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Tuesday
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Presenter(s): Kirk V and Sneyd J (Auckland)
Nugget (S1): Spatially separated feedback systems
Abstract: An enormous amount of work has been done on oscillations and waves in feedback systems in well-mixed systems (think of, for example, lambda-omega systems or the FitzHugh-Nagumo model). We even understand well how to use homogenisation to model oscillations and waves in systems where the spatial heterogeneities are on a small scale (calcium oscillations, for example). However, a recent problem to do with calcium oscillations and waves in parotid acinar cells has led us to consider feedback systems where the various feedback components are separated by significant distances. From numerical results we know that such spatial separation imposes drastic constraints on the type of model that can be used to describe such oscillations, but we have no theoretical understanding of why this is so.
Can bifurcation theory and continuation be used in such cases? Is there any existing general theory on spatially separated oscillating systems?
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Presenter(s): Roberts AJ (Adelaide)
Nugget (S1): Model slowly varying, nonlocal, heterogeneous systems in long `cylindrical' domains
Abstract: The slowly-varying/thin-layer assumption empowers understanding of many physical scenarios. Let's explore the case when the microscale is nonlocal and/or heterogeneous. Extant mathematical approximation methodologies are typically self-consistency or limit arguments as the aspect ratio becomes unphysically infinite. The new approach is to rigorously analyse the dynamics local to every cross-section, via Lagrange's remainder theorem, and expressed in terms of the local moments of the microscale heterogeneity. Centre manifold theory supports the local modelling of the system's dynamics with coupling to neighbouring locales treated as a non-autonomous forcing. The union over all cross-sections then provides powerful new support for the existence and emergence of a centre manifold model global in the long domain, albeit finite sized. The approach promises to quantify the accuracy of known approximations, extend such approximations to mixed order modelling, and open previously intractable modelling issues to new tools and insights.
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Presenter(s): Operator Convergence for Slow-Fast Systems
Nugget (S2): Hammerlindl A (Monash)
Abstract: For slow-fast systems, some rigorous results have been established
in the limit as the time separation tends to infinity. For instance, in some
cases, paths converge in a weak sense to those of the reduced systems. In many
applications, however, we would like to know convergence properties of the
spectrum and eigenfunctions of operators associated to the system. I will
discuss what is known and open problems in this setting.
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Presenter(s): Lustri C (Macquarie)
Nugget (S2): Nonlocal solitary waves in singularity-perturbed systems
Abstract: If a singularly perturbed problem has a leading-order solution that contains travelling waves, the full solution often contains exponentially-small non-decaying solitary waves, known as nanoptera. I will show two problems which demonstrate these features: the singularly-perturbed fifth-order KdV equation, and discrete period-2 dimer chains.
I will outline how this behaviour can be captured using exponential asymptotics. This requires analytically continuing the leading-order travelling wave, and extracting information from singular points of the analytic continuation. Finally, I will describe some challenging systems that I am currently studying, and I will present two key questions arising from these systems: the first relates to finding analytically-continued singular points of functions that must be approximated, and the second involves explaining this behaviour by studying spectral properties of the system.
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Wednesday
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Presenter(s): Vasil G (Sydney)
Nugget (S3): On mostly disordered behaviour:
Abstract: Fluid mechanics is notable for the rich range of dynamics found in seemingly simple setups. On the one hand, fluids can remain coherent and laminar. These states allow for efficient transport and persistent large-scale features. On the other hand, the natural state of many fluids is highly turbulent; with intricate multi-scale structure in both space and time. I will discuss some of the recent insights into systems that are not fully turbulent, and definitely not laminar. This middle ground between fully ordered and fully chaotic contains many surprises. I will discuss some of the statistical properties of such systems, as well as their ability to form order in surprising ways. The possible systems could include (i) Strong-field solar magnetoconvection; (ii) Active biological matter; (iii) Vortex-wave interactions in a stratified ocean; (iv) Thermalisation of a large pendulum chain (v) Nonlinear solitons on a topologically complex graph domain; and/or several others.
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Presenter(s): Denier J (Macquarie)
Nugget (S3): Local versus Global Behaviour in Hydrodynamic Stability
Abstract: Many physical and technologically important fluid flows exhibit a transition from a smooth (laminar) state to a chaotic (turbulent) state. Although there has been significant progress in understanding this transition process, a number of key challenges remain, presenting some significant mathematical challenges. One that we have been focussing on can be couched in terms of how, and when, do local changes to a fluid flow (through, for example, an small external disturbance) change the global flow structure. Many of the challenges here can be couched in terms of asymyptotic analysis of system varying both spatially and temporally (and not necessarily with slow variation in either time or both). I'll motivate these challenges through some recent work (both theoretical, computational and experimental) on transition in three dimensional fluid flows. This "nugget" will throw out more challenges than provides solutions!
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Presenter(s): Balasuriya S (Adelaide)
Nugget (S4): Expectation and variance of density fields due to stochastic perturbations
Abstract: There are many applications in which observed unsteady velocity data is processed to track coherent entities (e.g., ocean/atmospheric eddies and cyclones). A pertinent emerging question is how the presence of uncertainties in this data---which is inevitable because of resolution---affects conclusions. I have recently investigated an aspect of this uncertainty using a stochastic differential equations (SDE) approach, and quantified the expectation and variance of the divergence from the deterministic trajectory in the limit of small noise. Along with Georg Gottwald, I have also established via SDE simulations that this is linked to mixing across deterministic flow barriers. In this Problem Nugget, I speculate on whether/how the SDE expectation and variance results can inform us of transport characteristics of scalar fields such as a pollutant concentration or heat. In other words, what conclusions can be reached for the associated Fokker-Plank or advection-diffusion PDE, which model how a scalar density evolves?
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Presenter(s): Froyland G (UNSW)
Nugget (S4): FEM-based numerics for approximating the dynamic Laplacian and extracting coherent sets
Abstract: Transport and mixing properties of aperiodic flows are crucial to a dynamical analysis of the flow, and often have to be carried out with limited information. Finite-time coherent sets provide a skeleton of distinct regions around which more turbulent flow occurs. In the purely advective setting this is equivalent to identifying sets whose boundary interfaces remain small throughout their evolution. These sets manifest in geophysical systems in the forms of e.g. ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking even more challenging. I will describe an FEM-based numerical method to efficiently approximate the dynamic Laplace operator, and rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data.
This is joint work with O. Junge (TU Munich).
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Presenter(s): Gonzales-Tokman C (UQ)
Nugget (S5): Ergodic theory for non-autonomous dynamical systems
Abstract: The development of multiplicative ergodic theorems for transfer operators in the past decade has opened up the way for a spectral approach to investigate dynamical properties of a range of non-autonomous systems. For example, results include existence and stability of random physical invariant measures and various limit theorems. Current analytical and numerical challenges for discussion include: (i) expanding the scope of application of these results to a broader class of non-autonomous systems, and (ii) further understanding and developing the connection between so-called coherent structures and Oseledets ‘modes’.
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Presenter(s): Fish A (Sydney)
Nugget (S5): Quantitatively pleasant actions on Z^d
Abstract: TBA
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Thursday
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Presenter(s): Small M and Stemler T (UWA)
Nugget (S6): Structure dictates dynamics
Abstract: It is an open question how the edge structure of a network dictates the dynamics of oscillators representing the network's nodes. Over the recent years focusing on simple collective dynamics has made some progress. The collective dynamics here is the synchronization of the nodes' dynamics.
Schulz et al. investigated how edge addition in a simple tree network changes the Laplacian and consequently can change the control parameter regime for which synchronization is possible. They developed strategies and a classification of edges to choose the right kind of edge addition leading to an extended synchronization regime and determine its stability.
On the other hand Zou et al. studied explosive synchronisation in isolated star networks. They showed that coupling between the leaf nodes and the hub could be understood in such a way that instability and hysteresis emerge quite naturally — precisely the setting required for so-called ``explosive synchronisation’’. Alternatively we (Small et al.) found that almost all scale-free networks could be treated as an (almost) disjoint union of stars. But more than that, as ubiquitous agents within scale-free networks they offer a natural explanation for emerging dynamics. Coupled to that, even pathological network construction methods (such as that of Barabasi and Albert) — while not yielding an almost disjoint union of stars, do yield a rich-club of hubs, which we can perhaps model as a single hub.
These insights therefore result from quite different approaches to our problem. Schulz looks for microscopic manipulations changing the dynamics, while Zou explains the role of macroscopic network parts on the dynamics. Questions still remain on how to expand and maybe combine these techniques to understand more general kind of dynamics on networks.
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Presenter(s): Altmann E (Sydney)
Nugget (S6): Dynamics on networks: beyond mean field
Abstract: Some of the most paradigmatic dynamical systems on networks involve nodes that switch between two states depending on the state of their neighbours. An understanding of the properties of the system for large number of nodes is obtained approximating the dynamics of the network by a set of ordinary differential equations. I will discuss recently proposed methods to obtain such approximations, with particular interest in cases in which the traditional mean field approximations are inaccurate. Improved approximations include dynamical correlations beyond nearest neighbours and networks with heterogeneous degree distributions. A point for discussion is the effect of clustering and communities in the network, properties typically observed in real-world networks but that are not included in the approximations.
Reference:
"Binary-state dynamics on complex networks: Pair approximation and beyond", JP Gleeson, Physical Review X 3, 021004 (2013)
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Presenter(s): Gottwald G (Sydney)
Nugget (S7): Model Reduction in Complex Network Dynamics
Abstract: Model reduction of complex network dynamics such as the Kuramoto model are challenging as often real-world networks are finite in size and the finitude entails important nontrivial effects. I present a novel method to capture finite-size effects in a reduced model of a stochastic Kuramoto model using collective coordinates. In particular, I will show how to quantitatively describe the Brownian drift of a synchronized cluster (an effect which vanished in the thermodynamic limit). I will point towards some open questions in applying this framework to local synchonization where some but not all of the oscillators form a synchronized cluster.
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Presenter(s): Tzou J (Macquarie)
Nugget (S7): Narrow escape optimisation and localised pattern formation- are they related?
Abstract: Narrow escape problems ask how long on average a random walker takes before it first reaches a set of targets, the sizes of which are small in relation to the search domain. How should we optimally distribute stationary targets within the search domain in order to minimise the average search time? Do mobile targets lead to a smaller average search time, and if so, how should the targets move? Studies of localised patterns involve determining the dynamics and stability of spot patterns. For mutually repelling spots on a finite domain, at what locations do the spots settle in equilibrium? Is the equilibrium stable, and if not, how do the spots evolve in time? For special simple cases, we have found that there is a close correlation between the two sets of questions. Is there a physical reason behind it? If so, how can we exploit this connection
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Presenter(s): Osinga H and Krauskopf B (Auckland)
Nugget (S8): Is there more to chaos?
Abstract: We discuss two examples that feature higher-dimensional forms of chaos: (1) A hetero-dimensional cycle between a pair of saddle periodic orbits in a four-dimensional Atri model of intracellular calcium dynamics; (2) A so-called blender, in a family of three-dimensional Hénon-like maps, which has the property that its invariant manifolds behave as geometric objects of a higher dimension. Hetero-dimensional cycles and blenders are closely related and give rise to robust homoclinic tangencies. We illustrate these geometric objects in a parameter-dependent setting and discuss their relevance for
applications.
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Friday
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Presenter(s): Dullin H (Sydney)
Nugget (S9): Monodromy in the Kepler Problem
Abstract: What could possibly be said about the Kepler Problem that is new? It is well known that this superintegrable system can be separate in different coordinate systems, and each such separation defines a distinct Liouville integrable system. We show that for separation in prolate spheroidal coordinates the resulting integrable system has Hamiltonian monodromy, which means that the action variables cannot be globally smoothly defined. This has interesting consequences for the quantum mechanics of the problem, which we illustrate. Similar analysis can be done for many prominent superintegrable systems,
for example the harmonic oscillator, the free particle or the geodesic flow on the sphere.
https://arxiv.org/abs/1612.00823
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Presenter(s): Marangell R (Sydney)
Nugget (S9): PDEs on the Torus: Hill's equation and the Evans function
Abstract: TBA
