Controlling Chaotic Maps using Next-Generation Reservoir Computing

TL;DR

This work addresses chaotic-system control with a data-driven approach by coupling Sarangapani's discrete-time control framework to next-generation reservoir computing (NG-RC). The NG-RC learns a Brunovsky-form model ${\hat{Y}}_{i+1} = {\hat{W}}_X {\mathbb{O}}_{X,i} + {\hat{W}}_u {\bf u}_i$ and an associated control law $\mathbf{u}_i$ that cancels nonlinear dynamics while providing linear feedback, enabling robust stabilization of the Hénon map between unstable fixed points, to a four-period orbit, and to arbitrary states with as few as $M_{train}=10$ points. The method shows resilience to noise and modeling error, and achieves fast convergence with a small, seven-parameter weight vector, making it attractive for edge-computing hardware. Overall, NG-RC-based control offers a data-efficient, computationally lightweight route to real-time control of low- to moderate-dimensional chaotic systems and motivates extending to higher-dimensional dynamics.

Abstract

In this work, we combine nonlinear system control techniques with next-generation reservoir computing, a best-in-class machine learning approach for predicting the behavior of dynamical systems. We demonstrate the performance of the controller in a series of control tasks for the chaotic Hénon map, including controlling the system between unstable fixed-points, stabilizing the system to higher order periodic orbits, and to an arbitrary desired state. We show that our controller succeeds in these tasks, requires only 10 data points for training, can control the system to a desired trajectory in a single iteration, and is robust to noise and modeling error.

Figures (8)

• Figure 1: Learning and controlling a dynamical system (plant) using next-generation reservoir computing. (top) The NG-RC is trained to predict $x_{i+1}$, the next step of one of the plant's observable variable. (bottom) The controller is designed based on the NG-RC model estimation for the evolution of the observable variable to be controlled. Then, it applies a control signal $u_i$ to the plant, which is driven towards a desired state $x_{des,i+1}$.
• Figure 2: Phase portrait of the Hénon attractor for $a=1.4$, $b=0.3$ iterated for 500 points (blue xs). The orange square represents the unstable fixed points $P_{U1}$ and $P_{U2}$ with coordinate values $(x_{U1},y_{U1}) = (+0.63135,+0.18941)$ and $(x_{U2},y_{U2}) = (-1.13135,-0.33941)$, respectively. The red circles represent the points in the four-period orbit $P_1,P_2,P_3,P_4$ given by $(x_1,y_1)=(0.638194,-0.21203)$, $(x_2,y_2)=(0.217762,0.191458)$, $(x_3,y_3)=(1.12507,0.0653285)$, and $(x_4,y_4) = (-0.706767,0.337521)$, respectively. The dashed black line represents the trapping region: all points within the region remain within the region, and all points outside diverge to infinity.Henon1976
• Figure 3: Predicting the Hénon map system for different noise levels (color coded) for limited training data. From top to bottom, the average RMSE after ten training points is $1.09 \pm 0.01 \times 10^{-5}$, $1.05 \pm 0.01 \times 10^{-4}$, $1.04 \pm 0.01 \times 10^{-3}$, $1.02 \pm 0.01 \times 10^{-2}$, and $0.98 \pm 0.01 \times 10^{-1}$, and the average optimal $\alpha$ is $9.93 \pm 0.02 \times 10^{-8}$, $8.11 \pm 0.02 \times 10^{-7}$, $6.89 \pm 0.02 \times 10^{-6}$, $1.62 \pm 0.65 \times 10^{-4}$, and $1.62 \pm 0.34 \times 10^{-2}$.
• Figure 4: Controlling the Hénon map dynamics between the two unstable fixed points. (top) $x$, (middle) $y$, and (bottom) $u$ as a function of iterations for different values of $K$ (color code). The horizontal dashed line (top) indicates $x_{des}=x_{U2}$, and initial conditions for $x$ and $y$ are $x_{U1}$ and $y_{U1}$, respectively.
• Figure 5: Controlling Hénon map dynamics to a four-period orbit. (top) $x$, (middle) $y$, and (bottom) $u$ as a function of iterations for different values of $K$ (color code). Horizontal dashed lines (top) indicate the $x_{des}$ values, which change between $P_1$, $P_2$, $P_3$, and $P_4$ every iteration step starting at $x_{des} = x_1$ at iteration 0.