Bridging the gap between Koopmanism and Response Theory: Using Natural Variability to predict Forced Response

Complex chaotic systems exhibit nontrivial internal variability, with a power spectrum typically characterised by resonant broad peaks standing out on a continuous background of frequencies. These resonances correspond to nonlinear excitable modes of the system's evolution that can be generally attributed to long-lasting persistent events, weakly damped instabilities, or critical settings where the chaotic attractor is approaching a crisis [1].

In the context of the climate system, examples of such resonant behaviour are the El Niño Southern Oscillation, caused by a weakly damped instability of the atmosphere-ocean system [2], or the transition between the Warm and Snowball state of the Earth system due to a boundary crisis due to a change of the ice-albedo feedback [3].

On top of describing the relevant features of the internal variability of systems, such resonances also represent the fundamental modes shaping the resilience of the system to external perturbations. In particular, the linear response of the system to general forcing scenarios is solely determined by the Green’s function, for which a decomposition in terms of resonances can be obtained [4].

It is possible to establish a link between the system's nonlinear resonances and the spectral properties of the Koopman operator underlying the evolution in time of the system's observables.

Based on our work in [5], I will show that data-driven techniques developed to investigate the properties of the Koopman operator can be used to extract both resonances and dynamical modes from data. I will provide numerical evidence that the dynamical evolution of the statistical properties of the system can be interpreted as a superposition of such modes. In particular, by employing a projection of generic observables of the system onto the set of nonlinear modes, I will show that it is possible to reconstruct not only correlation functions but also the response of virtually any observable of interest.

Even though so far restricted to low dimensional systems, our results highlight the importance of such nonlinear modes in shaping the variability and response of chaotic systems and provide a way to (a) interpret the relevance of observables as a proxy to investigating dynamical properties of the system and (b) explain the difference between intrinsic variability of observables and their response to perturbations.

References

[1] Chekroun et al., Journal of Statistical Physics (2020) 179:1366–1402, https://doi.org/10.1007/s10955-020-02535-x

[2] Tantet et al., Journal of Statistical Physics (2020) 179:1449–1474, https://doi.org/10.1007/s10955-020-02526-y

[3] Tantet et al., 2018 Nonlinearity 31 2221, https://iopscience.iop.org/article/10.1088/1361-6544/aaaf42

[4] Manuel Santos Gutiérrez and Valerio Lucarini, 2022 J. Phys. A: Math. Theor. 55 425002, https://iopscience.iop.org/article/10.1088/1751-8121/ac90fd

[5] Zagli, Colbrook, Lucarini, Mezić, Moroney, “Bridging the gap between Koopmanism and Response Theory: Using Natural Variability to predict Forced Response”, https://doi.org/10.48550/arXiv.2410.01622