Workshop‎ > ‎


Keynote Addresses:

Speaker: Dr. James Atkinson
Title: Yang-Baxter maps, idempotent biquadratics and incidence
Abstract: A new interpretation of known Yang-Baxter maps and recently discovered integrable multi-quadratic quad equations will be described. Specifically these integrable discrete dynamical systems can be viewed as governing composition between idempotent biquadratic correspondences. This point of view provides a strikingly simple approach to understand previously unexplained symmetries that permute parameters and variables in these systems. Most importantly this exposes a rich generalisation of the consistency (integrability) property at the algebraic heart of these discrete models.

Speaker: Prof. Andy Hone
Title: Cluster integrable systems, tropical dynamics and entropy
Abstract: We describe a family of birational maps obtained from mutations in cluster algebras. A series of conjectures are presented which imply that the algebraic entropy of these maps can be determined explicitly from their tropical (or ultradiscrete) analogues. Details of both integrable and non-integrable maps within this family will be given. 

Speaker: Prof. Kenji Kajiwara
Title: Some explicit formulas in discrete differential geometry
Abstract: Some exact and explicit formulas in discrete differential geometry are presented. In particular, we consider (1) isoperimetric motion of plane/space discrete curves (2) discrete holomorphic functions. (1): We give explicit formulas for curve motions described by continuous/semi-discrete/discrete modified KdV equation in terms of the tau functions. We also consider the discrete analogue of hodograph transformations and present discretization of WKI elastic beam equation and related equations. (2): We construct an explicit formula for the discrete power function presented by Agafonov and Bobenko in terms of the hypergeometric tau functions of the Painleve VI equation. We also consider some generalizations by using this formula.

Speaker: Prof. Bernd Krauskopf
Title: Global manifolds in the transition to chaos in the Lorenz system
Abstract: The Lorenz system displays a transition from simple to chaotic dynamics via a state of pre-turbulence as the Raleigh parameter is changed. The talk discusses associated qualitative changes of the overall organization of the three-dimensional phase space. Specifically, we consider how two-dimensional global invariant manifolds change qualitatively at the two main bifurcations involved. This is joint work with Hinke Osinga, The University of Auckland,
and Eusebius Doedel, Concordia University.

Speaker: Prof. Wolfgang Schief 
Title: Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations
Abstract: TBA

Speaker: Prof. Vladimir Sokolov
Title: Integrable ODEs with matrix variables. Bi-Hamiltonian approach
Abstract: We consider integrable systems of ODEs with matrix dependent variables and special class of  Poisson brackets related to such systems. We investigate general properties of the brackets, present an example of a compatible pair  of quadratic and linear brackets and find the corresponding hierarchy of integrable models, which generalises the two-component Manakov's matrix system in the case of arbitrary number of matrices

Speaker: Prof. Ferdinand Verhulst
Title: Integrability and Chaos in Hamiltonian Systems
Abstract: Most time-independent Hamiltonian systems are not integrable. However, as we shall see, this is a very deceptive statement although it is mathematically correct. To get this in the right perspective, we shall start by outlining suitable approximation methods. These are canonical normal form methods, sometimes called after Birkhoff-Gustavson, and averaging performed in a canonical way. These methods admit precise error estimates and enable us therefore to determine local measures of regularity and chaos. We recall that two degrees of freedom systems near stable equilibrium can be normalised and that the normal form is always integrable to any order. The integrals are the Hamiltonian and its quadratic part. The integrable motion dominates phase-space and this result expresses that the amount of chaos near stable equilibrium is exponentially small. The situation is very different for more than two degrees of freedom. First we consider the genuine first order resonances of three degrees of freedom systems. It turns out that the normal form of the 1:2:2-resonance is integrable; this is caused by a hidden symmetry which reveals itself by normalisation. The 1:2:1-resonance to order 3 and the 1:2:3-resonance to order 4 on the other hand are not integrable for an open set of parameters of the Hamiltonian. We shall discuss the techniques involved to show this and aspects of the resulting dynamics of typical systems. Recently, O. Christov proved that the 1:2:4- and 1:2:3-resonances to order 3 are in normal form non-integrable. The n degrees of freedom Fermi-Pasta-Ulam problem can also be discussed by normalisation, thus explaining the famous recurrence phenomena.

Contributed Talks:

Speaker: Prof. Peter Forrester
Title: TBA
Abstract: TBA

Speaker: Prof. Partha Guha
Title: On determining elementary first integrals and Hamiltonization of non planar dynamical systems
Abstract: We apply the Darboux integrability method to determine  first integrals of motion of three-dimensional systems; whose associated vector fields are polynomials. We illustrate our methods with examples. The Jacobi Last Multiplier (JLM) is known to be a useful tool for deriving the Lagrangian of a planar dynamical system. in this talk we will demonstrate that for non planar dynamical systems the JLM continues to play a pivotal role in the context of the existence of the Poisson-Hamiltonian structures. 

Speaker: Dr. Peter van der Kamp
Title: On initial value problems on quad-graphs
Abstract: Conditions for well-posed, under-determined, and over-determined initial value problems for integrable equations on quad-graphs were given in [V.E. Adler and A.P. Veselov, Cauchy problem for integrable discrete equations on quad-graphs,
Acta Appl. Math. 84 (2004)]. I will show that these conditions on the initial value problem are neither necessary nor sufficient, that the integrability is not essential if the characteristic strips of the quad-graph do not close. I will present a method to 
construct well-posed initial value problems for not necessarily integrable quad-equations on not necessarily simply connected quad-graphs.

Speaker: Dr. Pavlos Kassotakis
Title: On non-multiaffine consistent-around-the-cube lattice models
Abstract: A brief review on the connection of integrable lattice equations with Yang-Baxter maps will be presented, followed by a procedure that one can associate integrable lattice models to these mappings. The lattice models associated to the F or H-list of Yang-Baxter maps will be presented.

Speaker: Prof. Masahiko Ito
Title: An introduction to q-calculus
Abstract: The q-calculus is an established topic within classical analysis. Much of the present day interest relates to the q-special functions. In this introductory talk, I'll explain Ramanujan's $_1\psi_1$ summation formula via the q-calculus.

Speaker: Richard Norton
Title: Existence of stationary solutions and local minima for 2D models of fine structure dynamics 
Abstract:  We consider two-dimensional models for the motion of a viscoelastic material with a non-monotone stress-strain relationship. I will give sketch of a proof to show there exist infinitely many stationary solutions to two model problems by constructing sequences of increasingly oscillatory functions, whose limit is a stationary solution. These equilibria may have arbitrarily small energy. Moreover, it is always possible to construct paths in phase space that strictly decrease the energy. This result negates the existence of local minima for the energy and asymptotically stable equilibria. These results are important first steps towards understanding the dynamics of fine structure in more than one dimension. 

Speaker: Dr. Christopher Ormerod
Title: A reduction of the discrete Schwarzian Korteweg-de Vries equation to $q$-$P(E_6^{(1)})$
Abstract: We discuss a systematic way of performing non autonomous reductions of lattice equations. As an application, we demonstrate how to obtain the $q$-Painleve system with $E_6^{(1)}$ symmetry as a reduction of the Schwarzian Korteweg de Vries equation.

Speaker: Prof. Reinout Quispel
Title: Geometric Properties of the Kahan-Hirota-Kimura Discretization
Abstract: We show that Kahan’s discretization of quadratic vector fields is equivalent to a Runge–Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge–Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings, explaining some of the integrable cases that were previously known. (Joint work with Elena Celledoni, Robert McLachlan & Brynjulf Owren).

Speaker: A. Prof. John Roberts
Title: Understanding Non-QRT maps
Abstract: In recent years, examples have been given of planar birational maps that preserve rational or elliptic fibrations but do not preserve each fibre as in the case of the QRT maps. We give some systematic results concerning these so-called non-QRT maps.

Speaker: Prof. Sergey Sergeev
Title: Quantization of modified KdV and sine-Gordon maps for (d,-1) traveling wave reduction
Abstract: We present the quantization scheme for the travelling wave reduction of modified KdV and sine-Gordon maps.

Speaker: Dr. Thi Dinh Tran
Title: Growth of degrees of lattice integrable equations
Abstract: In this talk  we investigate growth properties for lattice equations defined on quad graphs. Given a multi-affine lattice rule, one can write  the rule in projective coordinates.  We put initial values as polynomials in  $w$  along the horizontal and vertical axes in the first quadrant. We consider the gcd of the two polynomials  at each vertex. Loosely speaking, slow growth of degrees is associated with  very big gcd's. We give a conjecture that helps us to build a divisor of the gcd. It then gives us a recursive formula for an upper bound of the actual degree (after cancellations) at each vertex. Therefore, one can use it to prove that a certain class of integrable lattice equations including $Q_V$ has polynomial growth with some certain initial values.

Student Talks:

Speaker: Peter Langfield

Title: Isochrons in the Fitzhugh-Nagumo Model

Abstract: A point in the basin of attraction of a periodic orbit will converge to the periodic orbit with a particular asymptotic phase. An isochron is the set of all points in the basin with the same asymptotic phase. We will demonstrate how, via the continuation of suitable orbit segments, isochrons can be computed accurately and reliably. As a concrete example we determine the isochrons of the Fitzhugh-Nagumo model as studied by Arthur Winfree in 1980.

Speaker: Jennifer Creaser

Title: The Lorenz system near the loss of the foliation condition

Abstract: This talk discusses how advanced numerical methods can be used to compute and visualise the dynamics of the Lorenz system. The focus is on the point where the one-dimensional stable manifolds of the secondary fixed points flip sides and how this relates to the two-dimensional stable manifold of the origin. This occurs close to the point where the condition fails that there is a stable foliation of the return map, so that the reduction to the one-dimensional Lorenz map is no longer valid.

Speaker: Adam Rentsch

Title: $P_{III}$ and related lattice equations

Abstract: We consider discrete isomonodromic deformations as defining an equation on a lattice.