Global Dynamics and Applications
15 and 16 June 2009
room 209, SECaM,
University of Exeter
The workshop is organised by
Systems and Control group in
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.
Paul Glendinning (Univ. Manchester)
(Universidade Federal do Rio Grande do Sul, Brazil)
(Imperial College, London)
Joaquim Puig (Univ. Politecnica de Catalunya, Spain)
(University of Southampton)
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.
"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
"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
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
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
This is joint work with Àlex Haro (Universitat de Barcelona)
David Chillingworth: "Phase space geometry for an impact oscillator near a degenerate
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: