  Transition from chaotic to quasi-periodic phase in modified Lorenz model is
analyzed by performing the contact transformation such that the trajectory in
${\Vec R}^3$ is projected on ${\Vec R}^2$. The relative torsion number and the
characteristics of the template are measured using the eigenvector of the
Jacobian instead of vectors on moving frame along the closed trajectory.
  Application to the circulation of a fluid in a convection loop and
oscillation of the electric field in single-mode laser system are performed.
The time series of the eigenvalues of the Jacobian and the scatter plot of the
trajectory in the transformed coordinate plane $X-Z$ in the former and
$|X|-|Z|$ in the latter, allow to visualize characteristic pattern change at
the transition from quasi-periodic to chaotic. In the case of single mode
laser, we observe the correlation between the critical movement of the
eigenvalues of the Jacobian in the complex plane and intermittency.

Chaotic Dynamics