Designing Chaotic Attractors: A Semi-supervised Approach

Abstract

Chaotic dynamics are ubiquitous in nature and useful in engineering, but their geometric design can be challenging. Here, we propose a method using reservoir computing to generate chaos with a desired shape by providing a periodic orbit as a template, called a skeleton. We exploit a bifurcation of the reservoir to intentionally induce unsuccessful training of the skeleton, revealing inherent chaos. The emergence of this untrained attractor, resulting from the interaction between the skeleton and the reservoir's intrinsic dynamics, offers a novel semi-supervised framework for designing chaos.

Figures (4)

• Figure 1: Schematic diagram of the proposed method. (A) Periodic time series defining the overall shape. (B) Reservoir learning the skeleton. (C) Parameter exploration: Finding the parameter $\rho^{*}$. (D) Designed chaotic orbit. Left: Reservoir's output. Right: Plot of the internal state based on the principal components.
• Figure 2: Analysis of learning the Lissajous curve. (A) CLE of the driven system and MLE of the closed-loop system. (B) RMSE in $x$ component in open-loop prediction for 10,000 steps and predictions at $\rho\in\{1.20, 1.25, 1.30, 1.35\}$. (C) Time average of evaluation index $Q$ for the output of the closed-loop system over 2,000 steps, and trajectories at the corresponding points. (D) Bifurcation diagram based on extrema of node averages $\bar{x}^i_k=\frac{1}{N}\sum_{i=1}^{N}x^i_k$. Calculated for 10,000 steps, discarding the first 8,000 steps. (E) Pre- and post-training effective spectral radius labelled "pre" and "pst," respectively.
• Figure 3: Analysis of learning the Lissajous curve near the edge of chaos (the region meshed in Fig. \ref{['fig:macro']}). The blue dotted line f indicates $\rho^{(\mathrm{E})}$. (A) CLE and MLE. (B) Bifurcation diagram. Light green represents transients up to 2,000 steps. (C) Evaluation index $\langle Q \rangle$ in the closed-loop state. (D) Outputs of the autonomous system at points $\rho \in \{1.2800, 1.2850, 1.2900, 1.2910, 1.2925, 1.2940, 1.2995\}$ indicated by the dotted lines a--g.
• Figure 4: Examples of chaotic untrained attractors. The left panel shows skeleton$\bm{u}_k$, the middle panel shows $\bm{z}_k$, and the right panel shows the plot based on the first and second principal components of $\bm{x}_k$. (A) Van der Pol equation. $\mathrm{MLE}=0.002$ (B) Handwritten "at" symbol. $\mathrm{MLE}=0.004$. (C) Periodic orbit of the Rössler system. $\mathrm{MLE}=0.003$. (D) Time series created from the C-E-G keys of the piano. $\mathrm{MLE}=0.0003$. In all cases, $(a, \beta)=(0.5, 0.001)$. For A and B, $(N, \sigma)=(1000, 0.2)$; for C, $(1000, 0.02)$; and for D, $(1500, 0.2)$. The realizations of $\bm{W}$ are the same for A, B and C. For A and B, the realizations of $\bm{W}_{in}$ are also the same.