Dia: Dimecres, 19 de setembre de 2012

Lloc: Aula S05, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Piotr Zgliczynski, Jagiellonian University

Títol: Rigorous numerics for delay equations

Resum: We will discuss a possible approach to rigorous numerics for delay equations. We try to develop a scheme, that will make possible the application of topological tools like: covering relations, topological horseshoes.

This should be constrasted with some other work on rigorous numerics for delay equations, for example by Lessard and Zalewski, where they get periodic orbits by considering it as boundary value problem, and then transform it to the fixed point problem in the space of periodic functions.

This is a joint work with Ferenc Bartha and Robert Szczelina

Dia: Dimecres, 26 de setembre de 2012

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Florin Diacu, Department of Mathematics and Statistics, University of Victoria, Canada

Títol: Singularities of the curved N-body problem

Resum:
In this talk we will show how the concept of Newtonian
gravitation can be extended from the Euclidean space to spaces
of nonzero constant curvature *k*, namely to 2D and 3D spheres,
for *k> 0*, and to 2D and 3D hyperbolic spheres, for *k<0*,
and analyze the singularities of the equations of motion.
We will show that for *k>0* there occur some new types of
singularities of the equations of motion when at least two
bodies are antipodal. Consequently we will find out whether
there exist solutions that can reach such singularities.
Finally, we will extend some results on singularities due
to Painleve in flat space to spheres and to hyperbolic spheres.

Dia: Dimecres, 17 d'octubre de 2012

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Carles Simó, UB

Títol: From steady solutions to chaotic flows in a Rayleigh-Bénard problem at moderate Rayleigh numbers

Resum:
The dynamics of a Rayleigh-Bénard convection
problem in a cubical cavity at moderate values of the Rayleigh number
(*Ra*≤10^{5}) and a Prandtl number of *Pr*=0.71 (with extensions to
*Pr*=0.75 and 0.80) was investigated. The cubical cavity was heated
from below and had perfectly conducting sidewalls and uniform
temperature distributions on the two horizontal walls. A system of
ordinary differential equations with a dimension of typically
*N≈*11,000 was obtained when the conservation equations were
discretized by means of a Galerkin method. Previous knowledge of the
bifurcation diagram of steady solutions, reported in the literature,
was used to identify the origin of several branches of periodic orbits
that were continued with *Ra*. Half a dozen of such periodic orbits
were found to be stable within narrow ranges of *Ra* (at most, some
5000units wide). An attracting two torus, also restricted to a very
narrow region of *Ra*, was also identified. It was found that the
instabilization of periodic orbits quite often resulted into the
development of complex dynamics such as the creation of homoclinic and
heteroclinic orbits. Instances of both types of global bifurcations
were analyzed in some detail.

One particular instance of chaotic dynamics (a strange attractor) was
also identified. Chaotic dynamics has been found at *Pr*=0.71 in a
flow invariant subspace, which can be interpreted as a fixed-point
subspace in terms of equivariant theory; this subspace is not
attracting. However, some regions of attracting chaotic dynamics for
moderate Rayleigh numbers (9×10^{4}≤*Ra*≤10^{5}) were
found at values of *Pr* slightly above 0.71. The role of a
particular homoclinic solution found at *Pr*=0.71 in the generation of
these chaotic regions was analyzed.

This is a joint work with Dolors Puigjaner, Joan Herrero and Francesc Giralt. See Physica D, 240, (2011), 920-934.

Dia: Dimecres, 21 de novembre de 2012

Lloc: Aula S05, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Sergey Gonchenko, Research Institute for Applied Mathematics and Cybernetics, Nizhny Novgorod State University

Títol: Mixed dynamics as a new paradigm for dynamical chaos

Resum:
Mixed dynamics is a type of the Newhouse phenomenon connected with the
existence of open (in ^{2}

Dia: Dimecres, 28 de novembre de 2012

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Pau Martin, Dept. Matemàtica Aplicada IV, UPC

Títol: Oscillatory solutions in the Restricted Three Body Problem

Resum: In this work we consider the circular restricted three body problem which models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which move along circular planar Keplerian orbits. In a suitable system of coordinates, this system has two degrees of freedom and the conserved energy is usually called the Jacobi constant. In 1980, J. Llibre and C. Simó proved the existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume the ratio between the masses of the two primaries to be exponentially small with respect to the Jacobi constant. In the present work, we generalize their result proving the existence of oscillatory motions for any value of the mass ratio.

To obtain such motions, we show that, for any value of the mass ratio and for big values of the Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity which guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. The main achievement is to rigorously prove the existence of these orbits without assuming the mass ratio small since then this transversality can not be checked by means of classical perturbation theory respect to the mass ratio. Since our method is valid for all values of mass ratio, we are able to detect a curve in the parameter space, formed by the mass ratio and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.

Dia: Dimecres, 28 de novembre de 2012

Lloc: Aula S05, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Daniel Wilczak, Department of Mathematics and Computer Science, Jagiellonian University, Krakow, Poland

Títol: Uniformly hyperbolic attractors for ODEs - rigorous validation

Resum: In recent years there were proposed several algorithms for validation of hyperbolicity of chain recurrent sets. All of them were applied to the (real or complex) Henon map - in this case the chain recurrent set is strongly disconnected.

We propose a new algorithm for computer assisted verification of
uniform hyperbolicity for maps. The method has been successfully
applied a nonautonomous ODE in R^{4}. The model equation has been
proposed by Sergey Kuznetsov and it is the system of coupled
non-autonomous van der Pol oscillators. We proved that suitable
Poincare map exhibits hyperbolic attractor which is of the
Smale-Williams type. Computer assisted proof required massive parallel
computations.

The new algorithm has shown its supremacy over the existing algorithms when applied to the Henon map.

Reference: D. Wilczak, Uniformly hyperbolic attractor of the Smale-Williams type for a Poincare map in the Kuznetsov system, SIAM Journal on Applied Dynamical Systems, Vol. 9, No. 4, 1263-1283 (2010).

Dia: Dimecres, 19 de desembre de 2012

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Alejandro Luque, Universitat de Barcelona

Títol: The parametrization method in KAM theory towards CAPs

Resum: Persistence of quasi-periodic solutions (aka invariant tori) has been for long time a subject of remarkable importance in Dynamical Systems. The not-so-recent approach of the parametrization method consists in correcting an approximation of an invariant torus by adding (iteratively) a small function to the known approximation. This correction is obtained by solving (approximately) the linearized invariance equation around the torus (yes, this is a Newton method). The advantages have been highlighted several times in this seminar and can be summarized as follows: we don't need NANANPES! (no actions - no angles - no perturbed system or epsilon small)

Of course, this topic is not new for many of you! Nevertheless, we believe that there are some points that can be of your interest. Recently, in collaboration with Alex Haro we have simplified and generalized slightly the exposition of the results that allows us to obtain a more general construction of the adapted symplectic frame to solve the linear equation (this can be very interesting in applications). In addition, in the proof of the theorem we focus in obtaining explicit (and as sharp as possible) estimates of the involved constants. A very naive "phylosophical" trick allows also to reduce the size of the constants in the statement of the theorem. We will focus in the case of Lagrangian tori for exact symplectic maps but the construction directly extends to other typical contexts.

We will illustrate the results by considering a toy example: the golden curve in the standard map. At the present moment, we are working towards a computer-assisted-proof also in collaboration with Jordi-Lluis Figueras.

Dia: Dimecres, 30 de gener de 2013

Lloc: Aula S05, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Abed Bonemoura, CRM

Títol: A Diophantine duality and applications to perturbation of quasi-periodic motions

Resum: In this talk, we will explain a duality in Diophantine approximations, and as a consequence, we will develop a new approach to the perturbation theory for quasi-periodic motions, dealing only with periodic approximations and avoiding classical small divisors estimates. This is a joint work with Stéphane Fischler (Orsay).

Dia: Dimecres, 27 de febrer de 2013

Lloc: Aula S05, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Xavier Jarque, Universitat de Barcelona

Títol: Connectivity of the Julia set: from polynomial to meromorphic maps

Resum:
It is well-known the dichotomy for the Julia set for
quadratic polynomials. That is, the *filled Julia set* for the
quadratic family *P _{c}(z)=z^{2}+c*,

Dia: Dimecres, 6 de març de 2013

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Pau Rabassa, Universitat de Barcelona

Títol: Extreme value laws in dynamical system under physical observables

Resum: Classical extreme values theory concerns with the maximum over a collection of random variables. This theory can be applied to a process generated by (chaotic) deterministic system composed with an observable (a cost function). This is the basic idea behind the extreme value theory for chaotic deterministic dynamical systems, which is a rapidly expanding area of research. The observables which are typically studied in the literature are expressed as functions of the distance with respect a point within the attractor. This is at odd with the structure of the observable functions typically encountered in physical applications, such as windspeed or vorticity in atmospheric models. We consider extreme value limit laws for observables which are not necessarily functions of the distance from a density point of the dynamical system. The talk will discuss the extension of extreme value theory to this more general class of observables.

Dia: Dimecres, 13 de març de 2013

Lloc: Aula S05, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Marina Gonchenko, Dept MA-I UPC i TU Berlin

Títol: Exponentially small splitting of separatrices associated to whiskered tori with several frequencies

Resum:
We study the splitting of invariant manifolds of whiskered
(hyperbolic) tori with two or three frequencies in nearly-integrable
Hamiltonian systems. In the two-dimensional case, we consider tori
whose frequency ratios are quadratic irrational numbers. We deal with
numbers whose continued fractions satisfy certain arithmetic
properties which give us 24 cases for consideration. In the
three-dimensional case, we consider tori with 3 cubic frequencies with
a special attention to the cubic golden number, the real root of
*x ^{3} + x = 1*.
We show that the Poincaré-Melnikov method can be applied to
establish the existence of homoclinic orbits to the whiskered tori and
prove that these homoclinic orbits are transverse. Thereby, we
generalize the results obtained by A. Delshams and P. Gutiérrez for
the golden number
(√5 - 1)/2
and other few quadratic numbers.
This is a joint work with A. Delshams and P. Gutiérrez

Dia: Dimecres, 20 de març de 2013

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Albert Granados, INRIA Rocquencourt-Paris

Títol: The scattering map in a piecewise-smooth Hamiltonian system: energy accumulation in two coupled rocking blocks

Resum: In this talk we consider the coupling of two piecewise-defined Hamiltonian systems, each obtained as a generalization of a model of the rocking block, and study certain properties related to instabilities caused by energy accumulation under periodic perturbations.

The rocking block is not only a paradigm of a mechanical system with impacts, but also it is used to model the behaviour of slender structures under an external forcing, such as water tanks or nuclear fuel rods under earthquake excitation. In addition, the stacked coupling of such blocks is also of interest for the modeling of structures in civil engineering or nano carbon tubes under small vibrations.

When coupling two rocking blocks through a generic Hamiltonian perturbation
which also includes the non-autonomous periodic forcing, we obtain a
5-dimensional piecewise-defined Hamiltonian system with two switching
manifolds. We then focus on the configuration given by large amplitude
oscillations for one block while the other one oscillates with higher frequency
and smaller amplitude. For the unperturbed case, this mode of operation is
associated with 4-dimensional C^{0} heteroclinic manifolds between
3-dimensional manifolds that are only continuous.

By means of the impact map onto the switching manifold associated with the fast rocking block, we prove the persistence of these manifolds, derive sufficient conditions for the existence of heteroclinic transversal intersections and construct the scattering map. It associates asymptotic dynamics on the 3-dimensional manifolds through heteroclinic connections. The properties of this map allow us to show that, under certain conditions, for any arbitrarily ing block at every large oscillation of the slow motion rock. This allows us to construct an heteroclinic skeleton that, when shadowed, the system becomes unstable in large time scales by further accumulating energy, hence leading to Arnold diffusion.

We also illustrate the theoretical results with numerical computations of heteroclinic connections between tori whose averaged energies fit with the given by first order properties of the scattering map.

This is a joint work with S.J. Hogan and T. Seara

Dia: Dimecres, 10 d'abril de 2013

Lloc: Aula S05, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Vassili Gelfreich, University of Warwick

Títol: Oscillating mushrooms: ergodic adiabatic theory without ergodicity

Resum: We study the dynamics of a particle inside a mathematical billiard with moving walls. In contrast with an autonomous billiard, the energy of the particle is no longer preserved. If the particle moves fast, the changes in the energy are relatively small and its behaviour depends on the underline dynamics of the frozen billiard. Can islands contribute to sustained energy growth or decay? We show that a mushroom billiard with a periodically oscillating stem accelerates the particle inside exponentially fast. We provide an estimate for the rate of acceleration. Our numerical experiments confirm the theory.

We suggest that a similar mechanism applies to general systems with mixed phase space.

Dia: Dimecres, 8 de maig de 2013

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m
- Sardanyés, J., Martínez, F., Daròs J-A., Elena, S.F. Dynamics of alternative modes of RNA replication for positive-sense RNA viruses. J. Roy. Soc. Interface 9: 768-776 (2012)
- Martínez, F., Sardanyés, J., Elena, S.F., Daròs J-A., Dynamics of a plant RNA virus accumulation: stamping machine versus geometric replication. Genetics 188: 637-646 (2011)

A càrrec de: Josep Sardanyés, Complex Systems Lab, Universitat Pompeu Fabra

Títol: Modeling the replication mode of positive-sense RNA viruses using dynamical systems

Resum: In this talk we study nonlinear mathematical models describing the intracellular time dynamics of viral RNA accumulation for positive-sense single-stranded RNA viruses. Our models consider different replication modes ranging between two extremes represented by the geometric replication (GR) and the linear stamping machine replication (SMR). We first analyse a model that quantitatively reproduced experimental data for the accumulation dynamics of both polarities of turnip mosaic potyvirus RNAs. We identify a non-degenerate transcritical bifurcation governing the extinction of both strands depending on three key parameters: the mode of replication (α), the replication rate (r) and the degradation rate (δ) of viral strands. Our results indicate that the bifurcation associated with ?? generically takes place when the replication mode is closer to the SMR, thus suggesting that GR may provide viral strands with an increased robustness against degradation. This transcritical bifurcation, which is responsible for the switching from an active to an absorbing regime, suggests a smooth (i.e. second-order), absorbing-state phase transition. Finally, we also analyse a simplified model that only incorporates asymmetry in replication tied to differential replication modes

Related publications:

Dia: Dimecres, 15 de maig de 2013

Lloc: Aula S02, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Daniel Peralta-Salas, Instituto de Ciencias Matemáticas

Títol: Existence of knotted vortex tubes in steady Euler flows

Resum:
In this talk I will review recent results, obtained in
collaboration with A. Enciso, on the existence of knotted and linked
thin vortex tubes for steady solutions to the incompressible Euler
equation. More precisely, given a finite collection of (possibly
linked and knotted) disjoint thin tubes, we will see that they can be
transformed with a *C ^{m}*-small diffeomorphism into a set of vortex tubes
of a steady solution to the Euler equation that tends to zero at
infinity. In particular, we will recover and improve our previous
theorem on the existence of knotted and linked vortex lines (Ann. of
Math. 175 (2012) 345-367). The proof combines fine energy estimates
for a boundary value problem associated to the curl operator with KAM
theory and a Runge-type global approximation theorem. The problem of
the existence of knotted and linked thin vortex tubes can be traced
back to Lord Kelvin, and in fact these structures have been recently
realized experimentally by Irvine and Kleckner at Chicago (Nature
Phys. 9 (2013) 253-258).

Dia: Dimecres, 22 de maig de 2013

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Heinz Hanssmann, Universiteit Utrecht

Títol: Families of hyperbolic Hamiltonian tori

Resum: In integrable Hamiltonian systems hyperbolic tori form families, parametrised by the actions conjugate to the toral angles. The union over such a family is a normally hyperbolic invariant manifold. Under Diophantine conditions a hyperbolic torus persists a small perturbation away from integrability. Locally around such a torus the normally hyperbolic invariant manifold is the centre manifold of that torus and persists as well.

We are interested in 'global' persistence of the normally hyperbolic invariant manifold. An important aspect is how the dynamics behaves at the (topological) boundary. Where the manifold extends to infinity this boundary is empty - this case makes clear that we need the persistence theorem of normally hyperbolic invariant manifolds in the non-compact setting.

If the normal hyperbolicity wanes as the boundary is approached we need to ensure that the perturbed dynamics does not come closer to the boundary. This provides the necessary uniform lower bound of normal hyperbolicity to still ascertain persistence under small perturbations. Making use of energy preservation and of Diophantine tori persisting by KAM theory this can be achieved for families of two-dimensional hyperbolic tori.

Dia: Dimecres, 12 de juny de 2013

Lloc: Aula S02, FME, UPC.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Lara Sabbagh, University of Warwick

Títol: Windows and estimates of diffusion times

Resum:
We set out a geometric framework for estimating the drifting
time of orbits along normally hyperbolic annuli. More precisely, we
develop an explicit construction of windows for proving the existence
of shadowing orbits along a chain of invariant circles contained in a
normally hyperbolic manifold *N* for a diffeomorphism *f*, where *f* is an
integrable twist map that admits a specific normal form near *N*.
Moreover, we prove that the time needed to drift along our transition
chain splits into three characteristic parameters: the ergodization
time associated with each circle of the chain, the straightening time
given by a lambda-lemma that we will state, and the torsion time on
each circle.

Dia: Dimecres, 26 de juny de 2013

Lloc: Aula T2 (2n pis), Facultat de Matemàtiques, UB.

- Hora: 16h Café i galetes
- Hora: 16h15m

A càrrec de: Andrey Shilnikov, Neuroscience Institute Department of Mathematics & Statistics Georgia State University

Títol: Key bifurcations of bursting polyrhythms in three-cell central pattern generators

Resum: We identify and describe the principal bifurcations of bursting rhythms in multi-functional central pattern generators (CPG) composed of three neurons connected by fast inhibitory or excitatory synapses. We develop a set of computational tools that reduce high-order dynamics in biologically relevant CPG models to low-dimensional return mappings that measure the phase lags between cells. We examine bifurcations of fixed points and invariant curves in such mappings as coupling properties of the synapses are varied. These bifurcations correspond to changes in the availability of the network's phase locked rhythmic activities such as periodic and aperiodic bursting patterns. As such, our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of, and switching between, motor patterns generated by various animals.

Last updated: Thursday, 18-Jul-2013 08:49:40 CEST