1. Gronwall-Bellman estimates for the solutions of some linear integral inequalities with delay. Sebastián Buedo Fernández (Universidade de Santiago de Compostela)

2. Existence, uniqueness and computation of the solution in linear fractional integral equations of arbitrary real order with constant coefficients. Daniel Cao Labora (Universidade de Santiago de Compostela)

3. Global connections in a Lorenz-like system. A. Algaba, C. Domínguez-Moreno, M. Merino, A. J. Rodríguez-Luis (Universidad de Huelva)

4. On some global bifurcations regarding periodic orbits in piecewise-linear systems with hysteresis. Marina Esteban (Universidad de Sevilla)

5. Existence and localization of fixed points for compressions or expanssions of a cone by using star convex sets. Cristina Lois Prados (Universidade de Santiago de Compostela)

6. Resonance tongues in the linear Sitnikov equation. Mauricio Misquero Castro (Universidad de Granada)

7. Some results on impulsive differential equations. José Manuel Uzal (Universidade de Santiago de Compostela)

8- Phase-Locked States in the Oscillatory Regime of Neuronal Networks. Alberto Pérez Cervera (Universitat Politècnica de Catalunya)

• Abstracts of Posters

Outline and description

We discuss some of the mathematical theory for bifurcations of dynamical systems, with application to network dynamics. The presence of special structures in the form of network symmetries constrain but give a much richer set of possible generic bifurcations that the non-symmetric case. We introduce the tools of generic bifurcation theory for equivariant (symmetric) systems and apply this to examples in neural network dynamics where many of these bifurcations are associated with onset or loss of various types of synchrony.

Topics to be covered

1. Bifurcation theory for ODEs, bifurcations of equilibria and genericity

2. Centre manifolds, normal forms and symmetries

3. Symmetric (equivariant) dynamics: group representation and bifurcations

4. Network dynamical systems and applications of symmetric bifurcation theory

5. Some examples in oscillatory neural network dynamic 

References

P Ashwin, S Coombes, R Nicks. Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience, J. Math. Neurosci. 2016: available from https://mathematical-neuroscience.springeropen.com/articles/10.1186/s13408-015-0033-6

M Golubitsky, I Stewart. The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Birkhauser 2002: http://www.springer.com/gp/book/9783764366094

J Moehlis and E Knobloch. Equivariant dynamical systems – Scholarpedia: http://www.scholarpedia.org/article/Equivariant_dynamical_systems

Handouts:

handouts and notes of the course

Outline and description

Often people informally confuse random and chaotic motion. Yet, the first is probabilistic while the latter is deterministic. It follows that to understand in which precise sense a deterministic motion can be called random is a non trivial issue. In fact, our theoretical understanding of such question is limited to very simple models. Also, there exists some disagreement on the role of randomness in the study of dynamical systems. I will describe some classes of systems for which something rigorous can be said and discuss future perspective. 

More precisely, I will try to give some ideas on how to investigate the statistical properties and establish limit theorems (Central Limit Theorem, Large deviations, averaging, ….) for expanding maps, hyperbolic systems, partially hyperbolic systems and, possibly, systems with small noise. 

References

C. Liverani. Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Pubblicazioni della Classe di Scienze, Scuola Normale Superiore, Pisa. Centro di Ricerca Matematica "Ennio De Giorgi" : Proceedings. Published by the Scuola Normale Superiore in Pisa (2004). 

Jacopo De Simoi, C. Liverani. The Martingale approach after Varadhan and Dolpogpyat. In "Hyperbolic Dynamics, Fluctuations and Large Deviations", Dolgopyat, Pesin, Pollicott, Stoyanov editors, Proceedings of Symposia in Pure Mathematics, 89, AMS (2015). 

Blumenthal, Alex; Xue, Jinxin; Young, Lai-Sang. Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. of Math. (2) 185 (2017), no. 1, 285–310. 

Notes 

Notes of the course

Outline and description

The course will first give an overview of the different types of resonances appearing in celestial mechanics with numerous examples in the Solar System. The basic fundamental models will be presented and analysed, as well as the way to isolate (by averaging) the long-term resonant dynamics. The Hamiltonian formalism will be used to build the differential equations, and consequently its properties will be reminded.

Then the different types of resonances will be presented, with specifications and examples taken in various regions.

The captures into resonance or the crossings of a resonance can be modelled in some cases with the adiabatic invariant, in presence of dissipative forces. Probabilities of capture can be deduced in very simple models or through huge and complex numerical integrations.

Resonances could be associated with stability or with chaos, especially when they cross each other, to create a real web. Numerically and analytically these situations could be described or simply quantified, using well-known tools as chaos indicators (MEGNO or FLI or frequency analysis).

The understanding of resonant phenomena often implies to push the integrations on much longer time scales. They are then associated to numerical techniques, trying to avoid the increase of the energy, as symplectic integrators.

Topics to be covered

1. Resonances : surveys

1.1 Resonances In our solar system: examples, effect, history, a first panorama

1.2 General principles: closeness of two frequencies, isolation of a resonant angle, calculation of the equilibria, their stability, the separatrices.

1.3 Definition, models, examples and  simulations for different resonances: mean motion, secular, secondary, Kozai-Lidov, spin-orbit, gravitational, etc.

2. Capture into resonance in presence of dissipation

2.1 Adiabatic invariant theory

2.2 Main dissipative contributions

2.3 Capture or escapes

2.4 Simulations and applications

3. Chaos

3.1 Web of resonances

3.2 Chaos: measurement

3.3 Megno, FLI, frequency analysis

4. Long term integrations

4.1 Symplectic integrators

4.2 Validity and stability

4.3 Comparisons

References

Henrard J., 1982, Capture into resonance - an extension of the use of adiabatic invariants, Celestial Mechanics, 27, p. 3-22.

Jancart S. and Lemaitre A., 2001, Dissipative forces and external resonances, Celestial Mechanics and Dynamical Astronomy, 81, p. 75-80.

D’Hoedt S. and Lemaitre A , 2004, The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model, Celestial Mechanics and Dynamical Astronomy, 89, p. 267-283.

Lemaitre A., 2010, Resonances: Models and Captures, Dynamics of Small Solar System Bodies and Exoplanets, Eds. Souchay and Dvorak, Lecture Notes in Physics, 790 , p. 1-62.

Verheylewegen, E. and Lemaitre A., 2014, The 3:1 mean motion resonance between Miranda and the inner Uranian satellites, Cressida and Desdemona, Celestial Mechanics and Dynamical Astronomy, 119, p. 283-299.

Sansottera M., Lhotka Ch. And Lemaitre A., 2015, Effective resonant stability of Mercury, Monthly Notices of the Royal Astronomical Society, 452, 4, p. 4145-4152.