Kevin Stitely and Rodrigues Bitha (University of Auckland)

"Mapping chaotic switching in quantum and laser systems"

Coupled optical systems offer a flexible platform to observe a diverse range of rich nonlinear dynamics. Often, the vector fields that describe the dynamics of such systems have a natural symmetry which can lead to either delocalization or symmetry breaking. For example, it was recently shown that a wide range of interesting behaviors including bistability, symmetry breaking, and chaotic switching, can be observed in these systems.

We consider symmetry reformation leading to delocalized chaotic attractors, which are manifested as complex patterns of switching between symmetry-related regions of phase space. We present two case studies: a laser-driven quantum gas coupled to an optical resonator and a ring resonator with coupled light. A symbolic dynamics approach is used to identify various connecting orbits that organise the different observed switching patterns.

Gary Froyland (University of New South Wales, Sydney)

"What is turbulence and how does it happen?"

I will briefly report on recent work that can detect dynamic regime changes from chaotic/mixing dynamics to regular/coherent dynamics, including those regions of phase space that are undergoing change, in a single spectral computation. This leads to a broader question of what is turbulence and how does it happen?

Kamil Bulinsky (University of Sydney)

"ε-dense images of large sets under a semigroup action"

A theorem of Glasner states that if Y is an infinite subset of the torus (real numbers mod 1) then for each ε>0 there exists an integer n such that nY is ε-dense. One can ask whether this holds for other semigroup actions: We say an action of a semigroup G on a compact metric space X is Glasner if whenever Y is an infinite subset of X then for each ε>0 there exists g in G such that gY is ε-dense in X. In particular, many matrix groups acting on the multi-dimensional torus are known to be Glasner. We pose a problem asking whether a cartesian product of Glasner actions has a natural Glasner-like property for sets where no two points are on a common vertical or horizontal line. We provide some examples where this is true.

Claire Postlethwaite (University of Auckland)

"Regular and irregular behaviour near heteroclinic networks"

Heteroclinic networks can be thought of an embedding of a directed graph into a dynamical system, where vertices correspond to equilibria, and directed edges to heteroclinic trajectories. The dynamics of trajectories near heteroclinic networks can be incredibly rich: trajectories may visit all the equilibria in the network, or only a subset of them; the order in which equilibria are visited may be regular, or apparently chaotic. In this talk I will present some results showing intricate behaviour near heteroclinic networks, with both `regular' and `irregular' sequences of visits to the equilibria. Questions of interest include: (a) can we compute the basins of attraction of different types of regular or irregular behaviour? (b) what are possible ways in which the irregular behaviour might be analysed?

Courtney Quinn (University of Tasmania)

"Finite-time analysis of regime shifts and crises in chaotic attractors"

In many applications, particularly those of weather and climate, there is a greater interest in "local" behaviour as opposed to asymptotic. For instance, if a chaotic attractor has imbedded regimes, a transition between regimes could have drastic impacts on the tangible effects in the physical system. In order to understand such transitions, we have begun to consider various local dynamical measures such as finite-time Lyapunov exponents, alignment of covariant Lyapunov vectors, and dimension measures based on the former two. Such measures are perhaps not only useful in regime transitions, as recent study shows they may indicate crises in systems of multiple chaotic attractors.

Rua Murray (University of Canterbury, Christchurch)

"Thoughts on the role of Reproducing Kernel Hilbert Spaces in computational dynamics"

RKHS are in common use in a variety of areas of machine learning, and are attractive for both theoretical and practical reasons. They are prevalent in many variants of Dynamic Mode Decomposition (underpinning computations with Koopman operators), and some authors have explored their use for Perron-Frobenius operators. I'd like to explain why I think they hold such promise (easy to handle duality between measures and observables, stochastic dynamics defined via joint probabilities or data, alternate path around the "curse of dimensionality").

Lauren Smith (University of Auckland)

"Data assimilation for networks of coupled oscillators"

Many natural phenomena and engineering applications can be described as networks of coupled oscillators, for example, neurons in the brain and the dynamics of the power grid. Here we discuss the challenges and some recent progress towards applying data assimilation methods (commonly used for weather forecasting) to such networks. The main aim is to use noisy observational data to improve estimates of both state variables and model parameters in the case where not all nodes are observed.

Andy Hammerlindl and Warwick Tucker (Monash University)

"Blenders – A computational approach"

Our aim is to develop methods to rigorously detect certain geometric structures in systems that are known to imply chaos, and that are robust under C¹-perturbations. Such structures include blenders and robust heterodimensional cycles and homoclinic tangencies. An important step is to formulate the necessary conditions in an explicit (yet flexible), computer-friendly manner. Another challenge is to make the computations as automated as possible, with minimal need for human tweaking of the problem. We will then have the potential to handle non-trivial examples including Poincaré maps of high-dimensional ODEs.

Tony Humphries (McGill University, Montreal)

"State-dependent delay differential equations with threshold delay"

Threshold delays arise naturally in a wide variety of dynamical systems, including maturation and transport processes. When the speed of the process depends on the state of the system the delay is state-dependent. Moreover, even though the delay may be discrete, its evaluation depends on the solution over the whole delay interval. These can therefore be thought of as distributed delay problems, or functional differential equations, and standard off the shelf numerical methods cannot be directly applied. They can also be thought of as Delay Differential Algebraic Equations, by differentiating the threshold condition. We will discuss some of the issues that arise in numerical and analytical treatments.

Neil Broderick and Bernd Krauskopf (University of Auckland)

"Delay dynamics of coupled photonic devices"

Applications of photonics, including memory storage and information processing, require optical devices to be coupled in small to medium-sized networks. The development here is towards on-chip integration and the use of very few photons. We will present examples of such systems near the boundary between the classical and quantum worlds, and demonstrate that delays in the coupling play an important role for their dynamics. From a mathematical perspective, DDE models of photonic systems emerge as motivating case studies for the exploration of resonance phenomena arising from an interplay between intrinsic oscillations and delayed feedback.

Chantelle Blachut and Sanjeeva Balasuriya (University of Adelaide)

"Tracking the evolution of coherent structures during anomalistic conditions"

Effective characterisations of the evolution of Lagrangian coherent structures require the ability to track structures as they shift through time. This becomes difficult under the occurrence of anomalistic events, such as Sudden Stratospheric Warmings (SSWs) which drastically change the spatial structure of the Southern Polar Vortex. Crucially, these are often the periods of greatest meteorological interest. This Nugget poses the question of how best to consistently identify these structures – obtained here in terms of singular vectors of transition matrices – through time using a sliding window approach. Two distinct SSW events (2002 and 2019) are examined along with a baseline (1999) case.

Carlo Laing (Massey University)

"Periodic solutions in theta neuron networks"

We present a mean field description of a network of spiking neurons which takes the form of a complex-valued integro-differential equation on a periodic spatial domain. For some parameter values this network supports time-periodic solutions. We show a computationally efficient way to study these solutions, using both the form of the network coupling, and the fact that the dynamics are governed by a Ricatti equation. We do this by deriving a self-consistency equation for the current applied to neurons, rather than the dynamic variables of the network. Our approach can also be used to find periodic solutions in networks with delayed interactions at no extra computational cost, and applied to other mathematically similar networks.

Holger Dullin (University of Sydney)

"The full n-body problem for symmetric bodies"

In the "full" n-body problem instead of point masses extended bodies are considered. Thus each body is a rigid body, that interacts with all others through their gravitational interaction, which depends on the shape and orientation of the bodies, not just the distance and the masses. Already the simples possible questions about the relative equilibria of the system where all bodies are evolving by the same rotation is not understood for more than 2 bodies. We recently considered a slightly simpler case where each body is assumed to be rotationally symmetric. I will discuss how to write down nice equations of motion for this symmetric system, report on some results regarding relative equilibria, and point out some open problems.

Priya Subramanian (University of Auckland)

"Some new questions about imperfect patterns"

The lens of pattern formation allows us to understand, predict and analyse self-organisation/emergence in diverse dissipative physical systems. I want to use my nugget talk to discuss some new avenues that will allow us to
( i )   obtain, i.e., obtain patterns with defects as solutions to our PDE model,
( ii)   characterise, i.e., quantitatively measure local order/disorder for a given pattern, and
(iii)   analyse, i.e., explain the observed existence/stability range of parameters,
patterns with imperfections using this approach.

Jason Atnip (University of New South Wales, Sydney) and Cecilia Tokman (University of Queensland, Brisbane)

"Thermodynamic formalism for random open and closed dynamical systems"

In this nugget we summarize our work building a complete thermodynamic formalism for random open and closed dynamical systems. We construct equilibrium states for both open and closed random systems for which we prove an exponential decay of correlations. We then use our thermodynamic tools to prove an extreme value law which gives the distribution of the maxima of an observable exceeding a (not necessarily) random threshold. We also give formula for the Hausdorff dimension of the surviving set with applications to number theory. Finally, we discuss future directions and challenges in the field.

Ian Lizarraga and Martin Wechselberger (University of Sydney)

"Existence and stability of shock waves in regularised reaction-nonlinear diffusion models"

Wave fronts with sharp interfaces are ubiquitous in nature, and reaction-nonlinear diffusion (RND) PDEs are one way to model this front formation. Indeed, such RND models can exhibit shock-fronted travelling wave solutions, but there is a major caveat: these solutions are not unique! Luckily, higher order regularisations come to save the day which come in two flavours: non-local and viscous relaxation.

In this nugget, we will discuss (a) the utility of geometric singular perturbation theory (GSPT) in constructing such solutions with sharp interfaces when high-order regularising terms are present, and (b) how GSPT-style ideas (like fast-slow splittings of the eigenvalue problem) can be used to study the spectral stability of such waves. We will highlight the many challenges that one faces with these regularised RND models, which hopefully will lead to a lively discussion.

Anna Aksamit (University of Sydney)

"Incomplete information in optimal stopping games"

I will consider a couple of models related to an optimal stopping problem where a gain process is driven by Brownian motion or a Poisson process. I will describe problems that arise from considering incomplete/asymmetric information between the players in these models.

Nathan Duignan (University of Sydney)

"Measures of (non)-integrability"

Orbits of a dynamical system can be deemed chaotic or regular. Many computational methods have been devised to distinguish between regular and chaotic dynamics. If the phase space consists entirely of regular orbits, this is a sign that the dynamical system is integrable. In the design of magnetic confinement devices for fusion energy, it is often a requirement that the magnetic field lines form an integrable system, or are as close to integrable as possible. Consequently, designing a magnetic confinement device relies on devising a useful measure of how (non)-integrable a given magnetic field is. In this nugget, we will discuss some of the methods for distinguishing between regular and chaotic dynamics. Ultimately, we pose the question on how to turn these methods into a measure of (non)-integrability.

Jae Min Lee (University of Sydney)

"Geometric investigations of hydrodynamic stability"

One can investigate the dynamics of a conservative systems of continuum mechanics in a geometric perspective by recasting the corresponding PDE as a geodesic on an infinite-dimensional Lie group endowed with an invariant metric. This program, so-called geometric hydrodynamics, was initiated by the pioneering work of V.I. Arnold in 1966. Geometry helps us to study the qualitative behavior of the solutions by using our intuition from finite dimensions. For example, Arnold gave a heuristic argument of the impossibility of weather prediction because the atmospheric configuration space is negatively curved, which will create an exponential growth of errors in the initially measured velocity field. In this talk, I will share recent progress on the relationship between conjugate points along the geodesics of 2D stationary Euler flows and their stability.