The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as a coupled phase oscillators. In this talk, an infinite dimensional Kuramoto model is considered, and Kuramoto's conjecture on a bifurcation diagram of the system will be proved.

A linear operator obtained from the infinite dimensional Kuramoto model has the continuous spectrum on the imaginary axis, so that the usual spectrum does not determine the dynamics. To handle such continuous spectra, a new spectral theory of linear operators based on Gelfand triplets is developed. In particular, a generalized eigenvalue (resonance) is defined. It is proved that a generalized eigenvalue determines the stability and bifurcation of the system.

A self-propelled systems have attracted broader interest. I will introduce following two systems in which a droplet exhibits spontaneous motion induced by interfacial tension gradient. We consider mathematical model based on Stokes equation, and compare the theoretical results with experimental ones.

(i) We proposed a theoretical framework for the spontaneous motion of a droplet coupled with pattern formation inside it in a three-dimensional system. The nonlinearity of the chemical process inside the droplet induces inhomogeneous concentration profile, which leads to interfacial tension gradient at the droplet interface. The droplet is driven by the gradient through the surrounding flow due to the Marangoni effect. We also demonstrated experiments in which Belousov-Zhabotinsky (BZ) reaction droplet is moving coupled with chemical waves inside it.

(ii) We analyze spontaneous rotation of a droplet induced by the Marangoni flow in a two-dimensional system. The droplet with the small particle which supplies a surfactant at the interface is considered. We calculated flow field around the droplet using the Stokes equation and found that advective nonlinearity breaks symmetry for rotation. Theoretical calculation indicates that the droplet spontaneously rotates when the radius of the droplet is an appropriate size.

Reference: H. Kitahata, N. Yoshinaga, K. H. Nagai, and Y. Sumino, Phys. Rev. E, 84, 015101 (2011). H. Kitahata, N. Yoshinaga, K. H. Nagai, and Y. Sumino, Chem. Lett., 41, 1052 (2012). K. H. Nagai, F. Takabatake, Y. Sumino, H. Kitahata, M. Ichikawa, and N. Yoshinaga, Phys. Rev. E, 87, 013009 (2013).

The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.

This work is joint work with U Thiele, A J Archer, M J Robbins and H Gomez.

Ferro-solitons and patches of localised patterns have been observed experimentally on the surface of a magnetic fluid in the presence of a uniform vertical magnetic field by Richter and Barashenkov (PRL 2005). In this talk, I will present some recent experimental results of Richter and link these with some analysis and numerical results of the equations governing the free-surface. It turns out that the system possesses an energy formulation and for the linear magnetisation law case a Hamiltonian formulation exists. This Hamiltonian formulation is then used to do Normal form analysis for one-dimensional interfacial localised patterns. We then present numerical results for these patterns and trace-out a two-parameter bifurcation diagram depicting the existence of one-dimensional interfacial localised patterns. Finally, we present some preliminary results looking at a general nonlinear magnetisation law and two-dimensional interfacial localised patterns.

Some brief examples of systems where the motion of vortices play an important role in the dynamics will first be discussed to motivate the remainder material to be presented. Following this, the equations of motion governing the motion of point vortices in multiply-connected domains (for inviscid, incompressible fluids) will be introduced. A method for numerically calculating equilibria in multiply connected domains, applied to a Kasper Wing configuration, will then be presented and various properties of the flow (for example, the lift on the airfoils) will be analyzed. Time permitting, the motion of finite patches of constant vorticity in multiply-connected domains will then be introduced and used to study the robustness of the Kasper wing equilibria.

Many planar spatially extended systems exhibit localized standing structures such as pulses and oscillons. In particular, such structures can emerge at Turing and forced Hopf bifurcations. In this talk, I will give an overview of these mechanisms and show how geometric blow-up techniques can be used to analyze them: among the findings is the bifurcation of localized structures that have significantly larger amplitudes than expected from formal considerations. This is joint work with Kelly McQuighan, David Lloyd, and Scott McCalla.

We present results for an advection-reaction-diffusion model describing malignant tumour (i.e. skin cancer) invasion. Numerical solutions indicate that both smooth and shock-fronted travelling wave solutions exist for this model. We verify the existence of both type of these solutions using techniques from geometric singular perturbation theory and canard theory. Moreover, we provide numerical results on the stability of the waves and the actual observed wave speeds.

This is joint work with K. Harley, G. Pettet, R. Marangell and M. Wechselberger.