Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method

Area of Science:

• Applied Mathematics
• Computational Physics
• Dynamical Systems Theory

Background:

• Chaotic dynamics in complex systems are underpinned by unstable periodic orbits (UPOs).
• Accurate computation of UPOs is crucial for understanding and predicting chaotic phenomena like fluid turbulence.
• Existing methods for computing UPOs in high-dimensional systems suffer from poor convergence or high computational cost.

Purpose of the Study:

• To introduce a new, efficient, and robust method for computing UPOs in high-dimensional spatiotemporally chaotic systems.
• To overcome the limitations of existing computational techniques, specifically addressing poor convergence and computational expense.

Main Methods:

• A novel matrix-free, adjoint-based variational method is proposed.
• This method constructs an initial value problem in the space of closed loops, making UPOs attracting fixed points.
• The approach is demonstrated on the one-dimensional Kuramoto-Sivashinsky equation, a model for spatiotemporal chaos.

Main Results:

• The proposed method exhibits robust convergence for computing UPOs.
• It is unaffected by exponential error amplification common in time-marching methods.
• Convergence is independent of the orbit's period and does not require accurate initial guesses.

Conclusions:

• The new matrix-free method provides a powerful framework for analyzing chaotic dynamics in high-dimensional systems.
• This advancement facilitates the prediction of statistics for chaotic flows, including fluid turbulence.
• The method's global convergence and efficiency make it applicable to a wide range of complex dynamical systems.