oliveira, recently published in the Chinese Annals of Mathematics, Series B, on the Cauchy problem concerning a quasilinear system which describes a quartic anharmonic interatomic interaction with an external potential involving a pair electron-phonon.
I will speak about a joint work with J. L. Dias, J. P. Gaivão, G. Del Magno and D. Pinheiro, on ergodic and hyperbolic a properties of a class of dissipative convex polygonal billiards. Dissipative means the billiard map is defined based on a reflexion law where the angle of reflexion is strictly smaller than the incidence angle. In contrast with the extensively studied class of conservative polygonal billiards, these dissipative systems tend to be hyperbolic and mixing.
For an \(n\)-dimensional Nicholson’s blowflies model with multiple discrete delays, \[x'_i (t) = −d_i x_i (t) + \sum_{j=1,j\neq i}^{n} a_{ij} x_j (t) + \sum_{k=1}^{m} \beta _{ik} x_i (t − \tau _{ik} )e^{−xi (t−\tau _{ik} )} , i = 1, \dots , n,\] where \(d_i > 0, a_{ij} ≥ 0\) for \(j \neq i, \beta _{ik} ≥ 0, \sum_{k=1}^{m} \beta _{ik} \gt 0, \tau _{ik} \gt 0\), we give conditions for its persistence, permanence, boundedness of all positive solutions, and the existence and global attractivity of equilibria. For the existence of a stable positive equilibrium, some results in [2] are used. Criteria for the absolute global attractivity of both the zero solution and the positive equilibrium are given. When the community matrix is irreducible, sharper results are obtained; the case of a reducible community matrix (generally not treated in the literature) is however addressed. The present work improves and further pursues the study by the author in [1].
References
[1] T. Faria, Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays, Nonlinear Anal. 74 (2011), 7033-7046.
[2] J. Hofbauer, An index theorem for dissipative systems, Rocky Mountain J. Math. 20 4 (1990), 1017–1031.
Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutations. These permutations relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.
We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.
This is joint work with Pablo Castaneda, Juliette Hell, Carlos Rocha, Brian Smith, and Matthias Wolfrum.
Assume that \( W : \mathbb{R}^m \to \mathbb{R} \) is a symmetric potential with exactly two minima:
\[ W (\gamma u) = W (u), \text{ for } u \in \mathbb{R}^m, \]
\[0 = W (a) \lt W (u), \text{ for } u \in \mathbb{R}^m \setminus \{a, γ a\}, \]
for some \(a \in \mathbb{R}^m\) with \(a_1 > 0\). Here \(γ : \mathbb{R}^k → \mathbb{R}^k , k = m, n\) is the reflection \(\gamma(z_1 ,\dots, z_k ) = (−z_1 , \dots, z_k )\) in the plane \(z_1 \equiv 0\). Assume that the domain \(\Omega \subset \mathbb{R}^n\) is convex-symmetric in the sense that
\[x \in \Omega ⇒ (t x_1 , x_2,\dots, x_n ) \in \Omega \text{ for } |t| ≤ 1.\] Let \(u : \Omega → \mathbb{R}^m\) be a global minimizer of the energy \(\int_Ω \frac{1}{2} |\nabla u|^2 + W (u)\) in the set of symmetric maps \( (u(\gamma x) = \gamma u(x))\). Under the assumption that there exists a unique symmetric nondegenerate connection \( \overline{u} : \mathbb{R} \to \mathbb{R}^m \) between the minima \(a\) and \(γa\) of \( W \) , we show that there are constants \(k, K > 0\) such that
\[ \left|u(x) − \overline{u}(|x_1 |)\right| ≤ K e^{k d(x,\partial \Omega)} , x \in \Omega.\]
In this talk we consider stationary and time dependent mean-field games, which consist in systems of Hamilton-Jacobi equations coupled with transport of Fokker-Plank equations. These systems arise in the study of symmetric games with large numbers of players. We will discuss a class of mean-field games which have a variational form and are equivalent to the Euler-Lagrange equation of a suitable functional. For these mean-field games we establish various a-priori estimates which then allow to prove existence of smooth solutions. A number of possible extensions, will be also discussed if time permits.
The asymptotic behavior of a family of singular perturbations of a non-convex second order functional of the type \[\int_\Omega f(x,u(x),\nabla u(x), \nabla^2 u(x)) \, dx \] is studied through \(\Gamma\)-convergence techniques as a variational model to address two-phase transition problems.
This is a joint work with Ana Cristina Barroso, Margarida Baía and Milena Chermisi.
In this seminar I will expose an analog of Shilnikov Lemma (also called strong \(\lambda\)-lemma) for hamiltonian systems possessing a nondegenerate normally hyperbolic symplectic critical manifold \(M\). This joint research with S. Bolotin is motivated by applications to the Poincaré second species solutions of the 3 body problem.
The objective of the present paper is the modeling and analysis of the dynamics of macrophages and certain growth factors in the inﬂammatory phase, the ﬁrst one of the wound healing process. It is the phase where there exists a major difference between diabetic and nondiabetic wound healing, an effect that we will consider in this paper. We will propose and analyze a partial differential equation as a model for the interaction of some of the crucial elements involved in this phase of the wound healing. This model will generalize a previous existing ODE model. The ﬁnal model is a system of 3 PDE equations with nonlinear boundary conditions and we will show that it is a well posed problem both from a mathematical and biological point of view. More concretely, we will prove there exists a bounded invariant set where all the solutions are global and positive.
Work done in collaboration with Neus Consul and Marta Pellicer.
Lothar Collatz proposed in 1937, with a slightly different formulation, that any natural number under the iteration \[C(n) =\begin{cases}3n+1, \text{ if } n \text{ is odd}\\ \frac{n}{2},\text{ if }n \text{ is even.}\end{cases}\] will eventually reach the cycle \(\{ 1,2\}\). Many authors studied this problem and the Collatz conjecture is also named the \(3n+1\) problem, Ulam conjecture, Kakutani’s problem, Hasse’s algorithm, Thwaites conjecture or the Syracuse’s problem.
In this talk we will prove that under the forward (resp. backward) iterations of \(C( n)\) for any initial \(c\), for \(m\) integer, is uniform for the even numbers and uniform for the odd numbers. The probability distributions for the forward iteration and backward iterations are different due to the non invertible character of the iteration scheme.
I will first discuss some abstract equivariant bifurcation results for variational problems. Then I will present some
applications, including bifurcation of constant mean curvature embeddings, bifurcation of solutions of the Yamabe problems in product manifolds and in some special Riemannian submersions, and of solutions of the \(\sigma_2\)-Yamabe problem in product of Einstein manifolds.
In this talk we present the definition of a golden sequence \( \{{r_i } \}_{i \in \mathbb{N}} \). These golden sequences have the property of being Fibonacci quasi-periodic and determine a tiling in the real line. We prove a one-to-one correspondence between:
References and Literature for Further Reading
[1] A. A. Pinto, J. P. Almeida and A. Portela, Golden tilings. Trans. Amer. Math. Soc. 364, 2261-2280 (2012).
[2] A. A. Pinto, D. Rand and F. Ferreira, Fine Structures of Hyperbolic Diffeomorphisms. Springer Monograph in Mathematics, (2009).
[3] A. A. Pinto, D. Rand, Solenoid functions for hyperbolic sets on surfaces, Recent Progress in Dynamics. MSRI Publications, 54, 145-178, (2007).
[4] A. A. Pinto, D. Rand, Smoothness of holonomies for codimension 1 hyperbolic dynamics, Bull. London Math. Soc. 34, 341-352, (2002).
[5] A. A. Pinto, D. Rand, Rigidity of hyperbolic sets on surfaces, J. London Math. Soc. 2, 1-22, (2004).
[6] A. A. Pinto, D. Sullivan, The circle and the solenoid, Dedicated to A. Katok on the occasion of his 60th birthday, DCDS-A, 16 (2), 463-504, (2006).
A mean-field model consisting on a pair of coupled Smoluchowsky-like equations for the distribution of the size of the clusters of two species with intraspecific coagulation and interspecific (complete) annihilation is considered. In a recent paper, P. Laurençot and H. van Roessel studied the scaled behavior of this system assuming equal coagulation strength for both species by using Laplace transform techniques. In this talk, after some introductory considerations about self-similarity in coagulation equations, an extension of the above study performed as a joint work by F. P. Costa, H. van Roessel, R. Sasportes and the speaker will be presented. The withdrawn of the above restriction on the coagulation coefficients implies that besides the Laplace techniques used before, we have to charaterize some asymptotic properties of solutions of a planar Lotka-Volterra competition system in several regions of the parameter space.
A classical result of A. Weinstein states that a \(2n\)-dimensional Hamiltonian system has at least n periodic orbits in each energy level set if the Hamiltonian function is convex and the level set is sufficiently close to an equilibrium. In the case of Hamiltonian systems in dimension \(4\) we will present the following dichotomy: each of those level sets either has exactly two or infinitely many periodic orbits. Conditions to distinguish between each case will be presented.
We consider the ground state solutions of the Lane-Emden system with Hénon-type weights \[ \begin{cases}−\Delta u = |x|^\beta |v|^{q−1} v, \\ −\Delta v = |x|^\alpha |u|^{p−1} u\end{cases}\] in the unit ball \(B\) of \(\mathbb{R}^N\) with Dirichlet boundary conditions, where \(N \geq 1\), \(\alpha, \beta \geq 0\), \(p, q \gt 0\) and \(1/(p + 1) + 1/(q + 1) \gt (N − 2)/N\). We show that such ground state solutions \(u, v\) always have definite sign in \(B\) and exhibit a foliated Schwarz symmetry with respect to a unit vector of \(\mathbb{R}^N\). We also give precise conditions on the parameters \(\alpha, \beta, p\) and \(q\) under which the ground state solutions are not radially symmetric.
This is joint work with Ederson Moreira dos Santos (Universidade de São Paulo) and Denis Bonheure (Université Libre de Bruxelles) to appear in J. Functional Analysis.
An overview of their professional lifes from the point of view of a close personal friend
This lecture is based in the paper [*].
Some classical results of nonlinear oscillations. (See J. K. Hale, Ordinary Differenial Equations.)
A More General Class of ODEs.
Reference
[*] P. E. Kloeden and H. M. Rodrigues: Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis 74 (2011) 2695-2719.
A version of Poincaré-Birkhoff theorem for Reeb flows on \(S^3\) will be discussed. The existence of a pair of closed orbits forming a Hopf link implies the existence of infinitely many other closed orbits if a non-resonant condition is satisfied. This result applies to Hamiltonian flows with two degrees of freedom. In particular, it implies the existence of (p,q)-sattelites for Finsler geodesic flows on \(S^2\), generalizing a theorem of S. Angenent.
This is a joint work with Al Momin and U. Hryniewicz.
A model for the formation of prices in a simplified market proposed by J.-M. Lasry and P.-L. Lions in 2007 (Mean field games, Jpn. J. Math. 2 (2007), no. 1, 229–260) presents the evolution of such prices as a motion of a free boundary in a problem for a parabolic equation. The analysis of these equations has been receiving attention in the last years, as in the works of Chayes, L., González, M., Gualdani, M.-P. and Kim, I. (Global existence and uniqueness of solutions to a model of price formation, SIAM J. Math. Anal. 41 (2009), no. 5, 2107–2135) and of Caffarelli, L., Markowich, P. and Wolfram, M.-T. (On a price formation free boundary model by Lasry and Lions: the Neumann problem. C. R. Math. Acad. Sci. Paris 349 (2011), no. 15-16, 841–844). Somehow, the dynamics that is typically obtained is a smooth evolution into a stable time-independent price.
Inspired by the older work by Guidotti, P. and Merino, S. on the appearance of stable oscillations in a thermal control problem (Hopf bifurcation in a scalar reaction diffusion equation, J. Differential Equations 140 (1997), no. 1, 209–222) we have seen how to modify the previous equations in order to give raise to stable periodic solutions. We believe that, as it happens in fluid mechanics models, Hopf bifurcations can be seen as a preliminary but necessary first step towards complicated or chaotic dynamics, at least from a deterministic approach.
Joint work with M. González and M.-P. Gualdani
An updated version of the results of R. Garcia in Principal curvature lines near Darbouxian partially umbilic points of hypersurfaces immersed in \(\mathbb{R}^4 \) (Journal of Computational and Applied Mathematics, v. 20, n. 1-2, p. 121-148, 2001), extending to \(\mathbb{R}^4 \) results obtained by C. Gutierrez and J. Sotomayor for \(\mathbb{R}^3 \) in an aproximation theorem for immersions with stable configurations of lines of principal curvature (Springer Lectures Notes in Mathematics, v. 1007, p. 332-368, 1983) will be presented.
Misiurewicz and Zieman introduced in 89 the concept of rotation sets for torus homeomorphisms homotopic to the identity, a topological invariant generalizing the rotation number of orientation preserving homeomorphisms of the circle. This concept proved to be very useful in describing several features of the dynamics of such homeomorphisms. In particular, whenever the rotation set of an homeomorphism has nonempty interior (an open and dense condition in the area preserving case) the dynamics of the map must be very rich.
In this work we will present, for non wandering homeomorphisms whose rotation set has nonempty interior, a topological partition of the torus into an essential chaotic region and periodic bounded topological disks. Furthermore, we show that this chaotic region is externally transitive, has an abundance of periodic points and the local rate of linear diffusion in the lift is everywhere the same.
Joint work with A. Koropecki (UFF-BR)
We consider \(C^{1+\epsilon}\) area-preserving diffeomorphisms of the torus \(f,\) either homotopic to the identity or to Dehn twists and suppose that \(f\) has a lift \(\widetilde{f}\) to the plane such that its rotation set has interior. If zero is an interior point of the rotation set, then there exists a hyperbolic \(\widetilde{f}\)-periodic point \(\widetilde{Q}\)\(\in \mathbb{R}\) such that \(W^u(\widetilde{Q})\) intersects \(W^s(\widetilde{Q}+(a,b))\) for all integers \((a,b)\), which implies that \(\overline{W^u(\widetilde{Q})}\) is invariant under integer translations. Moreover, \(\overline{W^u(\widetilde{Q})}=\overline{W^s(\widetilde{Q})}\) and \(\widetilde{f}\) restricted to \(\overline{W^u(\widetilde{Q})}\) is invariant and topologically mixing. Each connected component of the complement of \(\overline{W^u(\widetilde{Q})}\) is a disk with uniformly bounded diameter. If \(f\) is transitive, then \(\overline{W^u(\widetilde{Q})}=\mathbb{R}\) and \(\widetilde{f}\) is topologically mixing in the whole plane.