Tuesday
******** Session 3 ********
Presenter: 
Nick Cranch (University of Sydney) 
EPoster: 
"Random feature maps" 
Abstract: 
Random feature maps have emerged in the last 15 years as a cheap yet accurate machine learning method. Arbitrary functions can be learned via linear regression after applying a randomised affinelinear transformation and nonlinear activation to the training data. These mappings possess the universal approximation property and can be applied to a wide variety of supervised learning tasks. This research is concerned in the long term about extending applications of random feature maps to climate forecasting, however we became interested in the question of exactly which features were being selected by models. We observe high consistency from features whose parameters drawn from a simple convex region, allowing training on general problems to be performed without expensive trialanderror tuning of hyperparameters. 
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Presenter: 
Dana C'Julio (University of Auckland) 
EPoster: 
"Transition to a blender in a threedimensional Hénonlike map: in search of wild chaos" 
Abstract: 
Wild chaos is a new type of chaotic dynamic that can only arise in systems of sufficiently high dimensions. The characterising feature of blenders is that they admit invariant manifolds that behave as geometric objects of higher dimensions than expected from the dimensions of the manifolds themselves. This may imply the existence of complicated and robust heterodimensional connections between different parts of phase space, which can lead to wild chaos.
We construct an (non)orientable threedimensional Hénonlike map that has the ingredients to exhibit a blender. With advanced numerical techniques, we compute the onedimensional stable and unstable manifolds of this map and characterise the geometric properties of intersections of the manifolds with a plane. The intersections feature a selfsimilar structure that brings about the generation or disappearance of blenders. This allows us to estimate the parameter values for which a blender arises. 
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Presenter: 
Lloyd Lee (University of Auckland) 
EPoster: 
"The effect of calcium influx on intracellular signalling" 
Abstract: 
Cells utilise calcium signals to communicate and perform cellular functions. The generation of calcium signals is the result of interactions between various calcium fluxes, with calcium influx across the cell's plasma membrane being one of the most important. I will talk about how mathematical modelling coupled with bifurcation analysis can be used to investigate the role of calcium influx in cell signalling.

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Presenter: 
Andrew Cook (Monash University, Melbourne) 
EPoster: 
"Fixed points of 1dimensional neural networks" 
Abstract: 
Motivated by the study of fixed points of neural networks, my research studies a class of functions on the real line that is closed under composition and contains all nonconstant affine functions. It turns out that any function in this class either has at most three fixed points, or its set of fixed points is an interval. We use Schwarzian derivatives to show that functions such as the hyperbolic tangent function, or the logistic sigmoid function are members of this class. 
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Presenter: 
Damien McLeod (University of Sydney) 
EPoster: 
"Spherical Harmonics of S^{3}" 
Abstract: 
Spherical harmonics are a well studied area with fundamental application in physics and engineering. While for S^{2} effectively only one natural spherical coordinate system arises, in higher dimensions there is an abundance of choice. We compare coordinate system choices in S^{3} – the standard hyperspherical and the Hopf coordinate system – and investigate the distribution of eigenvalues in the joint spectrum of commuting operators induced by the coordinate system. 
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Presenter: 
Tom Miller (University of South Australia, Adelaide) 
EPoster: 
"Shocks and negative diffusion in reaction diffusion equations" 
Abstract: 
Reactiondiffusion equations have many applications in areas such as chemical physics, population dynamics and biomedical processes, and are used to describe how the density or concentration of something varies in space and time. Usually, the diffusion is positive which causes the concentration to disperse, but in practice the density can sometimes aggregate. This can be modelled by including a nonlinear diffusion that is negative for some values of the concentration.
While negative diffusion can cause problems such as making the problem illposed and requiring regularisation to solve numerically, it can also lead to features we want such as shocks in the solution.
Using a nonclassical symmetry, we can construct an implicit multivalued solution to an example problem that has a region of negative diffusion. We can fix the multivalued part of the solution by inserting a shock, but how do we pick the right position? 
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******** Session 4 ********
Presenter: 
Thiago de Paiva Souza (Monash University, Melbourne) 
EPoster: 
"Knot theory from flows" 
Abstract: 
I am interested in knots that appear in dynamics as periodic orbits of flows. More precisely, I am interested in applying knot theory to study knots that appear in this context. An example of these types of knots are Lorenz knots. Lorenz knots are periodic orbits of the Lorenz system. Most Lorenz knots are believed to be hyperbolic, meaning that their complement in the 3sphere don't have any essential annuli and tori. When a knot is known to be hyperbolic, then we can study its hyperbolic invariants, such as volume, from hyperbolic knot theory. However, these hyperbolic invariants are currently difficult to compute for these types of knots. So one of our goals is to find ways to make their computation concrete, with the main goal of classifying all knots that appear in this family of knots. 
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Presenter: 
Sam Walker (University of Auckland) 
EPoster: 
"Bifurcations in the SuarezSchopf model with seasonal forcing" 
Abstract: 
The SuarezSchopf model is one of the earliest delay differential equation (DDE) models of the El NiñoSouthern Oscillation (ENSO) – yet its overall dynamics had not been explored fully. We present the bifurcation analysis of both the original SuarezSchopf model and of an extension that includes seasonal forcing. We find structures of repeating torus bifurcations that demonstrate the ability of quite simple and tractable DDEs with forcing to produce complex and intriguing dynamics. 
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Presenter: 
Lachlan Burton (University of Sydney) 
EPoster: 
"Escape time statistics in dissipative scattering" 
Abstract: 
In Hamiltonian systems for which escape of the trajectory from some central region of phase space is a possible outcome (through a hole, leak etc) the decay of the distribution of trajectory lifetimes is welldescribed by existing transient chaos literature. However, the addition of weak dissipative effects, such a velocitydependent drag, to such open systems is not wellexplored. I will discuss some results obtained in models from dynamical astronomy that are distinct from the usual picture in conservative systems. 
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Presenter: 
Nic Lam (University of Canterbury, Christchurch) 
EPoster: 
"Experimental design protocol for practical parameter estimation" 
Abstract: 
Parameter identification is a common process in science and mathematics that aims to find the best fit for a set of parameters within a model, often a dynamical system, for some observed behaviour. It is common to search for parameters that yield the smallest leastsquares error on an objective surface. Unfortunately, highly parametrised models often have identifiability issues: parameter identification of these models can yield a large distribution of parameters that exhibit the same or similar modelled behaviour to observations.
Structural identifiability issues, where there are mathematical redundancies in the model, have been explored and analysed extensively in literature. In contrast, practical identifiability issues, where experimental conditions and/or model structure cause variation in parameter estimates, has been infrequently employed to support modelbased analysis. The focus of my research is to quantify metrics for practical identifiability and use this measure to improve experimental design protocol in contexts such as diabetes modelling. 
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Presenter: 
Juan Patiño Echeverría (University of Auckland) 
EPoster: 
"Transitions to wild chaos in a 4D Lorenzlike system" 
Abstract: 
Wild chaos is the name given to a higherdimensional form of chaotic dynamics that can only arise in vector fields of dimension at least four. Recently, Gonchenko, Kazakov and Turaev (2021) showed that a fourdimensional Lorenzlike vector field with an additional parameter has a wild chaotic attractor. This means that every orbit in the attractor is unstable and that specific conditions hold and guarantee the persistence of this instability property.
We investigate in a twoparameter setting how this wild chaotic attractor arises geometrically. As a starting point, we continue the bifurcation structure of the equivalent threedimensional classic Lorenz equations when this additional parameter is "switched on". In particular, we find that the homoclinic explosion point of the classic Lorenz equations unfolds and gives rise to different types of global connections of the fourdimensional system. Due to the new feature of spiralling near the origin, these connections are of Shilnikov type, and we also find fold, perioddoubling and torus bifurcations of limit cycles. The overall bifurcation diagram provides new insight into the organisation of the fourdimensional phase space. 
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Presenter: 
Tim Lapuz (University of Sydney) 
EPoster: 
"Quasisteadystate approximations and singular perturbation" 
Abstract: 
Equations arising from enzyme kinetics have long been analysed by applied mathematicians. A classic paper by Heineken et al. in 1967 investigates one example of these equations. There they translate the biochemist's standard quasisteadystate approximation (sQSSA) to a reduction via singular perturbation analysis, leading to the product formation rate known as the MichaelisMenten equation. A multitude of papers then arose that looks at these enzyme kinetics equations through the lens of singular perturbation theory.
In this talk, I will discuss QSSA from a geometric singular perturbation theory (GSPT) pointofview. A comparison will be made between the literature definitions of several QSSAs (standard, reverse, total) and the geometric definitions arising from GSPT. I will also highlight that all these QSSAs can be treated uniformly via a coordinateindependent GSPT approach which, in particular, differs from the approach found in literature for the total QSSA case. 
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******** Session 5 ********
Presenter: 
John Bailie (University of Auckland) 
EPoster: 
"Connecting resonance tongues in a conceptual climate model" 
Abstract: 
The Atlantic Meridional Overturning Circulation (AMOC) is a branch of the global ocean circulation system driven by salinity and temperature differences. We analyse a seasonally forced twobox model for salinity and temperature with three parameters: the ratio μ of salinity and temperature fluxes into the surface ocean box, the threshold density η between the two boxes, and the seasonal freshwater influx strength c.
In the (μ, c)plane, a branch of torus bifurcations bounds a compact region where a stable invariant torus exists. Resonance tongues emanate from resonance points from this branch and also an interval of the caxis. We explore how these resonance tongues connect different resonance points, which is organised by local maxima and minima of the rotation number ρ. This can be explained by using singularity theory to describe all possible connections in the (μ, ρ)plane. 
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Presenter: 
Natalia McAlister (Monash University, Melbourne) 
EPoster: 
"Stable ergodicity on 3manifolds" 
Abstract: 
In this talk I will introduce the following:
 Theorem 1.
 Generically, for f a volume preserving C^{1} diffeomorphism in a 3dimensional manifold, if there exists a minimal contracting or expanding foliation, then f is stably ergodic.
This was published in 2020 by G. Núñez and J. RodriguezHertz along with the conjecture that generically, a volume preserving C^{1} diffeomorphism that presents dominated splitting has a minimal contracting or expanding foliation. If this conjecture were true, the theorem proves, for 3dimensional manifolds, a conjecture from 2012: Generically, a volume preserving C^{1} diffeomorphism that presents dominated splitting is stably ergodic. 
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Presenter: 
Behnaz Rahmani (University of Auckland) 
EPoster: 
"Analysing calcium dynamics using geometric singular perturbation techniques" 
Abstract: 
Mathematical models of intracellular calcium dynamics are known to exhibit a wide variety of complex oscillations. This project is interested in determining the mathematical mechanisms underlying socalled broad spike oscillations seen in some calcium dynamics models. I will discuss progress towards using geometric singular perturbation theory to explain the broad spike oscillations that occur in a model of calcium dynamics in hepatocytes. 
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