Spatial Van Der Pol Oscillator. Technical Applications in Gyroscopy

The Van der Pol equations describing self-oscillations in the quasilinear one-dimensional oscillator are generalized to the case when the generating isotropic oscillator has an arbitrary number of degrees of freedom. Two-dimensional (planar) and three-dimensional (spatial) cases are considered specifically.In contrast to the classical problem in which a given amplitude of oscillations is stabilized, in the general case it is possible to stabilize not only the energy of oscillations but also the area of a planar elliptical trajectory, its orientation in space, the frequency of the oscillatory process and precession. Promising technical applications of the corresponding mathematical models are indicated.