nuggets

$tuesday Presenter: Gary Froyland (University of New South Wales) Nugget: Efficient transfer operator analysis for fast/slow dynamics Abstract: The spectrum of the transfer operator carries important timescale information concerning relative speeds of motions in dynamical systems. The level sets of the corresponding eigenfunctions (or Oseledets functions in the time-dependent case) determine fast directions in the phase space. By projection along the fast directions, one can obtain reduced dynamics in slow directions. One practical issue is the efficient computation of the transfer operator spectrum and eigenfunctions (at least that part of the spectrum corresponding to the slowest timescales) in high dimensions. Mesh-free constructions can help, perhaps auxiliary geometric computations can lead to pre-processing methods to make the computations more efficient. *** Presenters: Cecilia Gonzales-Tokman (University of Queensland) Nugget: Advances and challenges in the detection of dominant large scale features in dynamical systems via transfer operator and related methods Abstract: We discuss recent advances in the ergodic theoretical study of non-autonomous dynamical systems, towards the detection of large-scale features responsible for the dominant long-term statistical behaviour of the system. The possible topics of discussion range from applied to theoretical questions: (i) Can we improve over existing algorithms to achieve more efficient and/or faster methods for the detection of coherent structures in e.g. models of geophysical flows? (ii) Can we quantify (or qualify) the sensitivity of existing algorithms, for example with respect to space and/or time scale resolution? (iii) Can we devise new strategies to expand the current theory to provide a wider range of rigorous examples of stability (or instability) of coherent structures and their corresponding time-scales under model/numerical/random perturbations? *** Presenters: Sanjeeva Balasuriya (University of Adelaide) Nugget: Effect of stochasticity on coherent flow structures Abstract: Fluid flows often have dominant flow structures such as Jupiter's Great Red Spot, the Gulf Stream, and blobs of unmixed fluid in well-mixed areas. Effects such as stochasticity or diffusion usually operate at spatial scales which are significantly smaller than these structures. How do these small-scale phenomena effect the large-scale structures, and in particular the transport of quantities such as heat? This issue has become particularly important in climate modeling, in which the global circulation models used are typically of low resolution (approx 100 km), and there is considerable interest in being able to incorporate effects occurring at subgrid scales into these large-scale models; this is the problem of ‘stochastic parametrisation’. In this problem nugget, I will discuss some thoughts on ways in which we might approach this issue. *** Presenter: Michael Small (University of Western Australia) Nugget: Stability and Robustness of Engineered and Natural Systems - What is the role of the designer? Abstract: Several models of complex network evolution and growth explain the formation of scale-free networks in social or naturalistic systems. The archetypal example is preferential attachment which sees the “rich get richer” as nodes are added to a network preferential attaching to the existing high degree nodes. A surprising empirical observation is that similar structures also arise in engineered complex systems. In systems with design rules and design constraints guiding the formation of local and large scale structure, scale-free and/or small-world features may still arise. A natural question then is how are these different types of systems similar or different? In particular, how do many of the properties observed in social networks (robustness/fragility and dynamical formation of chimera or cluster states, for example) manifest in engineered or designed systems? *** Presenter: Anthony Roberts (University of Adelaide) Nugget: Rigorous macroscale modelling of pattern formation in multi-D Abstract: Spatial patterns are ubiquitous. For example, they may arise in nonlinear reaction-diffusion systems, and chaotically in turbulence. But their rigorous macroscale modelling has been an open challenge for over fifty years. A microscale model of reaction-diffusion pattern resolves all details of the field u(x,t) in multi-D space. The macroscale phase equation assumes that near each point in space the structure of the pattern is approximately a ‘roll’ (\exp[i\theta(x,t)]) and the macroscale model is typically that the phase diffuses. But often the scenario is that near each point in space there is a ‘continuum’ of structures all with a critical wavenumber. The challenge is to see if the developing rigorous theory on making macroscale models in wide but thin domains can apply in multi-D as it does in 1D space. wednesday Presenters: Anmar Khadra (McGill) Nugget: Multiscale rhythms in neural systems: From stochastic single cell dynamics to network behaviour Abstract: Many neuronal systems display several hormonal and electrophysiological rhythms that occur at different time scales ranging from hours (e.g., hormone release) to minutes (e.g., calcium oscillations) to milliseconds (e.g., electrical activities). These rhythms maintain their time scales while interacting with each other to produce the proper signaling response during healthy conditions. Disruption of the time scale of any of these rhythms can lead to devastating outcomes. Burst firing in membrane potential is one of these rhythms of electrical activities exhibited by some neuronal systems. It is regulated by the slow and fast dynamics of ion channels expressed on these neurons, and mediated by noise induced by synaptic inputs and/or the intrinsic properties of these neurons. Analyzing how noise in such systems (at the single cell level) affects bursting rhythms and determining how the latter (at the network level) interacts with other rhythms that occur at a much slower time scale, remain open problems that need to be resolved. *** Presenters: Vivien Kirk and James Sneyd (University of Auckland) Nugget: Multiple timescales in models of calcium oscillations Abstract: Oscillations of calcium concentration are a vital signalling mechanism in almost all cell types, but despite decades of research, both theoretical and experimental, many fundamental questions remain unanswered. One such is the question of how long-period oscillations arise in a system where there are no obvious long time scales. Another is the question of how calcium influx from outside the cell can affect the intracellular dynamics. Our group is studying these, and related, questions in a variety of cell types. We’ll briefly talk about our latest results, both theoretical and experimental, from parotid cells, HSY cells and airway smooth muscle cells, and then discuss some hypotheses for how the shape of the cell can influence the shape of the calcium responses, and how some current widely-accepted hypotheses (from other groups, of course!) are almost certainly incorrect. *** Presenter: Bernd Krauskopf & Hinke Osinga (University of Auckland) Nugget: Slow and fast and global Abstract: Numerical methods based on two-point boundary value problem continuation have the major advantage that they remain well posed in parameter regimes where geometric singular perturbation theory does not necessarily apply. We utilise numerical findings in such parameter regimes to identify special events that characterise transitions between evolutions on slow and fast time scales. These techniques are particularly useful when studying changes in the global system dynamics, such as mixed-mode oscillations, global re-injection mechanisms, transient bursting, and phase sensitivity. As a specific example we consider the interaction between a repelling slow manifold and a global unstable manifold of an equilibrium. *** Presenters: Andy Hammerlindl (Monash University) Nugget: How prevalent is robust dynamical behaviour Abstract: n recent years, many complicated dynamical objects, such as heterodimensional cycles and homoclinic tangencies, have been shown to be robust under perturbation. This gives a chance at least that these complicated behaviours might actually be discovered in physical processes in the "real world". Is this indeed the case? *** Presenters: Holger Dullin (University of Sydney) Nugget: Regularisation in the N-body problem Abstract: It is a classical result that binary collision in the gravitational N-body problem can be regularised but that triple collisions can not. I will review theses results, along with the (non-regularising) blow-up transformation due to McGehee that is used to study orbits close to triple collision. Open problems in this area include the study of simultaneous multiple binary collision (say (12) and (34) in the 4-body problem), and the simultaneous (in space, not in time) blow-up of triple collisions. *** Presenters: Vladimir Gaitsgory (Macquarie University) Nugget: On control of systems with slow variables Abstract: Dynamical systems with slow observables have been studied by a number of researchers. However problems of control of such systems have never been discussed in the literature (to the best of our knowledge). In this talk, we will discuss results of a pilot study of the way how problems of optimal control of systems with slow observables can be approached. We will consider a possibility of approximating such problems with problems of optimal control of the averaged system, the construction of the latter being a key element of our consideration. thursday Presenters: TBA Nugget: Abstract:$