The workshop is organised by the Dynamical Systems and Control group in the Mathematics Research Institute and will take place at the School of Engineering, Computing and Mathematics (SECaM), in the Harrison Building (see the map). The workshop aims to bring together researchers with an interest in theoretical developments and applications of homoclinic bifurcations and resonant dynamics.

Confirmed speakers:

Paul Glendinning (Univ. Manchester)

Jason Gallas
(Universidade Federal do Rio Grande do Sul, Brazil)

Jeroen Lamb
(Imperial College, London)

Joaquim Puig (Univ. Politecnica de Catalunya, Spain)

David Chillingworth
(University of Southampton)

11.00 - 11.50 | Paul Glendinning | Global bifurcations: from smooth to hybrid dynamical systems |

11.50 - 12.40 | Jason Gallas | Phase diagrams of autonomous flows: Infinite cascades of hubs and their hierarchical structuring |

12.40 - 13.30 | Jeroen Lamb | Homoclinic bifurcation with symmetry |

13.30 - 15.00 | Lunch | |

15.00 - 16.00 (room 101) | Joaquim Puig | Harper operators, equations and maps: a laboratory for Strange Nonchaotic Attractors |

16.00 - 16.45 (staff room) | Tea and coffee | |

16.45 - 17.35 | David Chillingworth | Phase space geometry for an impact oscillator near a degenerate graze |

The talks will take place in room 209 unless otherwise specified.

Now, we present numerically obtained phase diagrams that, quite surprisingly, show hubs not to be isolated organizers but in fact to exist abundantly in regular networks and to "cooperate collectively" in the organization of periodic and aperiodic phases in parameter space. Hubs arise in infinitely nested hierarchies, or cascades. We find hub cascades: (i) to accumulate along very interesting paths in parameter space, and (ii) to accumulate towards a characteristic parameter point of great dynamical significance, a sort of "hub of all hubs".

The intricate phenomena at hand are clear cases of global bifurcations in flows. We recall some important bifurcations associated with the birth of periodic orbits from a homoclinic (separatrix) loop to a saddle, and from a separatrix to a saddle-node. Descriptions based on linearizations contain some aspects of the phenomenon, but not hub cascades. In sharp contrast with previous work, the present phenomena involves infinite cascades of stable periodic and aperiodic orbits. Hubs and hub cascades formed by stable periodic orbits have not been theoretically anticipated. Their explanation seems to transcend currently available knowledge concerning homoclinic orbits and the structuring induced by them in nearby orbits.

In this talk we prove the existence of SNAs for a family of one-dimensional maps with quasiperiodic forcing: Harper maps. This family arises from a model in mathematical physics and it is a paradigm of a 1D quasiperiodically forced map. Our approach connects the spectral theory of Schrödinger operators and the theory of nonuniformly hyperbolic systems. So, even if the proof is made for a concrete family, many of the arguments apply to other families.

This is joint work with Àlex Haro (Universitat de Barcelona)

If you are interested in participating or to have more
information, please contact the organisers:

Alejandra
Gonzalez, m.a.gonzalez-enriquez(at)ex.ac.uk

Peter
Ashwin, p.ashwin(at)ex.ac.uk

Renato
Vitolo, r.vitolo(at)ex.ac.uk