Stability properties of solutions of linear second order differential equations with random coefficients

Abstract

The equation x ″ + a 2 ( t ) x = 0 , a ( t ) : = a k > 0 if t k − 1 ⩽ t < t k ( k ∈ N ) is considered where { a k } k = 1 ∞ is given and { t k } k = 1 ∞ is a random sequence. Sufficient conditions are proved which guarantee either stability or instability for the zero solution. Stability means that all solutions almost surely tend to zero as t → ∞ . By instability we mean that the sequence of the expected values of the amplitudes of every solution tends to infinity as k → ∞ . It turns out that a k ↗ ∞ ( k → ∞ ) implies stability for all absolutely continuous distributions and for the “overwhelming majority” of the singular distributions. The instability theorem is applied to the problem of random swinging, when { a k } k = 1 ∞ is periodic with two different terms (Meissner's equation) and { t k − t k − 1 } k = 1 ∞ are independent identically distributed random variables. The application gives conditions for stochastic parametric resonance.