Regime identification and control of extremes in the non-autonomous Lorenz model with chaos and intransitivity

Adaptive chaos control has been studied extensively for autonomous systems. For real world, non-autonomous systems, such as the planetary weather, observations of the system state in response to seasonally and diurnally varying forcing are available only at discrete times and locations, over which system trajectories are likely to have diverged given uncertainties in initial conditions. We consider a stochastic representation of such systems, as a building block for adaptive control, and develop and test control strategies in an idealized setting. We present the first example of finite time adaptive chaos control for a seasonally forced and noise-perturbed Lorenz84 model. We demonstrate two strategies for triggering control: (1) local Lyapunov exponents (LLE), and (2) transition probabilities for the latent states of a non-homogeneous Hidden Markov Model (NHMM). The second approach is motivated by thinking of future applications to a latent embedding space of planetary atmospheric circulation that would get us closer to real world analyses. The NHMM triggers are found to coincide with strongly positive LLE regimes, confirming their dynamical interpretability. These results provide a conceptual bridge towards the use of deep learning based weather and climate foundation models, whose hidden states could be leveraged for adaptive control to mitigate extreme weather events.

💡 Research Summary

This paper tackles the problem of adaptive chaos control in a non‑autonomous, seasonally forced and noise‑perturbed Lorenz‑84 (L84) model, which serves as a low‑dimensional analogue of extratropical atmospheric circulation. Classical chaos‑control literature has focused almost exclusively on autonomous systems, leaving a gap for real‑world applications where external forcings vary in time and observations are sparse, noisy, and often divergent due to uncertainty in initial conditions. The authors propose and evaluate two distinct trigger mechanisms for initiating control interventions: (1) a physics‑based local Lyapunov exponent (LLE) monitor, and (2) a data‑driven non‑homogeneous Hidden Markov Model (NHMM) that learns latent dynamical regimes and their seasonally modulated transition probabilities.

Model formulation
The L84 equations describe the evolution of a zonal jet (x) and two planetary‑eddy components (y, z). Seasonal forcing is introduced through a time‑varying equator‑to‑pole temperature gradient (F(t)=F_0+F_1\cos(\omega t)) with (F_0=7), (F_1=2) and (\omega) set to a yearly frequency. Multiplicative white noise (\varepsilon_t\sim\mathcal N(0,m|x_{t-1}|)) is added to mimic observational uncertainty and the divergence of trajectories that would arise from imperfect initial states. The system is integrated with a fourth‑order Runge–Kutta scheme (Δt = 0.01) for 2000 steps, using standard parameters (a=0.25), (b=4.0), (G=1.0). Four seasonal settings (F = 5–9) are examined, with the winter configuration producing the strongest chaotic response.

Control strategy 1 – LLE trigger
At each model time step the local Lyapunov exponent is estimated from the Jacobian of the L84 flow. A long‑run reference distribution of LLE values is built, and a high‑percentile threshold (e.g., 95th percentile) defines an “instability window”. When the instantaneous LLE exceeds this threshold, a bounded perturbation vector (\delta\mathbf x=(\delta x,\delta y,\delta z)) is computed by solving a constrained optimization that minimizes the Euclidean norm of the perturbation while keeping the eddy‑energy measure (|y|+|z|) below a prescribed bound. The perturbation is applied over a short prediction horizon (h) and the system is then allowed to evolve freely. This approach is directly linked to classic OGY‑type control and provides a clear physical interpretation of when the system is about to diverge.

Control strategy 2 – NHMM trigger
The observed state vector (\mathbf X_t) is modeled as emissions from a hidden Markov chain with (K) latent regimes (S_t). Conditional on regime (k), each component follows a Gaussian AR(1) process with parameters ((\mu_{d,k},\phi_{d,k},\sigma_{d,k})). Crucially, the transition matrix (\mathbf P(t)) is allowed to depend on a seasonal covariate (\mathbf C(t)=(\sin 2\pi t/365,\cos 2\pi t/365)) via a multinomial logistic regression. The EM algorithm (implemented with the depmixS4 package) estimates all emission and transition parameters; model selection uses BIC to choose the optimal number of regimes. After fitting, the Viterbi algorithm reconstructs the most likely past regime sequence, while the forward algorithm propagates the regime probability vector (\boldsymbol\alpha_{t+h}) over a user‑defined horizon (H).

A “dangerous” regime set (\mathcal D) is defined as those latent states whose empirical eddy‑energy exceeds a threshold (E^\ast). For each dangerous regime (i) a weight (w_i=P_i(|y|+|z|>E^\ast)) quantifies how often extreme eddy amplitudes occur within that regime. When a control action is considered, a candidate perturbation (\mathbf u_t) is added to the current state, producing a perturbed state (\tilde{\mathbf X}_t). The forward recursion is then re‑run from (\tilde{\mathbf X}t) to obtain a new future regime distribution (\boldsymbol\alpha{t+h}(\mathbf u_t)). The instantaneous danger score at step (h) is
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