Chaos as an Intermittently Forced Linear System (1608.05306v1)

Abstract: Understanding the interplay of order and disorder in chaotic systems is a central challenge in modern quantitative science. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines Takens' delay embedding with modern Koopman operator theory and sparse regression to obtain linear representations of strongly nonlinear dynamics. The result is a decomposition of chaotic dynamics into a linear model in the leading delay coordinates with forcing by low energy delay coordinates; we call this the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the canonical Lorenz system, as well as to real-world examples such as the Earth's magnetic field reversal, and data from electrocardiogram, electroencephalogram, and measles outbreaks. In each case, the forcing statistics are non-Gaussian, with long tails corresponding to rare events that trigger intermittent switching and bursting phenomena; this forcing is highly predictive, providing a clear signature that precedes these events. Moreover, the activity of the forcing signal demarcates large coherent regions of phase space where the dynamics are approximately linear from those that are strongly nonlinear.

• The paper presents a novel HAVOK approach that linearizes chaotic dynamics by combining time-delay embedding with sparse regression.
• It employs Koopman operator theory to reveal underlying linear structures in canonical chaotic systems like the Lorenz and Rössler attractors.
• The method demonstrates broad applicability by accurately forecasting events in diverse domains, from physiological signals to epidemiological outbreaks.

Overview of "Chaos as an Intermittently Forced Linear System"

The paper "Chaos as an Intermittently Forced Linear System" presents a novel approach to decompose chaotic systems using a data-driven methodology, termed HAVOK (Hankel Alternative View of Koopman) analysis. This approach provides a linear representation of chaos in combination with intermittent forcing, yielding a more interpretable framework to predict and control chaotic dynamics.

The core idea leverages Takens' embedding theorem, Koopman operator theory, and sparse regression to convert the inherently nonlinear and complex dynamics of chaos into a linear framework. The authors illustrate this technique's efficacy across a variety of scenarios, including canonical chaotic systems like the Lorenz and Rössler equations, a delay differential equation (Mackey-Glass), as well as experimental and simulated data such as electrocardiograms, electroencephalograms, and measles outbreaks.

Key Methodologies

1. Time-Delay Embedding: Using Takens' embedding theorem, chaotic attractors are reconstructed from delay coordinates. This technique translates a chaotic time series into a higher-dimensional space to expose the dynamics' underlying structure.
2. Koopman Operator Theory: The paper leverages the Koopman operator to find linear representations for nonlinear systems within delay-embedded data. This operator acts on observables of the system, facilitating a global linearization of the dynamics.
3. Sparse Regression: Sparse identification methodologies, like SINDy (Sparse Identification of Nonlinear Dynamics), are employed to distill the most significant dynamics from data. This aids in identifying the sparse terms that contribute most to the system's evolution.
4. Hankel Matrix Decomposition: Singular Value Decomposition (SVD) of Hankel matrices, constructed from time-series data, is used to extract predominant time-delay modes. This hierarchical decomposition allows efficient projection onto a reduced-order linear dynamical system.

Numerical and Real-World Applications

The HAVOK analysis demonstrates its effectiveness across several applications:

• Canonical Systems: In the Lorenz system, HAVOK accurately predicts lobe switching events, where chaotic trajectories hop between different regions of the attractor. In the Rössler system, it captures transient bursts.
• Practical Systems: The method is applied to stochastic models of Earth's magnetic field reversals, indicating its capability to predict rare and significant geomagnetic events.
• Experimental Data: By analyzing ECG and EEG data, HAVOK uncovers latent structures and transitions within these physiological signals. Additionally, it provides forecasts for real-world epidemiological data by identifying patterns in measles outbreak dynamics.

Implications and Future Directions

The paper's framework offers a significant leap towards understanding and controlling chaotic systems using linear proxies. This implies a potential shift in tackling prediction challenges across various scientific fields, from engineering systems to biological and ecological models. Significant implications for control theory emerge, as linear systems are much easier to regulate compared to their nonlinear counterparts.

Future developments could extend this framework's applicability to more complex and higher-dimensional systems, potentially integrating HAVOK analysis with reinforcement learning and other adaptive algorithms to refine control strategies in real-time.

Overall, by reformulating chaos as an intermittently forced linear system, this research holds promise for advancing our ability to model, predict, and manage complex systems across diverse domains.