李雅普诺夫指数对应的特征方向

洛伦兹系统的李雅普诺夫指数的符号为 $(+，0，-)$

• Any continuous time-dependent dynamical system without a fixed point will have at least one
  zero exponent, corresponding to the slowly changing magnitude of a principal axis tangent to the flow
• The sum of the Lyapunov exponents is the time-averaged divergence of the phase space
  velocity; hence any dissipative dynamical system will have at least one negative exponent, the sum of all of the exponents is negative, and the posttransient motion of trajectories will occur on a zero volume limit set, an attractor.
• The exponential expansion indicated by a positive Lyapunov exponent is incompatible with motion on a bounded attractor unless some sort of folding process merges widely separated trajectories. Each positive exponent reflects a “direction” in which the system experiences the repeated stretching and folding that decorrelates nearby states on the attractor.

上面三段话如醍醐灌顶，可谓直达李雅普诺夫指数的本质