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Workshop on Network DynamicsJuly 15, 2009 – Exeter |
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This one-day workshop will be held in University of Exeter, SECAM. The aim of the workshop is to bring together researchers in Network Dynamics. There will be lectures and short talks about new mathematical results in this field. Participants are invited to present their posters at a poster session which will be run during the buffet lunch. This workshop is partially sponsored by the EPSRC Mathematical Neuroscience Network and London Mathematical Society (Scheme 2 visitor grant for M Field).
Confirmed Speakers:
Nikita Agarwal (Houston), Manuela Aguiar (Porto), Ana Dias (Porto), Mike Field (Houston), Alastair Rucklidge (Leeds), Hiroko Kamei (Dundee), Stephen Gin (Warwick)
Provisional Programme: (Room 101)
11.00-11.30 |
Arrival, Coffee (Room 101) |
11.30-12.15 |
Mike Field, Dynamical Equivalence of Coupled Dynamical Systems (I) |
12.15-13.00 |
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13.00-14.00 |
Lunch (and poster session) |
14.00-14.45 |
Alastair Rucklidge, Switching in a Heteroclinic Network |
14.45-15.30 |
Ana Dias, Hopf Bifurcation and Synchrony in Coupled Cell Networks |
15.30-16.00 |
Tea |
16.00-16.30 |
Nikita Agarwal, Dynamical Equivalence of Coupled Dynamical Systems (II) |
16.30-17.00 |
Hiroko Kamei, Bifurcation Analysis Using Lattice Generators and Lattice Indices |
17.00-17:30 |
Organizers:
Registration Fee:
There will be a small registration fee that will be waived for members of the Mathematical Neuroscience Network and the network can refund some travel/accommodation costs for members of that network. Contact Özkan Karabacak to register and for more details.
Directions:
The workshop will take place in University of Exeter, School of Engineering, Computing & Mathematics, Harrison Building, Room 101. For Harrison building see #23 on the university map, and for the university see the city map. For more information on directions to the campus click here.
Accommodation:
Exeter has some good accommodation facilities for visitors. Below is a list of bed and breakfast accommodation close to the University and city centre.
The Bendene (£30, £48), www.bendene.co.uk
Dunmore Guest House (£40), www.dunmorehotel.co.uk
Park View Hotel (£28-£45), www.parkviewexeter.co.uk
Road Lodge(£30), http://www.roadlodge.co.uk
Telstar Hotel (£30-£40), www.telstar-hotel.co.uk
Woodbine Guesthouse(£37-£45), www.woodbineguesthouse.co.uk
Attendees:
Abstracts of Seminars:
Mike Field and Nikita Agarwal, Dynamical Equivalence of Coupled Dynamical Systems (I) and (II):
We
start by describing some of the motivation for an approach to
coupled dynamical systems that is inspired by linear systems theory
and transfer function methodology. The set-up is broad enough to
encompass continuous dynamical systems (ODEs) with general phase
spaces, discrete dynamical systems and hybrid systems.
The
remainder of the talk will be concerned with dynamical equivalence
of coupled dynamical systems and the concepts of input and output
equivalence. We present some of the main results which include a
simple characterization of dynamical equivalence for continuous
dynamical systems together with explicit algorithms for realizing
the equivalence. (This work is joint with Nikita Agarwal and will be
continued by her in part 2 of the talk.)
Manuela Aguiar, Bifurcations from Regular Quotient Networks
We consider regular (identical-edge identical-node) networks whose cells can be grouped into classes by an equivalence relation. The identification of cells in the same class determines a new network - the quotient network. In terms of the dynamics this corresponds to restricting the coupled cell systems associated with a network to flow-invariant subspaces given by equality of certain cell coordinates. Assuming a bifurcation occurs for a coupled cell system restricted to the quotient network, we ask how that bifurcation lifts to the overall space. Surprisingly, for certain networks, new branches of solutions occur besides the ones that occur in the quotient network. To investigate this phenomenon we develop a systematic method that enumerates all networks with a given quotient. We also prove necessary conditions for the existence of solutions branches not predicted by the quotient. This is a joint work with Ana Dias (University of Porto), Martin Golubitsky (Ohio StateUniversity) and Maria Leite (Purdue University).
Alastair Rucklidge, Switching in a Heteroclinic Network
We describe an example of a structurally stable heteroclinic network for which nearby orbits exhibit irregular but sustained switching between the various cycles in the network. The mechanism for switching is the presence of spiralling due to complex eigenvalues in the flow linearised about one of the equilibria common to all cycles in the network. We construct and use return maps to investigate the asymptotic stability of the network, and show that switching is ubiquitous near the network. A simple numerical example illustrates the rich dynamics that can result from the interplay between the various cycles in the network.
Ana Dias, Hopf Bifurcation and Synchrony in Coupled Cell Networks
In this talk we address some issues concerning three classes of networks: interior-symmetric networks, symmetric networks and lattice networks with nearest neighbour coupling architecture. For the class of interior-symmetric networks, we present the full analogue of the Equivariant Hopf Theorem for networks with symmetries. Concerning symmetric networks, using representation theory of abelian groups, we show that in codimension-one, very degenerate behaviour concerning Hopf bifurcation of the coupled cell systems can occur, when taking into account the network architecture. Finally, for lattice networks with nearest neighbour coupling architecture, we recall that a pattern of synchrony is a finite-dimensional flow-invariant subspace for all lattice dynamical systems with the given network architecture. These subspaces correspond to a classification of the cells into $k$ classes, or colours, and are described by a local colouring rule, named balanced colouring. In the last part of the talk, we state a necessary and sufficient condition for the existence of a spatially periodic pattern of synchrony, given an $n$-dimensional lattice network with nearest neighbour coupling architecture, and a local colouring rule with $k$ colours. Joint works with Antoneli (São Paulo, Brasil), Paiva (Leiria, Portugal) and Pinho (Porto, Portugal).
Hiroko Kamei, Bifurcation Analysis Using Lattice Generators and Lattice Indices
In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. We consider a class of coupled cell network, called regular homogeneous networks (identical cell, identical coupling and all cells have the same number of input arrows). In particular, we consider regular homogeneous networks in which the internal dynamics of each cell is one dimensional, and whose associated adjacency matrices have simple eigenvalues. All balanced equivalence relations of such networks are determined solely by the network structure (the adjacency matrix), and they have an ordered structure, termed a lattice. We show that all possible forms of lattice associated with such networks can be explicitly constructed by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. Lattice elements with nonzero index and their positions in the lattice indicate the existence of synchrony-breaking bifurcating branches and the number of synchronised clusters. Using four-cell regular networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type and some of these networks even share the same generic bifurcations.
Stephen Gin, Synchronisation of Non-Autonomous Oscillators
I will first introduce the oscillator which has a linearly attracting limit cycle - the simplest example of a normally hyperbolic invariant manifold. Subject to time dependent forcing, the oscillation is generally Non-Periodic. However we will see that for weak time dependent forcing, not necessarily periodic, the oscillation survives - in a sense that can be viewed from the theory of Normal Hyperbolicity. We will see, under this view, that under certain conditions the dynamics can be collapsed to an attracting normally hyperbolic trajectory. We will then look at the coupling of two, not necessarily identical, oscillators and see that the theory in the one oscillator case extends to this case. I will present sufficient conditions for them to synchronise - which will be defined in the non-autonomous setting. Finally, I will outline the routes that we can take to study synchronisation of many oscillators in a network.