Applied Nonautonomous Dynamical Systems: Theory, Methods and Examples

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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.