Applied Nonautonomous Dynamical Systems: Theory, Methods and Examples

In climate modelling, there is international consensus on the need for action to mitigate the effects of climate change due to anthropogenic forcing. The United Nations Framework Convention on Climate Change (UNFCCC) has the objective to "stabilize greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system", and to do so "within a time-frame sufficient to allow ecosystems to adapt naturally to climate change....". Policy discussions generally focus on the first part of this objective, which is about defining a "dangerous" level of global warming which must be avoided - most recently through the 2015 Paris agreement which aims to hold the increase in the global average temperature to well below 2C. This proposal aims to develop methodologies that address the second part of this objective, namely to understand how changes to the system may interact with time-frame of response for the system.

The presence of sudden changes in past climate (as evidenced in paleoclimate records) and nonlinear feedbacks between components has highlighted the likelihood that parts of the climate systems may "tip" from one state to another - for example, runaway ice-loss due to the positive albedo feedback, or major changes in ocean heat transport patterns. Slowly changing forcing may already lead to relatively fast changes in climate state, and passing through threshold value of a forcing parameter (for instance, in atmospheric greenhouse gas concentration) has been implicated in tipping points in the past.

These are questions about nonautonomous dynamical systems, i.e. dynamical systems that evolve in time but where parameters also change with time. Although there has been a concerted attempt to gain a theoretical underpinning of nonautonomous dynamical systems in recent years, these results can be hard to apply or not particularly useful in specific applications: this is because most results in this area are either for very general, or for very specific systems. There are especially few applicable tools for cases where a change to the system (forcing/input) occurs on timescale similar to those within the system. This proposal aims to rectify this problem by developing such tools.

We intend to develop new methods to understand "typical behaviour" in terms of local pullback attractors of Milnor type and their instabilities to forcing (that may be non-stationary), guided by applications in climate modelling. Particular problems we will focus on are the response of global mean temperature and ocean circulation by greenhouse gases to anthropogenic forcing, especially where the forcing timescale coincides with ocean circulation timescales.

We expect the findings to have potential for impact in a number of areas as outlined below.

Impact on Climate Science: We envisage a possible outcome of this project will be improved protocols for investigating rapid changes in climate that are path-dependent rather than threshold-dependent. In particular, the proposal has been developed in collaboration with MOHC partners, and potential impacts will be discussed at an early stage with these partners. The TiPES consortium aims to better quantify the tipping elements that may be present in the climate system and to ensure that climate projections include these. It is a consortium of many of the major climate science institutes within the EU and its focus will be primarily on underpinning climate science of tipping points. Hence the proposed project (which focuses on mathematical questions) will complement and generate synergies with TiPES. The University of Exeter and MOHC host a significant number of lead authors for previous IPCC reports. Prof Lucarini's position on the board of the Walker Institute will be an additional route for rapid, high quality feedback from climate scientists on the relevance of the work undertaken within the project.

Impact on Climate Policy, Industry and Commerce: The partnership with the MOHC and the planned Policy Briefing will provide a route for rapid dissemination to operational climate forecasters and government scientists interested in climate services. The work in this project will lead to improved methodologies for understanding sudden changes in climate projections in the face of anthropogenic emissions. In particular, the identification of scenarios that lead to instabilities of attracting behaviour is of direct relevance to tipping points that may appear even if the system is forced out-of-equilibrium. Identifying these should lead to an improved understanding and quantification of climate risk, of direct benefit to policymakers in government as well as financial institutions, insurers and those working on climate services. In addition to impacts from these partnerships and indirect impacts (eg via publications), the Policy Briefing in London towards the end of the project will summarise findings of relevance and to engage in dialogue with policymakers. We will invite project members, partners, selected academics, representatives from insurance and climate services industries. We will also invite policy advisors from UK government offices and departments.

Impact on Public Understanding of Science: Although much of the progress will be mathematical advances, there is likely to be a high degree of societal and public interest in this research. The recent emergence of grass-roots movements such as "Extinction Rebellion" and "School strike for climate" highlights a growing general concern about climate futures, and the importance and responsibility of scientists to communicate clear, accurate and unbiased information about climate projections to the general public. Experience from outreach work of the PI within "CliMathNet" and "ReCoVER" suggests that there is (a) much to be done to improve awareness of the fundamental importance of mathematical methodologies in climate science and (b) the latter can provide a clear and compelling motivation for young people to understand the societal importance of mathematics. We will use opportunities presented by this project to promote this understanding. The PI is active in outreach to schools; he runs student group projects for the "Exeter Mathematics School" (a specialist school in Exeter that takes mathematically highly able 16-18 year olds) on an annual basis since 2014, and will incorporate aspects of the research challenges and findings into future group projects. These group projects will provide an opportunity to develop and test school resources that will made more widely available as part of a project blog.