The backward phase flow method for the Eulerian finite time Lyapunov exponent computations.

Area of Science:

• Numerical analysis
• Dynamical systems theory
• Computational fluid dynamics

Background:

• Lyapunov exponents are crucial for characterizing the stability and predictability of dynamical systems.
• Existing methods for computing Lyapunov exponents, such as phase flow and backward phase flow methods, are often limited to autonomous systems.
• Long-time propagation of level sets in dynamical systems presents computational challenges.

Purpose of the Study:

• To develop a generalized Eulerian approach for computing the Lyapunov exponent of dynamical systems.
• To extend the applicability of flow map computation to time-dependent (periodic and aperiodic) flows.
• To analyze the stability and demonstrate the effectiveness of the proposed numerical method.

Main Methods:

• A simple Eulerian approach is proposed, generalizing a backward phase flow method.
• The method is designed for moderate to long time flow map computation.
• The approach is extended to handle time-dependent dynamical systems, overcoming limitations of previous methods.

Main Results:

• The proposed Eulerian method effectively computes the Lyapunov exponent for both periodic and aperiodic dynamical systems.
• The algorithm demonstrates applicability to time-dependent flows, a significant advancement over autonomous-only methods.
• Numerical examples confirm the stability and effectiveness of the developed approach.

Conclusions:

• The generalized Eulerian method provides a robust and versatile tool for Lyapunov exponent approximation.
• This approach enhances the analysis of complex dynamical systems, including those with time-varying behavior.
• The method offers a stable and effective numerical solution for long-time flow map computation.