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.
10:30
Global Stability Properties of the Climate: Melancholia States, Invariant Measures, and Phase Transitions
Valerio Lucarini | University of Reading and University of Hamburg | United Kingdom
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Valerio Lucarini | University of Reading and University of Hamburg | United Kingdom
For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. In a previous work, we defined the Melancholia states that sit between the two climates. Such states are embedded in the boundaries between the two basins of attraction and feature extensive glaciation down to relatively low latitudes. Here, we explore the global stability properties of the system by introducing random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a phase transition.
11:30
Randomly Switched Vector Fields and Bifurcations
Tobias Hurth | Université de Neuchâtel | Switzerland
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Tobias Hurth | Université de Neuchâtel | Switzerland
Switching at random times between finitely many integrable vector fields gives rise to an interesting class of random dynamical systems whose long-term behavior is governed by both the stochastic effects of the switching and the deterministic dynamics of the individual vector fields. In this talk, we will focus on invariant probability measures for the Markov semigroups of such systems. The talk will be example-driven, with particular emphasis on systems characterized by switching the value of the bifurcation parameter for classical bifurcations such as pitchfork or transcritical. It is based on a joint project with Christian Kuehn as well as ongoing work with Yuri Bakhtin, Sean Lawley, and Jonathan Mattingly.
12:00
Computation of Sensitivities for the Invariant Measure of a Parameter Dependent Diffusion
Tony Lelievre | Ecole des Ponts ParisTech | France
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Tony Lelievre | Ecole des Ponts ParisTech | France
We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter \lambda, and admitting a unique invariant measure for any value of \lambda around \lambda =0. Our aim is to compute the derivative with respect to \lambda of averages with respect to the invariant measure, at \lambda =0. We analyze a numerical method which consists in simulating the process at \lambda =0 together with its derivative with respect to \lambda on a long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to \lambda of the mean of an observable through Monte Carlo simulations.