Workshop on

Global Dynamics and Applications

15 and 16 June 2009

room 209, SECaM, University of Exeter


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)

Programme

15 June:

 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.

16 June:

An informal meeting will take place in room H171 starting at 10am, with mathematical discussions (Paul Glendinning, Jason Gallas and Joaquim Puig will be there). All are welcome.

Abstracts

Paul Glendinning: "Global bifurcations: from smooth to hybrid dynamical systems"

I will describe a range of bifurcation phenomena involving homoclinic orbits, starting with classical results on smooth systems and ending with new results on bifurcations in hybrid dynamical systems (the latter allow for discontinuous switching between smooth systems and are relavant in some models of digital computer control). The recurring theme is that the existence of one bifurcation point implies the existence of a countable set of bifurcations converging on this point.

Jason Gallas: "Phase diagrams of autonomous flows: Infinite cascades of hubs and their hierarchical structuring"

In a recent work, we reported phase diagrams for a certain autonomous flow (an electronic circuit controlled by linear and piecewise-linear resistive elements) to contain a characteristic "periodicity hub", a key point responsible for organizing in a very systematic way all periodic and aperiodic dynamics over wide parameter regions around it. For more details: Phys. Rev. Lett. 101, 054101 (2008).
    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.

Jeroen Lamb: "Homoclinic bifurcation with symmetry"

In this talk I will discuss some results and open problems concerning the study of homoclinic bifurcations in equivariant and reversible systems.

Joaquim Puig: "Harper operators, equations and maps: a laboratory for Strange Nonchaotic Attractors"

In the last two decades the so-called strange nonchaotic attractors (SNA) in forced dynamical systems have been widely investigated. These attracting invariant objects of dynamical systems capture the evolution of a large subset of the phase space and are very relevant for their description. SNA are geometrically complicated (they are strange) and their dynamics is regular (in most of the examples, quasiperiodic). In contrast to the vast amount of numerical and experimental work in the area, there are only few rigorous proofs.
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)

David Chillingworth: "Phase space geometry for an impact oscillator near a degenerate graze"

Using tools from singularity theory we classify local geometry for the discontinuity set of a generic 1-paramer family of impact oscillators, and describe the associated low-velocity dynamics. In particular we show how stable manifolds for chatter orbits develop wild oscillations at generic bifurcation through a degenerate graze.


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