Geometrical constraints on finite-time Lyapunov exponents in two and threedimensions

Constraints are found on the spatial variation of finite-time Lyapunov expo nents of two- and three-dimensional systems of ordinary differential equati ons. In a chaotic system, finite-time Lyapunov exponents describe the avera ge rate of separation, along characteristic directions, of neighboring traj ectories. The solution of the equations is a coordinate transformation that takes initial conditions (the Lagrangian coordinates) to the state of the system at a later time (the Eulerian coordinates). This coordinate transfor mation naturally defines a metric tensor, from which the Lyapunov exponents and characteristic directions are obtained. By requiring that the Riemann curvature tensor vanish for the metric tensor (a basic result of differenti al geometry in a flat space), differential constraints relating the finite- time Lyapunov exponents to the characteristic directions are derived. These constraints are realized with exponential accuracy in time. A consequence of the relations is that the finite-time Lyapunov exponents are locally sma ll in regions where the curvature of the stable manifold is large, which ha s implications for the efficiency of chaotic mixing in the advection-diffus ion equation. The constraints also modify previous estimates of the asympto tic growth rates of quantities in the dynamo problem, such as the magnitude of the induced current. (C) 2001 American Institute of Physics.