ABSTRACT
As an example of the applications of the lemma on stability, we consider the problem of parametric resonance for the equation
d2x
dt2 + ω2[1 + εf(t)]x = 0, (4.1)
where ω is a real parameter, ε > 0 is a small parameter, f(t) is an almost periodic or periodic function. If the parameter ω is such that the zero solution of equation (4.1) is unstable, then this equation has unbounded solutions. In order to find such values of the parameter ω, we use the lemma on stability. We shall assume that f(t) = A cosλt, i.e. we consider the Mathieu equation
d2x
dt2 + ω2[1 + εA cosλt]x = 0. (4.2)
Rewrite equation (4.2) as
d2x
dt2 + ω2x = F (t), (4.3)
where F (t) = −εω2Ax cosλt. By means of a change
x = a cos νt + b sin νt, dx dt = −aν sin νt + bν cos νt,
(4.4)
where a, b are new variables, the frequency ν being chosen later, we transform equation (4.3) into the system
da dt cos νt +
db dt sin νt = 0
−dadt ν sin νt + dbdtν cos νt− ν2(a cos νt + b sin νt) = −ω2(a cos νt + b sin νt)−εω2A cosλt)(a cos νt + b sin νt).
