Oseledets' splitting of standard-like maps.

Area of Science:

• Dynamical Systems and Chaos Theory
• Geometric Mechanics
• Nonlinear Dynamics

Background:

• Differentiable maps of the plane, particularly standard-like maps (McMillan form), are fundamental in studying chaotic behavior.
• Finite-time Lyapunov exponents (FTLE) quantify the rate of separation of nearby trajectories in phase space, crucial for characterizing chaos.
• Invariant manifolds, stable and unstable, define the local geometric structure around fixed points and periodic orbits in dynamical systems.

Purpose of the Study:

• To establish a direct relationship between the geometric properties of local invariant manifolds and an orbit's contribution to FTLE.
• To conduct a comparative analysis of local Lyapunov exponents, manifold curvature, and the splitting angle of stable/unstable manifolds.
• To investigate the statistical behavior of FTLE contributions in relation to local manifold structures.

Main Methods:

• Analysis of differentiable maps, with a focus on standard-like maps (McMillan form).
• Computation of point-wise curvature of local invariant manifolds.
• Calculation and analysis of finite-time Lyapunov exponents (FTLE) for associated orbits.
• Comparative study involving local Lyapunov exponent, manifold curvature, and manifold splitting angles.

Main Results:

• A simple relation is demonstrated between the directions of local invariant manifolds and their contribution to FTLE.
• The Chirikov-Taylor standard map analysis indicates that positive FTLE contributions predominantly arise from points with locally hyperbolic manifold structures.
• Regions where manifolds are flat and transversal exhibit predominantly positive and large one-step exponents, in a statistical sense.

Conclusions:

• The geometric structure of local invariant manifolds directly influences the finite-time Lyapunov exponents of orbits.
• Hyperbolic manifold configurations are statistically linked to significant positive contributions to FTLE, providing a new perspective on chaotic dynamics.
• Analytic arguments explain this phenomenon and suggest methods for approximating manifold splitting.