General disagreement between the Geometrical Description of Dynamical In-stability -using non affine parameterizations- and traditional Tangent Dynamics

TL;DR

This paper reveals that geometrical Lyapunov exponents can significantly differ from tangent dynamics exponents when non-affine parameterizations are used, affecting the interpretation of dynamical stability.

Contribution

It demonstrates that non-affine parameterizations cause discrepancies between geometrical and tangent Lyapunov exponents, highlighting potential misinterpretations in stability analysis.

Findings

Geometrical Lyapunov exponents differ from tangent exponents under non-affine parameterizations.

Non-affine parameterizations can lead to false indications of chaos due to parametric resonance.

Results challenge the equivalence of geometrical and tangent stability measures in certain frameworks.

Abstract

In this paper, the general disagreement of the geometrical lyapunov exponent with lyapunov exponent from tangent dynamics is addressed. It is shown in a quite general way that the vector field of geodesic spread $\xi^k_G$ is not equivalent to the tangent dynamics vector $\xi^k_T$ if the parameterization is not affine and that results regarding dynamical stability obtained in the geometrical framework can differ qualitatively from those in the tangent dynamics. It is also proved in a general way that in the case of Jacobi metric -frequently used non affine parameterization-, $\xi^k_G$ satisfies differential equations which differ from the equations of the tangent dynamics in terms that produce parametric resonance, therefore, positive exponents for systems in stable regimes.