Table 1 Selected nonlinear system approaches that may be suitable for solving earth system tasks related to predictability, change detection, uncertainty, teleconnections, and prediction and predictive skill

Predictability

Quantifying sensitivity of ESM to ICs

Lyapunov exponents and Chaos Theory: Determine sensitivity to ICs by computing Lyapunov exponents, which measures the rate at which trajectories in a dynamical system diverge.

Determining predictability of climate

Bifurcation Analysis: Identify the conditions under which small changes in parameters can lead to qualitative changes in system behavior, helping to determine predictability.

Change detection

Identifying climate change

Attractor Reconstruction: Use techniques such as phase space and embedding theorems to reconstruct the attractor of the climate system, identifying changes in the attractor’s structure over time.

Identifying tipping points

Bifurcation Analysis: Detect tipping points by identifying early warning signals such as increased autocorrelation and variance, indicative of critical slowing down of components of the climate system.

Uncertainty

Characterizing irreducible uncertainty due to internal climate variability

Stochastic Resonance and Noise-Induced Transitions: Model the impact of internal variability using stochastic differential equations, exploring how noise can induce transitions and contribute to irreducible uncertainty.

Teleconnections

Characterizing spatial connectivity structures and teleconnections

Complex Network Analysis and Synchronization Phenomena: Utilize complex network theory to analyze teleconnections, identifying community structures, hubs and synchronized regions in the climate network.

Prediction and predictive skill

Nonlinear prediction at climate scales

Machine Learning with Nonlinear Dynamical Systems: Capture and predict the evolution of large-scale nonlinear climate dynamics using machine learning methods such as reservoir computing or recurrent neural networks