Tangent bundle viewpoint of the Lorenz system and its chaotic behavior

A new geometric viewpoint of the Lorenz system is proposed based on a theory of tangent bundle. By introducing the geometrical viewpoints of second order system governed by Euler–Poincaré equation or Lie–Poisson equation, geometrical invariants of the Lorenz system can be obtained. Especially, a torsion tensor as one of geometrical invariants relates to the chaotic behavior characterized by the Rayleigh number, and results from the decomposition from the second order system to the tangent space (state space) and base space (configuration space).