Randomness in differential equations is a key topic to build and understand mathematical models arising in applications. Yet, most real-world problems are nonlinear and may exhibit complicated dynamics. A key concept to understand transitions in nonlinear systems are bifurcations. In this minisymposium we aim to bring together the two communities from uncertainty quantification and nonlinear dynamics more closely. The goal is to foster the interface and interaction between nonlinear systems theory and the role of randomness. In particular, we expect many important open questions to arise out of this exchange of ideas.
14:00
Computing Invariant Sets of Random Differential Equations using Polynomial Chaos
Maxime Breden | École Polytechnique | France
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Maxime Breden | École Polytechnique | France
Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modelling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte-Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets of differential equations such as steady states, stable/unstable manifolds, periodic orbits, and heteroclinic orbits. We employ PC to compute representations of all these different types of invariant sets for RODEs. This allows us to obtain fast sampling, geometric visualization of distributional properties of invariants sets, and uncertainty quantification of dynamical output such as periods or locations of orbits. We apply our techniques to a predator-prey model, where we compute steady states and stable/unstable manifolds. We also include several benchmarks to illustrate the numerical efficiency of adaptively chosen PC depending upon the random input. Then we employ the methods for the Lorenz system, obtaining computational PC representations of periodic orbits, stable/unstable manifolds and heteroclinic orbits.
14:30
- CANCELED - Random Batch methods for interacting particle systems and for high dimensional global minimization problems in machine learning
Shi Jin | Shanghai Jiao Tong University | China
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Shi Jin | Shanghai Jiao Tong University | China
We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions. We will also use this method and improve the consensus-based model for global optimization for high dimensional machine learning problems.
15:00
A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillators
Maximilian Engel | Technical University of Munich | Germany
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Maximilian Engel | Technical University of Munich | Germany
For an attracting periodic orbit (limit cycle) of a deterministic dynamical sytem, one defines the isochron for each point of the orbit as the section with fixed return time under the flow. Isochrons can equivalently be characterized as stable manifolds foliating neighbourhoods of the limit cycle or as level sets of dominating eigenfunctions of the associated Koopman operator. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillators, i.e. limit cycles or heteroclinic orbits exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We will introduce a new rigorous definition of isochrons as random stable manifolds for random limit cycles and demonstrate how an appropriate invariance equation can link this random dynamical systems interpretation to the physicist's approaches by appropriate averaging.
15:30
- CANCELED - Dynamic Vulnerability in Oscillatory Networks and Power Grids
Xiaozhu Zhang | Technical University Dresden | Germany
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Xiaozhu Zhang | Technical University Dresden | Germany
Recent work found distributed resonances in driven oscillator networks and AC power grids. The emerging dynamic resonance patterns are highly heterogeneous and nontrivial, depending jointly on the driving frequency, the interaction topology of the network and the node or nodes driven. Identifying which nodes are most susceptible to dynamic driving and may thus make the system as a whole vulnerable to external input signals, however, remains a challenge. Here we propose an easy-to-compute Dynamic Vulnerability Index (DVI) for identifying those nodes that exhibit largest amplitude responses to dynamic driving signals with given power spectra and thus are most vulnerable. The DVI is based on linear response theory, as such generic, and enables robust predictions. It thus shows potential for a wide range of applications across dynamically driven networks, for instance for identifying the vulnerable nodes in power grids driven by fluctuating inputs from renewable energy sources and fluctuating power output to households.