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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 '' +a 2 (t)x=0,t0

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 [0,2T].

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:
34F05ODE with randomness
34C11Qualitative theory of solutions of ODE: growth, boundedness
70L05Random vibrations (general mechanics)
References:
[1]Almira, J.; Torres, P. J.: Invariance of the stability of meissner’s equation under perturbation of its intervals, Ann. mat. Pura appl. 180, 245-253 (2001) · Zbl 1105.34312 · doi:10.1007/s10231-001-8205-2
[2]Arnold, V. I.: Mathematical methods of classical mechanics, (1978)
[3]Bihari, I.: Asymptotic result concerning equation x″|x ' |n - 1+a(t)xn*=0. Extension of a theorem by armellini – tonelli – sansone, Studia sci. Math. hungar. 19, 151-157 (1984)
[4]Bobryk, R. V.; Chrzeszczyk, A.: Parametric resonance in coupled oscillators driven by colored noise, Europhys. lett. 68, 344-349 (2004)
[5]Chung, Kai Lai: A course in probability theory, (2001)
[6]Cooper, J.: Parametric resonance in wave equations with a time-periodic potential, SIAM J. Math. anal. 31, 821-835 (2000) · Zbl 0985.34076 · doi:10.1137/S0036141098340703
[7]Elbert, Á.: Stability of some differential equations, , 165-187 (1997) · Zbl 0890.39010
[8]Elbert, Á.: On damping of linear oscillators, Studia sci. Math. hungar. 38, 191-208 (2001) · Zbl 0997.34040 · doi:10.1556/SScMath.38.2001.1-4.13
[9]Fečkan, M.: Existence of almost periodic solutions for jumping discontinuous systems, Acta math. Hungar. 86, 291-303 (2000) · Zbl 0973.34038 · doi:10.1023/A:1006719608910
[10]Galbraith, A. S.; Mcshane, E. J.; Parrish, Gene B.: On the solutions of linear second-order differential equations, Proc. natl. Acad. sci. USA 53, 247-249 (1965) · Zbl 0133.34103 · doi:10.1073/pnas.53.2.247
[11]Gnedenko, B. V.; Kolmogorov, A. N.: Limit distributions for sums of independent random variables, (1968)
[12]Graef, J. R.; Karsai, J.: On the oscillation of impulsively damped halflinear oscillators, Electron. J. Qual. theory differ. Equ. 14 (2000)
[13]Graef, J. R.; Karsai, J.: Behavior of solutions of impulsively perturbed non-halflinear oscillator equations, J. math. Anal. appl. 244, 77-96 (2000) · Zbl 0997.34008 · doi:10.1006/jmaa.1999.6685
[14]Hartman, P.: On a theorem of milloux, Amer. J. Math. 70, 395-399 (1948) · Zbl 0035.18204 · doi:10.2307/2372337
[15]Hartman, P.: The existence of large or small solutions of linear differential equations, Duke math. J. 28, 421-430 (1961) · Zbl 0102.30301 · doi:10.1215/S0012-7094-61-02838-1
[16]Hartman, P.: Ordinary differential equations, (1982) · Zbl 0476.34002
[17]Hatvani, L.: On the existence of a small solution to linear second order differential equations with step function coefficients, Dyn. contin. Discrete impuls. Syst. 4, 321-330 (1998) · Zbl 0916.34013
[18]Hatvani, L.: On small solutions of second order linear differential equations with non-monotonous random coefficients, Acta sci. Math. (Szeged) 68, 705-725 (2002) · Zbl 1027.34058
[19]Hatvani, L.: The growth condition guaranteeing small solutions for a linear oscillator with an increasing elasticity coefficient, Georgian math. J. 14, 269-278 (2007) · Zbl 1131.34029
[20]Hatvani, L.; Stachó, L.: On small solutions of second order differential equations with random coefficients, Arch. math. (Brno) 34, No. 1, 119-126 (1998) · Zbl 0915.34051
[21]Hochstadt, H.: A special Hill’s equation with discontinuous coefficients, Amer. math. Monthly 70, 18-26 (1963) · Zbl 0117.05103 · doi:10.2307/2312778
[22]Kiguradze, I. T.; Chanturia, T. A.: Asymptotic properties of solutions of nonautonomous ordinary differential equations, Math. appl. (Soviet ser.) 89 (1993) · Zbl 0782.34002
[23]De L. Kronig, R.; Penney, W. G.: Quantum mechanics in crystal lattices, Proc. R. Soc. lond. 130, 499-513 (1931) · Zbl 0001.10601 · doi:10.1098/rspa.1931.0019
[24]Lakshmikantham, V.; Papageorgiou, N. S.; Vasundhara, J.: The method of upper and lower solutions and monotone technique for impulsive differential equations with variable moments, Appl. anal. 51, 41-58 (1993) · Zbl 0793.34013 · doi:10.1080/00036819308840203
[25]Macki, J. W.: Regular growth and zero-tending solutions, Lecture notes in math. 1032, 358-374 (1983) · Zbl 0529.34040
[26]Mcshane, E. J.: On the solutions of the differential equation y″+p2y=0, Proc. amer. Math. soc. 17, 55-61 (1966)
[27]Meissner, E.: Über schüttelschwingungen in systemen mit periodisch veränderlicher elastizität, Schweizer bauzeitung 72, No. 10, 95-98 (1918)
[28]Milloux, H.: Sur l’equation differentielle x″+A(t)x=0, Prace mat.-fiz. 41, 39-54 (1934)
[29]Ruijgrok, M.; Verhulst, F.: Parametric and autoparametric resonance, Nodea nonlinear differential equations appl. 19, 279-298 (1996) · Zbl 0846.34028
[30]Pucci, P.; Serrin, J.: Precise damping conditions for global asymptotic stability for nonlinear second order systems, Acta math. 170, 275-307 (1993) · Zbl 0797.34059 · doi:10.1007/BF02392788
[31]Pucci, P.; Serrin, J.: Asymptotic stability for ordinary differential systems with time-dependent restoring potentials, Arch. ration. Mech. anal. 132, 207-232 (1995) · Zbl 0861.34034 · doi:10.1007/BF00382747
[32]Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, World sci. Ser. nonlinear sci. Ser. A monogr. Treatises 14 (1995) · Zbl 0837.34003
[33]Seĭranyan, A. A.; Seĭranyan, A. P.: Decay of resonance zones for the meissner equation with the introduction of small dissipation, Vestnik moskov. Univ. ser. I mat. Mekh. 5, 53-59 (2003) · Zbl 1127.70310
[34]Tondl, A.; Ruijgrok, T.; Verhulst, F.; Nabergoj, R.: Autoparametric resonance in mechanical systems, (2000)