General method for the solution of some problems of stabilization and destabilization of motion

Abstract

We demonstrate a complete mathematical analogy between the description of motion of an electron in a periodic field and the phenomenon of parametric resonance. A band approach to the analysis of the phenomenon of parametric resonance is formulated. For an oscillator under the action of an external force described by the Weierstrass function, we calculate the increments of increase in oscillations and formulate a condition for parametric resonance. For the known problem of a pendulum with vibrating point of suspension, we find exact conditions for the stabilization of the pendulum in the upper (unstable) equilibrium position by using the Lamé equation.