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Stability properties of solutions of linear second order differential equations with random coefficients. (English) Zbl 1189.34106

The starting point is the non-autonomous equation

${x}^{\text{'}\text{'}}+{a}^{2}\left(t\right)x=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\ge 0$

in the hypotheses that the coefficient function ${a}^{2}$ is not a deterministic periodic function, and that the lengths of the time intervals between consecutive jump points are independent, positive and not necessarily identically distributed random variables with values in the interval $\left[0,2T\right]$.

The authors give sufficient conditions for both stability and instability of the equilibrium position $x=0$ in the most general case, and hence, they extend some results given in [L. Hatvani and L. Stachó, Arch. Math., Brno 34, No. 1, 119–126 (1998; Zbl 0915.34051)] or in others L. Hatvani’s papers.

The main results are given in Theorem 2.3, Theorem 2.4, Theorem 2.5, Theorem 2.10 and Theorem 2.11. We remark the complex and very explicitly proofs of these results. More examples and more remarks, which explain the possible applications of this equation type, continue the very important theoretical results. Some numerical results are presented.

##### MSC:
 34F05 ODE with randomness 34C11 Qualitative theory of solutions of ODE: growth, boundedness 70L05 Random vibrations (general mechanics)