Resolvent-based optimization for approximating the statistics of a chaotic Lorenz system

Area of Science:

• Fluid Dynamics
• Chaos Theory
• Computational Physics

Background:

• Traditional methods for analyzing turbulent flows and chaotic systems, such as cycle expansion, are computationally expensive for high-dimensional systems.
• Identifying unstable periodic orbits (UPOs) is crucial for understanding the statistical properties of chaotic trajectories.
• Existing techniques struggle with the computational demands of high-dimensional fluid dynamics.

Purpose of the Study:

• To develop a computationally efficient framework for approximating the statistical properties of turbulent flows.
• To overcome the limitations of traditional methods by leveraging dimensionality reduction techniques.
• To demonstrate the framework's efficacy on a well-known chaotic system, the Lorenz 1963 equations.

Main Methods:

• Combines variational methods for finding unstable periodic orbits with resolvent analysis for dimensionality reduction.
• Constructs approximate trajectories in a low-dimensional subspace using resolvent modes.
• Employs gradient-based optimization to adjust mode amplitudes, minimizing projected governing equation violations.

Main Results:

• Achieved an exact dimensionality reduction of the Lorenz 1963 equations from three to two dimensions using resolvent analysis.
• Averaged observables, probability distributions, and spectra rapidly converged to values from long chaotic simulations with limited iterations.
• Demonstrated that approximate trajectories provide a sufficient 'sketch' of the system's attractor.

Conclusions:

• The proposed framework effectively approximates the statistical behavior of chaotic systems.
• Exact solutions are not necessary for capturing essential statistical properties of turbulent flows.
• This approach offers a computationally feasible alternative for analyzing complex dynamical systems.