Reduced dimension and Rotation vector formula of ordinary differential equation
Résumé
The leader trajectory function defined in this article is an approximated solution of a differential equation. It is defined by some independent one-dimensional differential equations. The generalized main result of this article asserts that if the leader trajectory exists then it is at finite distance from the solution of the system. However, the differential equation can be controlled by a reduced one-dimensional differential equations. The application of the generalized main result is to control the trajectory of the periodic systems. We prove that for any periodic system and any initial condition there exists a leader trajectory which is a linear function of the time variable. In other words, we find an exact Rotation vector formula which is the relation between the rotation vector and the initial condition. In addition, we find easily a necessary and sufficient condition for the existence of a locally constant rotation vector under perturbation of the system, known by the the Arnold Tongue. The main result answer to many problems in natural sciences.
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