Table of contents

It is shown that there is an exact reduction from the full three-dimensional inviscid water wave problem with vorticity to a set of hyperbolic equations for the free surface horizontal velocity field. The key term in the reduced equations is g + (Dw/Dt)|^S where g > 0 is the gravitational constant and (Dw/Dt)|^S is the Lagrangian vertical acceleration at the surface. Vertical accelerations—both Eulerian and Lagrangian—at the surface are widely used as a diagnostic for wave breaking and this reduction gives a precise theoretical argument for how accelerations drive the surface dynamics. When the acceleration term is small, the surface equations reduce to the model equation proposed in Pomeau et al (2008 Nonlinearity21 T61–T79, Proc. R. Soc. Lond. A 464 1851–66) which provides a universal law for spreading of the crest during wave breaking, and when the acceleration term is negative the equations can switch type (hyperbolic to elliptic) and solutions can fail to exist. Addition of surface tension leads to dispersive regularization of the surface equations.

Consider the map T ↦ ω(T) on the collection of unimodal transformations, where ω(T) is the ω-limit set of T. The collection of unimodal transformations is endowed with the topology of uniform convergence, and the collection of subsets of [0, 1] is endowed with the Hausdorff metric. Conditions on a unimodal transformation T implying the continuity of the map S ↦ ω(S) at T are investigated.

We study finite time blow-up solutions of the Chern–Simons–Schrödinger system. In particular, explicit blow-up solutions are constructed by observing the pseudo-conformal invariance and finding solutions of the self-dual equations.

In this paper, a delayed neural network with unidirectional coupling is considered which consists of two two-dimensional nonlinear differential equation systems with exponential decay where one system receives a delayed input from the other system. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the centre manifold theorem. We also investigate the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay-differential equations combined with representation theory of Lie groups. Then the global continuation of phase-locked periodic solutions is investigated. Numerical simulations are given to illustrate the results obtained.

In this paper we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schrödinger equation at time scales α_h tending to infinity as the semiclassical parameter h tends to zero (when α_h = 1/h this is equivalent to considering solutions to the non-semiclassical Schrödinger equation). Some general results are presented, among which is a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with periodic geodesic flow) via averaging formulae relating the semiclassical measures corresponding to the evolution to those of the initial states. The case of the flat torus is also addressed; it is shown that non-classical behaviour may occur when energy concentrates on resonant frequencies. Moreover, we present an example showing that the semiclassical measures associated with a sequence of states no longer determines those of their evolutions. Finally, some results concerning the equation with a potential are presented.

We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painlevé II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair and the corresponding Riemann–Hilbert problem. In addition, we evaluate the total integral of a function related to a special solution to the Painlevé V equation. As a corollary, we obtain short proofs of the computation of the constant terms of the limiting gap probabilities in the edge and the bulk of the Gaussian Orthogonal and Gaussian Symplectic Ensembles that were obtained recently in (Baik et al 2008 Commun. Math. Phys.280 463–97, Ehrhardt 2007 Commun. Math. Phys.272 683–98). We also evaluate the total integrals of certain polynomials of the Painlevé functions and their derivatives. These polynomials are the densities of the first integrals of the modified Korteweg-de Vries equation. We discuss the relations of the formulae we have obtained to the classical trace formulae for the Dirac operator on the line.

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L² × H^−1/2 × H^−3/2. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L²-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schrödinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data—a result which is false for the cubic nonlinear Schrödinger equation in dimension two—and it is optimal because Glangetas–Merle's solutions blow up at that time.

When one considers a Poincaré return map on a general unbounded (n − 1)-dimensional section for a vector field in there are typically points where the flow is tangent to the section. The only notable exception is when the system is (equivalent to) a periodically forced system. The tangencies can cause bifurcations of the Poincaré return map when the section is moved even if there are no bifurcations in the underlying vector field. The interaction of invariant manifolds and the tangency loci on the surface gives rise to discontinuities of the Poincaré map and there can be open regions where the map is not defined. We study the case of a four-dimensional phase space . Specifically, we make use of tools from singularity theory and flowbox theory to present normal forms of the codimension-one tangency bifurcations in the neighbourhood of a tangency point.

Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark–Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of 'rotational' periodic solutions that display lens-chain structures for a general N-dimensional map. We then unfold the codimension-two, shrinking point bifurcation, where the tongues have zero width. A number of codimension-one bifurcation curves emanate from shrinking points and we determine those that form tongue boundaries.

In this paper the perturbations of F₄(x, y) = (y, −x² + 4x) are considered. The existence of infinitely many critical homoclinic orbits for F₄ makes this mapping a bifurcation point involving diverse dynamical structures. It is proved that there exist infinitely many codimension one submanifolds in the space of C² endomorphisms, each of which accumulates on F₄, is contained in the boundary of the set of hyperbolic mappings and is determined by the continuation of one of the homoclinic orbits of F₄.

This paper is devoted to the study of spatial dynamics of a class of periodic integro-differential equations which describe the population dispersal process via a dispersal kernel. By appealing to the theory of asymptotic speeds of spread and travelling waves for monotonic periodic semiflows, we establish the existence of the spreading speed c^* and the nonexistence of continuous periodic travelling wave solutions with wave speed c < c^*. We also prove the existence of left-continuous periodic travelling waves with wave speed c ⩾ c^*. In the autonomous case, the continuity of monotonic wave profiles with wave speed c ⩾ c^* is obtained.

We consider a one-parameter family of area preserving maps in a neighbourhood of an elliptic fixed point. As the parameter evolves hyperbolic and elliptic periodic orbits of different periods are created. The exceptional resonances of order less than 5 have to be considered separately. The invariant manifolds of the hyperbolic periodic points bound islands containing the elliptic periodic points. Generically, these manifolds split. It turns out that the inner and outer splittings are different under suitable conditions. We provide accurate formulae describing the splittings of these manifolds as a function of the parameter and the relative values of these magnitudes as a function of geometric properties. The numerical agreement is illustrated using mainly the Hénon map as an example.

The 2009 European Conference on Complex Systems will take place 21–25 September 2009 at the University of Warwick in the UK.

Local Organising Committee Markus Kirkilionis (Warwick, Chair), Francois Kepes (Genopole, Programme Chair), Robert MacKay (Warwick), Robin Ball (Warwick), Jeff Johnson (Open University).

International Steering Committee Markus Kirkilionis (Warwick; Chair 2008–10), Fatihcan Atay (Leipzig), Jürgen Jost (Leipzig), Scott Kirkpatrick (Jerusalem), David Lane (University of Modena and Reggio Emillia), Andreas Lorincz (Hungarian Academy of Sciences), Denise Pumain (Sorbonne), Felix Reed-Tsochas (Oxford), Eörs Szathmáry (Collegium Budapest, Hungary), Stephan Thurner (Wien), Paul Verschure (Barcelona), Alessandro Vespignani (Indiana, ISI), Riccardo Zecchina (Torino).

Main tracks and Organisers Policy, Planning & Infrastructure: Jeff Johnson (Open University, Chair), Arnaud Banos (Strasbourg) Collective Human Behaviour and Society: Felix Reed-Tsochas (Oxford, Chair), Frances Griffiths (Warwick), Edmund Chattoe-Brown (Leicester) Interacting Populations and Environment: TBA Complexity and Computer Science: András Lörincz (Eötvös Loránd University), Paul Verschure (Zürich) From Molecules to Living Systems: Mark Chaplain (Dundee, Chair), Wolfgang Marwan (Magdeburg) Mathematics and Simulation: Holger Kantz (Dresden, Chair), Fatihcan Atay (Leipzig), Matteo Marsili (Trieste).

Deadlines Paper submission: 31 March 2009 with decisions 15 May 2009. Paper submission deadline likely to be extended. See http://www.eccs09.info for more information. Meeting registration: early registration July 2009; last assured chance 1 Sept.

Further information For contacts and the most up-to-date information visit http://www.eccs09.info.