Valerie Jeong (University of Auckland)

"A noisy heteroclinic controller for a locomotion task"

It is known that trajectories near a heteroclinic cycle spend longer time near equilibria. When small noise is added, this slowing down no longer occurs. However, previous work showed that when there is a stable periodic orbit near a heteroclinic cycle, adding small noise can, counterintuitively, increases the time spent near equilibria.

In this talk, we consider an evolutionary robotics locomotion task for which a heteroclinic cycle is used as a controller. We use a genetic algorithm to optimise parameters in the system; faster average speed corresponds to better performing agents. A bifurcation analysis of the best performing agent shows different types of periodic solutions in which: (i) the average speed decreases as the period increase; (ii) the average speed increases as the period increases. The latter type is non-intuitive, and linking this to our previous result, we show that adding small noise to this system can increase the average speed, and hence the fitness of the agent.

Paco Casteñada Ruan (University of Auckland)

"Understanding the effect of mitochondrial intake on intracellular calcium oscillations"

Rapid changes in the concentration of calcium of a cell, caused by fluxes in an out of the cell and its organelles, is a signalling mechanism present in most cell types. For example, calcium intake by the mitochondria is vital to the regulation of energy production, as has been understood for some time now. However, what effect mitochondrial intake has on the whole cell calcium dynamics is not well understood. In this talk I will discuss modelling work that has attempted (but, so far, failed) to explain some recent experimental results regarding mitochondrial intake.

Seigan Hayashi (University of Canterbury, Christchurch)

"Control-based continuation for probing real systems"

Identification of nonlinear features can cause difficulties as many design conventions and experimental testing methods are derived from linear assumptions about models of interest. Systematically identifying bi-stable behaviour can be difficult due to hysteresis. Unstable solutions cannot be identified with open-loop experimental techniques. In micro-scale devices such as micro-electromechanical systems (MEMS) these nonlinearities can dominate, making design and analysis difficult to conduct. Control-based continuation (CBC) is becoming a promising approach in investigating dynamical systems. CBC allows the user to trace along the solution path of a dynamical system in a similar manner to a numerical continuation solver. A pseudo-arclength like method alongside a stabilising feedback controller allows for continuation through fold bifurcations between stable and unstable solution branches. Unstable behaviour is controlled and stabilised, and therefore become visible to the experiment. CBC techniques have been used to great success to identify nonlinear frequency response, S-curves, and backbone curves.

Mitchell Curran (University of Sydney)

"Hamiltonian spectral flows and the Maslov index"

I have been using the Maslov index – a winding number for a path of Lagrangian planes – to study the real spectrum of a Hamiltonian differential operator. In particular, an application of the Maslov index leads to a lower bound for the number of positive real eigenvalues. The bound includes a (somewhat) mysterious contribution to the Maslov index that is not computable by the standard means. Fortunately, it is computable using the topological properties of the Maslov index and an analysis of the eigenvalue curves, which represent the evolution of eigenvalues with respect to perturbation of the domain. Applying the theory leads to stability results for standing wave solutions to nonlinear Schrodinger equations, while comparison with existing lower bounds leads to some interesting connections between the Maslov index and constrained eigenvalue counts. If anyone has a Hamiltonian eigenvalue problem for a nonlinear wave with real spectrum: come talk to me!

Andrew Tyler (Monash University, Melbourne)

"Does accessibility imply ergodicity?"

Our main result is that accessibility implies ergodicity for volume preserving, partially hyperbolic, endomorphisms of the torus. We did this by adapting the Hopf argument to our setting. Our arguments are independent to our choice of manifold. Our next step is to generalise our result.

Renzo Mancini (University of Auckland)

"Analysis of a conceptual model for the Atlantic Meridional Overturning Circulation with two time delays"

We study a scalar delay differential equation (DDE) model with two time delays for the Atlantic Meridional Overturning Circulation (AMOC). Here, the time delays are associated with the temperature feedback between North Pole and Equator, and the salinity exchange between surface and deep water at the Pole. We perform a numerical bifurcation analysis of the DDE with the continuation package DDE-Biftool. It reveals rich behaviour of this AMOC model organized by homoclinic orbits, as well as resonance phenomena due to the interplay between the two delayed feedback loops.

David Groothuizen-Dijkema (University of Auckland)

"Switching near heteroclinic networks as a piecewise-smooth dynamical system"

A heteroclinic cycle is an invariant structure in a dynamical system composed a sequence of saddle equilibria and heteroclinic orbits connecting them in a cyclic manner. Near an attracting heteroclinic cycle, trajectories spend increasingly longer periods of time in the vicinity of one equilibria before making a transition to the next. A heteroclinic network is a connected union of heteroclinic cycles.

When a heteroclinic cycle is a part of a larger heteroclinic network, trajectories may switch from one cycle to another, and possibly even cycle between cycles. The usual approach to analyse the stability of heteroclinic cycles and networks is to construct return maps on Poincaré sections. In this talk, we will describe how we can use continuous, piecewise-smooth maps derived from these return maps to investigate the switching properties of heteroclinic networks.

Indranil Ghosh (Massey University, Palmerston North)

"Bifurcation structure of robust chaos in 2D piecewise-linear maps"

Piecewise-linear maps arise from mathematical models in diverse applications. Families of such maps readily exhibit chaos in a robust fashion and this was popularised by Banerjee, Yorke, and Grebogi in their 1998 Physical Review Letters paper. In this talk, I will show how the family of maps they considered naturally subdivides through the powerful technique of renormalisation, and how this concisely explains some features of the dynamics described by other authors. I will further show how the robust chaos extends more broadly to orientation-reversing and non-invertible piecewise-linear maps.

Morgan Meertens (University of Auckland)

"Folded Homoclinic Bifurcations"

Some systems of ordinary differential equations have the property that one or more variables in the system evolve on a slower timescale than other variables. Trajectories in such systems may exhibit "bursting", which occurs when intervals of rapid spiking of the amplitude of one or more variables are interspersed with quiescent periods during which the amplitude changes slowly. Over the years, there have been many studies of the mathematical mechanisms underlying bursting, particularly the mechanisms that are associated with a change in the number of spikes in a bursting solution.

One mechanism for the onset of bursting and spike-adding is the so-called folded homoclinic bifurcation. However, no systematic theoretical study of this phenomenon has been conducted and the generic dynamics associated with a folded homoclinic bifurcation is still unknown.

My talk will describe the progress towards understanding the nature and origin of complex oscillations that arise from folded homoclinic bifurcations.

Aaron Moston-Duggan (Macquarie University, Melbourne)

"Generalized solitary waves in Karpman equations: effects of discretization"

We consider generalizations of nonlinear Schrödinger equations, which we call "Karpman equations," that include additional linear higher-order derivatives. Singularly perturbed Karpman equations produce generalized solitary waves (GSWs) in the form of solitary waves with exponentially small oscillatory tails. Previous research on continuous Karpman equations has shown that GSWs occur in specific settings. We use exponential asymptotic techniques to identify GSWs in singularly-perturbed continuous Karpman equations. We then study the effect of discretization on GSWs by applying a finite-difference discretization to continuous Karpman equations. By comparing GSWs in these discrete Karpman equations with GSWs in their continuous counterparts, we show that the oscillation amplitudes and periods in the GSWs differ in the continuous and discrete equations. We also show that the parameter values at which there is a bifurcation between GSW solutions and solitary wave solutions depends on the choice of discretization. Finally, by comparing different higher-order discretizations of the fourth-order Karpman equation, we show that the bifurcation value tends to a nonzero constant for large orders, rather than to 0 as in the associated continuous Karpman equation.

Nick Cranch (University of Sydney)

"Random feature maps"

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 affine-linear transformation and non-linear 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 trial-and-error tuning of hyperparameters.

Dana C'Julio (University of Auckland)

"Transition to a blender in a three-dimensional Hénon-like map: in search of wild chaos"

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 three-dimensional Hénon-like map that has the ingredients to exhibit a blender. With advanced numerical techniques, we compute the one-dimensional stable and unstable manifolds of this map and characterise the geometric properties of intersections of the manifolds with a plane. The intersections feature a self-similar structure that brings about the generation or disappearance of blenders. This allows us to estimate the parameter values for which a blender arises.

Lloyd Lee (University of Auckland)

"The effect of calcium influx on intracellular signalling"

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.

Andrew Cook (Monash University, Melbourne)

"Fixed points of 1-dimensional neural networks"

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 non-constant 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.

Damien McLeod (University of Sydney)

"Spherical Harmonics of S³"

Spherical harmonics are a well studied area with fundamental application in physics and engineering. While for S² effectively only one natural spherical coordinate system arises, in higher dimensions there is an abundance of choice. We compare coordinate system choices in S³ – 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.

Tom Miller (University of South Australia, Adelaide)

"Shocks and negative diffusion in reaction diffusion equations"

Reaction-diffusion 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 ill-posed and requiring regularisation to solve numerically, it can also lead to features we want such as shocks in the solution.

Using a non-classical 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?

Thiago de Paiva Souza (Monash University, Melbourne)

"Knot theory from flows"

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 3-sphere 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.

Sam Walker (University of Auckland)

"Bifurcations in the Suarez-Schopf model with seasonal forcing"

The Suarez-Schopf model is one of the earliest delay differential equation (DDE) models of the El Niño-Southern Oscillation (ENSO) – yet its overall dynamics had not been explored fully. We present the bifurcation analysis of both the original Suarez-Schopf 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.

Lachlan Burton (University of Sydney)

"Escape time statistics in dissipative scattering"

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 well-described by existing transient chaos literature. However, the addition of weak dissipative effects, such a velocity-dependent drag, to such open systems is not well-explored. I will discuss some results obtained in models from dynamical astronomy that are distinct from the usual picture in conservative systems.

Nic Lam (University of Canterbury, Christchurch)

"Experimental design protocol for practical parameter estimation"

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 least-squares 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 model-based 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.

Juan Patiño Echeverría (University of Auckland)

"Transitions to wild chaos in a 4D Lorenz-like system"

Wild chaos is the name given to a higher-dimensional 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 four-dimensional Lorenz-like 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 two-parameter setting how this wild chaotic attractor arises geometrically. As a starting point, we continue the bifurcation structure of the equivalent three-dimensional 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 four-dimensional system. Due to the new feature of spiralling near the origin, these connections are of Shilnikov type, and we also find fold, period-doubling and torus bifurcations of limit cycles. The overall bifurcation diagram provides new insight into the organisation of the four-dimensional phase space.

Tim Lapuz (University of Sydney)

"Quasi-steady-state approximations and singular perturbation"

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 quasi-steady-state approximation (sQSSA) to a reduction via singular perturbation analysis, leading to the product formation rate known as the Michaelis-Menten 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) point-of-view. 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 coordinate-independent GSPT approach which, in particular, differs from the approach found in literature for the total QSSA case.

John Bailie (University of Auckland)

"Connecting resonance tongues in a conceptual climate model"

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 two-box 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 c-axis. 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.

Natalia McAlister (Monash University, Melbourne)

"Stable ergodicity on 3-manifolds"

In this talk I will introduce the following:

Theorem 1.

Generically, for f a volume preserving C¹ diffeomorphism in a 3-dimensional 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. Rodriguez-Hertz along with the conjecture that generically, a volume preserving C¹ diffeomorphism that presents dominated splitting has a minimal contracting or expanding foliation. If this conjecture were true, the theorem proves, for 3-dimensional manifolds, a conjecture from 2012: Generically, a volume preserving C¹ diffeomorphism that presents dominated splitting is stably ergodic.

Behnaz Rahmani (University of Auckland)

"Analysing calcium dynamics using geometric singular perturbation techniques"

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 so-called 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.