Time and location: 12pm, Thursday 25 October, WGB 304

All are welcome at the following seminar:

UCC Applied Mathematics Seminar

Speaker: Adrián Rodriguez-Sanjurjo (University College Cork)

Title: Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions

Abstract: In this talk I outline recent work providing a rigorous mathematical analysis of some nonlinear surface waves in the presence of a depth-invariant zonal current constituting a generalisation of Pollard's waves. These oceanic water waves are remarkable due to the fact that they represent exact, explicit nonlinear solutions of the geophysical f-plane equations accounting for rotational effects. By way of applying a mixture of analytical and degree-theoretical arguments, it is shown that the three-dimensional Lagrangian flow-map prescribing these solutions is a global diffeomorphism and the fluid motion is dynamically possible. This is achieved by imposing certain conditions on the physical and Lagrangian labelling parameters. Furthermore, it is also proven that parameter specifications not meeting those conditions can produce solutions that fail to be globally valid; thereby demonstrating the necessity of subjecting these Lagrangian solutions to such rigorous analytical considerations.    

Rodríguez-Sanjurjo, A. Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions. Annali di Matematica (2018) 197:1787. 

https://doi.org/10.1007/s10231-018-0749-5

Time and location: 12noon, Thursday 18 October, WGB 405

Speaker: Eoin Clerkin, TU Darmstadt

Title: Problems of stability determination for large time-periodic mechanical systems.

All are welcome to attend

Time and location: 12pm, Thursday 13 September, WGB 304

Speaker: Philipp Hövel (University College Cork)

Title: A safari tour guide for chimeras

Abstract: Systems of nonlocally coupled oscillators can exhibit surprisingly complex spatio-temporal patterns, called chimera states, that consist of coexisting domains of spatially coherent (synchronized) and incoherent (desynchronized) dynamics. First observed in systems of identical elements with symmetric coupling topology, these hybrid states have been intensively studied during the last decade.

In my talk, I will first present a number of examples of these peculiar states of partial synchrony including time-discrete maps and time-continuous models. Then, I will discuss an approach to control chimeras. The control scheme is based on targeted modifications of system parameters of a few elements. In detail, I will explore the influence of a block of excitable units on the existence and behavior of chimera states in a nonlocally coupled ring-network of FitzHugh-Nagumo elements. The FitzHugh-Nagumo system, a paradigmatic model in many fields from neuroscience to chemical pattern formation and nonlinear electronics, exhibits oscillatory or excitable behavior depending on the values of its parameters. In previous studies, chimera states have been realized in networks of coupled oscillatory FitzHugh-Nagumo elements. I will show that introducing a block of excitable units into the network may lead to several interesting effects. It allows for controlling the position of a chimera state as well as for generating a chimera state directly from the synchronous state.

Time and location: 12pm, Thursday 20 September, WGB 304

Speaker: Sajjad Bakrani (Imperial College London)

Title: Invariant Manifolds of a homoclinic orbit in a 4D system with a first integral and a Z_2 symmetry

Abstract: A homoclinic orbit is an orbit of a dynamical system which joins an equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable invariant manifold and the unstable invariant manifold of an equilibrium. These orbits play an important role in understanding the global behavior of a dynamical system. In this talk we describe the dynamics in a small neighborhood of a homoclinic orbit of a particular family of 4D systems of differential equations. In particular we provide necessary and sufficient conditions for the existence of the stable and the unstable invariant manifolds of the homoclinic orbit. We then pay a special attention to the case where these two invariant manifolds intersect.

Time and location: 12pm, Thursday 27 September, WGB 304

Speaker: Alan Compelli (UCC)

Title: Soliton propagation in a fluid of non-uniform depth

Abstract: A surface water wave over a bed of non-uniform depth is considered. The fluid is incompressible, inviscid and irrotational. The Hamiltonian is determined in terms of wave-only quantities using a Dirichlet-Neumann operator. By introducing an appropriate scaling regime, and considering the bottom to vary `slowly', a KdV equation with variable coefficients is derived. A one-soliton solution approaching a ramp on the seabed is then considered and numerical results demonstrate the effect the ramp shape has on the birth of new solitons as the soliton passes over it.

Time and location: 12pm, Thursday 4 October, WGB 304

Speaker: Bernd Krauskopf (The University of Auckland)

Title: Dynamic complexity of two coupled photonic nanocavities

Abstract: This is joint work with Andrus Giraldo, Neil Broderick and Alejandro Giacomotti.

We consider two coupled passive optical resonators in the form of photonic crystal nanocavities, which operate with only a few hundred photons. This low-photon system is described mathematically by the Bose–Hubbard model consisting of two complex ordinary differential equations for the slowly varying amplitudes of two electric fields. It was shown experimentally that this system exhibits spontaneous symmetry breaking and bistable behaviour, which is of particular interest for optical memories and logical switching. This type of dynamics has also been found in the Bose–Hubbard model, and previous theoretical work has concentrated on finding parameter regions where stable symmetric and asymmetric continuous-wave solutions exist.

We consider here the overall dynamics of the system as described by the Bose–Hubbard model. In particular, we focus on the existence of complex self-pulsations. As more energy is pumped into the system self-pulsations arise from Hopf bifurcations. These periodic solutions then change or disappear in sequences of homoclinic bifurcations. In particular, we find chaotic Shilnikov bifurcations and the appearance of chaotic attractors. Such complicated dynamics can take place in either an individual cavity or in both of them simultaneously. We present how local and global bifurcations bound regions of different dynamics in relevant parameter planes. Our global bifurcation analysis of the two coupled photonic crystal nanocavities predicts types of (chaotic) dynamics well within the range of future experiments.

Time and location: 12pm, Thursday 11 October, WGB 304

Speaker: Martin Stynes (Beijing Computational Science Research Center)

Title: Fractional-order derivatives and the numerical solution of time-fractional problems

Abstract: An introduction to fractional derivatives and some of their properties is presented. The regularity of solutions to Caputo fractional initial-value problems in one dimension is discussed; it is shown that typical solutions have a weak singularity at the initial time t=0. This singularity has to be taken into account when designing and analysing numerical methods for the solution of such problems. To address this difficulty we use graded meshes, which cluster mesh points near t=0, and answer the question: how exactly should the mesh grading be chosen? Next, initial-boundary value problems in one space dimension are considered, where the time derivative is a Caputo fractional derivative. (This is a fractional-derivative generalisation of the classical parabolic heat equation.) Once again a weak singularity appears at t=0, and the mesh in the time coordinate should be graded to compute satisfactory numerical solutions.