Program/Abstracts

Invited talks

Quenched limit laws and thermodynamic formalism for random dynamical systems

Cecilia Gonzalez Tokman (U Queensland)

Non-autonomous or random dynamical systems (RDS) provide flexible models to investigate systems whose evolution depends on external factors, such as seasonal forcing and/or noise. Recent developments on multiplicative ergodic theory and transfer operators allow us to get useful insights into the long-term behaviour of these systems. In this talk, we will present results in this direction, including (quenched) limit theorems and thermodynamic formalism for a class of RDS. Our results will be illustrated with examples, including random open and closed intermittent maps and non-transitive systems. This talk is based on collaborations with J. Atnip, D. Dragicevic, G. Froyland and S. Vaienti.

Kernel-based approximation of the Koopman generator and Schrödinger operator

Stefan Klus (University of Surrey)

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We will present a kernel-based method for the approximation of transfer operators and differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. We will illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics, fluid dynamics, and quantum mechanics.

The geometry of a blender: the case of spaghetti in a can

Bernd Krauskopf, Hinke Osinga and Dana C Julio (University of Auckland)

Blenders are a geometric tool to construct robust recurring dynamics in a theoretical setting; one also speaks of wild chaos. We consider an explicit family of 3D Hénon-like maps and show that it exhibits a blender for a specific parameter range where the underlying 2D Hénon map has a full horseshoe. The blender arises as the closure of the one-dimensional stable manifolds of two saddle points, which weave such a complicated pattern in space that these curves form an impenetrable carpet when seen from certain directions. We combine advanced numerical techniques with the compactification of phase space to compute extremely long pieces of the two global invariant manifolds, which allows us to determine when they form a blender and when not. In a nutshell, we study the geometry of infinitely long spaghetti in a can.

Phase resetting as a two-point boundary value problem

Hinke Osinga, Bernd Krauskopf (University of Auckland) and Peter Langfield (University of Bordeaux)

Phase resetting is used in experiments with the aim to classify and characterise different neurons by their responses to perturbations away from a periodic bursting pattern. The same approach can also be applied numerically to a mathematical model. Resetting is closely related to the concept of isochrons of a periodic orbit, which are the submanifolds in its basin of attraction of all points that converge to this periodic orbit with a specific phase. Until recently, such numerical phase resets were performed in an ad-hoc fashion, and the development of suitable computational techniques was only started in the last decade or so. We present an approach based on the continuation of solutions to a two-point boundary value problem that directly evaluates the phase associated with the isochron that the perturbed point is located on. We illustrate this method with the FitzHugh–Nagumo model and investigate how the resetting behaviour is affected by phase sensitivity in the system.

Data-Driven Prediction of Partially Observed Multiscale Systems

Krithika Manhoar (U Washington)

Complex systems with dynamics evolving on multiple timescales pose a tremendous challenge for data-driven modeling. For complex systems in oceans and climate exhibiting scale separation, the macroscopic (slow) dynamics are often modeled by treating fast variables as stochastic effects. Motivated by this idea, we use kernel methods in machine learning to approximate the Koopman evolution operator associated with the dynamical system based on only observing the slow variables. This method, called kernel analog forecasting, applies a Gaussian kernel to data points to build a Markov kernel operator and diffusion features from its eigenfunctions. Using these eigenfunctions as a basis, we construct an operator semigroup modeling the slow dynamics, and study its predictive skill on chaotic multiscale examples.

Extraction of small- and large-scale structures in dynamic ocean flows

Erik van Sebille (U Utrecht)

The ocean is in constant motion, with water circulating within and flowing between basins. As the water moves around, it carries heat and nutrients, as well as planktonic organisms and plastic litter around the globe.

The most natural way to study the pathways of water and the connections between ocean basins is using trajectories, either from observations of drifters or computed by simulating virtual particles in fine-resolution ocean models. But interpreting this Lagrangian data and teasing out information on structures is not trivial and an active field of research.

Here, I will show a few recent examples of how techniques from Set Oriented Numerics can be applied to dynamic ocean flows, and how these compare to techniques from e.g. machine learning.

Tensor-Based Computation of Metastable and Coherent Sets

Feliks Nüske (U Paderborn)

Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory -- with extended dynamic mode decomposition (EDMD) being a cornerstone of the field. On the other hand, low-rank tensor product approximations -- in particular the tensor train (TT) format -- have become a valuable tool for the solution of large-scale problems in a number of fields. In this work, we combine EDMD and the TT format, enabling the application of EDMD to high-dimensional problems in conjunction with a large set of features. We derive efficient algorithms to solve the EDMD eigenvalue problem based on tensor representations of the data, and to project the data into a low-dimensional representation defined by the eigenvectors. We extend this method to perform canonical correlation analysis (CCA) of non-reversible or time-dependent systems.

Stability under Uncertainty: Towards a Bifurcation Theory for Dynamical Systems with Bounded Noise

Kalle Timperi (Department of Mathematics, Imperial College London)

During the last decades, complex systems have emerged as a unifying conceptual framework for addressing topical challenges in a wide array of applications. These range from the climate system to the neuronal dynamics of the human brain and all the way to the atomic scale. In particular, there is a growing need to understand how the stability of patterns and behaviours in a given system is affected by uncertainty or noise in the system and its environment.

In the deterministic (non-noisy) setting, the classical bifurcation theory of dynamical systems studies how a (slow) change in system parameters affects the stability of system behaviours. However, a corresponding bifurcation theory for random dynamical systems is only in its very early stages of development. An important observation here is that assumptions on the type of noise make a difference for the theory. In particular, one may distinguish between bounded and unbounded (for instance Gaussian) noises.

In the first part of this talk I consider the notions of (i) stability/robustness and (ii) noise/uncertainty, and discuss some of the challenges in constructing a satisfactory bifurcation theory in the noisy setting. In the second part of the talk I describe some recent efforts towards building a bifurcation theory in the context of bounded noise, which offers a promising alternative to the unbounded approaches.

A Subspace Expanding Technique of Cell Mapping Method for Finding Zeros of Nonlinear Algebraic Equations

Jian-Qiao Sun, Zigang Li, Jun Jiang and Ling Hong (Department of Mechanical Engineering, School of Engineering, University of California Merced)

Stability analysis, bifurcation study and optimization of multi-degree-of-freedom (MDOF) engineering systems often lead to the problem of finding roots of transcendental equations or optimality conditions. There have not existed effective methods for finding roots of large set of algebraic equations of many variables. Without prior knowledge of the number and positions of the zeros, this is a highly challenging problem, particularly when the dimension of the system is high. In this talk, I shall present a subspace expanding technique (SET) to expand the ability of the cell mapping methods to efficiently discover and find all the zeros of nonlinear functions. The covering set of cells that may contain zeros is identified in parallel computing by the root bracketing method that is simple and efficient for continuous functions. The covering set can be found first in a low dimensional subspace, and then gradually extended to higher dimensional spaces as more equations and variables are brought into computation. The subspace expansion technique is found to be highly efficient for finding zeros in a high dimensional space. The subdivision technique of the cell mapping method can further be applied to refine the covering set, leading to accurate numerical results of zeros. Examples of zeros finding of nonlinear functions are presented to demonstrate the utility and power of the proposed method. It is believed that the proposed method will significantly enhance our ability to study stability, bifurcation and optimization problems of complex MDOF nonlinear dynamic systems.

Relative equilibria for the n-body problem

Warwick Tucker (Monash University), Piotr Zgliczynski (Jagiellonian University) and Jordi-Lluis Figueras (Uppsala University)

We will discuss the classical problem from celestial mechanics of determining the number of relative equilibria a set of planets can display. Several already established results will be presented, as well as a new contribution (in terms of a new proof) for the restricted 4-body problem. We will discuss its possible extensions to harder instances of the general problem.

Contributed talks

Random open dynamics and quenched extreme value theory via spectral perturbation

Jason Atnip (USNW), C. Gonzalez-Tokman, G. Froyland and S. Vaient

We consider the setting of random piecewise-monotonic interval map cocycles with contracting potentials and transfer operators which are perturbed by the introduction of small random holes, creating a random open dynamical system. The Birkhoff Ergodic Theorem ensures that Lebesgue almost every point will eventually be mapped into a hole, leaving only a measure zero set of points which survive. Via a perturbative approach, we establish the existence of equilibrium states and conditionally invariant measures supported on the surviving set. We obtain a first-order perturbation formula for the leading Lyapunov multipliers, and our new machinery is then deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics with general ergodic invertible driving, and random observations.

Using the first-order terms we derive an extreme value law, which is then used to further derive quenched hitting time statistics for sequences of random holes.

A Quantitative Evaluation of the Impact of Velocity Uncertainty on the Self–Consistency of Lagrangian Coherent Structure Detection Methods

Aleksandar Badza (U Adelaide)

Lagrangian coherent structures are objects which are used to visualise the most influential flow behaviour within a velocity system over a particular interval of time. Many computational methods have been developed for numerically detecting these structures within various flows, each with their own definition of a coherent structure from flow barriers along ridges of more chaotic particle advection to more cohesive flow regions or sets. While plenty of comparative research into these methods has been undertaken, the self–consistency of these methods against uncertainty or noise present within velocity data has seldom been investigated. This is an important consideration for each detection method, especially as such uncertainty is inevitable in all realistic flow systems due to errors or inaccuracies in data measurement, observation or modelling. Hence in this talk, we take eight different methods for Lagrangian coherent structure detection, implement these methods on two “real–world” type flows with stochastic noise applied to each flow, and using statistical analysis of the results, assess which of these methods are the most reliable under the influence of acute velocity uncertainty.

Ergodic invariant measures for multidimensional random dynamical systems

Fawwaz Batayneh (U Adelaide)

In the area of dynamical systems, a deterministic discrete dynamical system is given by a map f:X → X, where X ⊂ Rⁿ, n ≥ 1. The set X is usually taken to be compact. One main question is to statistically understand the long term behaviour of trajectories of the map f for a large set of initial conditions x ∈ X. The existence of invariant measures provides key information about the dynamical behaviour, especially when this invariant measure μ is absolutely continuous (ACIP) with respect to the Lebesgue measure.

In this talk, we focus on a more general type of discrete dynamical systems, called random. In our case, we deal with a collection of multidimensional maps {f_ω:X → X: ω ∈ Ω} where X ⊂ Rⁿ, n > 1, indexed by a probability space (Ω,P). In real life applications, the relevance of random dynamical systems is clear due to the fact that systems are influenced by external factors or noise.

This talk will focus on the ergodic properties of such systems. In addition, we investigate the existence and bounds on the number of ergodic ACIPs.

Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty

Carlos Ignacio Hernández Castelalnos (IIMAS-UNAM)

In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). For such problems, not much has been reported in terms of uncertainties. We focus on the set-based robustness which allows the decision maker to analyze a given solution from the worst-case perspective. In this kind of problems, each solution in decision space maps to a set that represents the trade-offs of the worst possible scenarios.

To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. To reduce the numerical cost of the offline phase -- which grows exponentially with the parameter dimension -- we exploit symmetries in the control problems. Furthermore, to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem.

We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Our results show that the method can design driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.

Computing higher eigenfunctions of the p-Laplacian

Alvaro de Diego (TU Munich)

The p-Laplacian is a nonlinear Differential Operator that naturally arises when optimizing the Rayleigh quotient of order p. When p goes to 1, the level sets of its eigenfunctions approximate solutions to certain isoperimetric problems.

I will motivate using the p-Laplace for detecting Lagrangian Coherent Structures and talk about the numerical difficulties that arise and our attempts to solve them.

Extracting coherent sets from dynamic Laplacian eigenfunctions

Christopher P Rock (UNSW)

Finite-time coherent sets in time-varying flows are parcels of fluid whose time-averaged boundary is minimised relative to the enclosed volume. For example, in mesoscale ocean dynamics, ocean eddies with low filamentation or distortion can be coherent sets. The dynamic Cheeger constants measure the coherence of the kth-best coherent set, for each k=1,2,…. Large relative gaps between successive dynamic Cheeger constants indicate ‘natural’ numbers of coherent sets in a system.

Coherent sets can be identified using the eigenfunctions and eigenvalues of a dynamic Laplace operator. We give upper and lower bounds on the dynamic Cheeger constants, in terms of the corresponding eigenvalues of the dynamic Laplace operator. We also show that some of the superlevel sets of each eigenfunction, or of each linear combination of eigenfunctions, are approximate coherent sets. This leads to a new justification for an approach to identifying coherent sets known as sparse eigenbasis approximation. Our results are even new in the static setting, where they describe the relationship between the higher Cheeger constants and the eigenfunctions and eigenvalues of the Laplace-Beltrami operator.

Open-flow mixing in terms of transfer operators

Anna Klünker (Leuphana U, Kathrin Padberg-Gehle, Jean-Luc Thiffeault

We study finite-time mixing in time-periodic open flow systems. The systems we consider contain an inlet and an outlet flow region as well as a mixing region and are characterized by constant in‐ and outflow of fluid particles. We describe the transport of densities in terms of a transfer operator, which is represented by the transition matrix of a finite-state Markov chain. The transport processes in the open system are organized by the chaotic saddle and its stable and unstable manifolds. We extract these structures directly from leading eigenvectors of the transition matrix and use different measures to quantify the degree of mixing.

Evolving Lagrangian sets in turbulent convection flows

Christian Schneide (Leuphana U)

The detection of Lagrangian coherent sets in turbulent flows is naturally limited to short time scales by turbulent dispersion. Applying evolutionary clustering methods which use historic information for the clustering and relaxing the material property of the sets enables the detection of persistent evolving sets which exist on much larger time scales.

In this talk, we show the dynamics of evolving sets. We present the application of evolutionary clustering on two- and three-dimensional Rayleigh-Bénard convection data and show that the detected evolving sets are significant for the analysis of heat transport in the system.

Turning an analysis technique into a tool: Identification and simulation of Hamiltonian systems using inverse modified equations

Christian Offen (U Paderborn)

If a system of ordinary differential equations forms a Hamiltonian system, then Hamiltonian structure guarantees important qualitative aspects of the dynamical system, such as a lack of attractors, energy conservation, and is related to further topological properties of the phase portrait and conservation laws. Learning Hamiltonian structure from trajectories is an important task in system identification theory. Another challenge is to simulate Hamiltonian dynamics using numerical methods while preserving important structural properties under discretisation. Inverse modified differential equations have recently been introduced as an analysis technique for Hamiltonian neural networks. In this talks I would like to show how to turn this analysis technique into a tool for system identification and structure preserving simulations.

Boundary dynamics on minimal invariant sets of set-valued map

Wei Hao Tey (Imperial College London)

The theory of dynamical system is pivotal in real-world application ranges from computer algorithm to weather prediction. Realistically, there exists uncertainties and random noises, which translated to the study of random dynamical systems. Consider a discrete time dynamical system with bounded noise which can be represented by a set-valued mapping. We are interested in changes of the minimal invariant sets under these set-valued mappings. In this talk, we investigating their boundary dynamics which would help in detecting discontinuous changes of the set. We then look at simple examples of linear maps where the minimal invariant sets are generally non-trivial for non-zero bounded noise.