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Noise expresses exponential growth under regime switching. (English) Zbl 1184.93027
Summary: Consider a given system under regime switching whose solution grows at most polynomially, and suppose that the system is subject to environmental noise in some regimes. Can regime switching and the environmental noise work together to make the system change significantly? The answer is yes. In this paper, we will show that the regime switching and the environmental noise will make the original system whose solution grows at most polynomially become a new system whose solution will grow exponentially. In other words, we reveal that regime switching and the environmental noise will express the exponential growth.

93B12Variable structure systems
93E03General theory of stochastic systems
60H15Stochastic partial differential equations
60G22Fractional processes, including fractional Brownian motion
Full Text: DOI
[1] Hasminskii, R. Z.: Stochastic stability of differential equations, (1981)
[2] Appleby, J. A. D.; Mao, X.: Stochastic stabilisation of functional differential equations, System control lett. 54, No. 11, 1069-1081 (2005) · Zbl 1129.34330 · doi:10.1016/j.sysconle.2005.03.003
[3] Appleby, J. A. D.; Mao, X.; Rodkina, A.: On stochastic stabilisation of difference equations, Discrete contin. Dyn. syst. Ser. A 15, No. 3, 843-857 (2006) · Zbl 1115.39008 · doi:10.3934/dcds.2006.15.843
[4] Arnold, L.; Crauel, H.; Wihstutz, V.: Stabilisation of linear systems by noise, SIAM J. Control optim. 21, 451-461 (1983) · Zbl 0514.93069 · doi:10.1137/0321027
[5] Bellman, R.; Bentsman, J.; Meerkov, S.: Stability of fast periodic systems, IEEE trans. Automat. control 30, 289-291 (1985) · Zbl 0557.93055 · doi:10.1109/TAC.1985.1103936
[6] Caraballo, T.; Garrido-Atienza, M.; Real, J.: Stochastic stabilisation of differential systems with general decay rate, Systems control lett. 48, 397-406 (2003) · Zbl 1157.93537 · doi:10.1016/S0167-6911(02)00293-1
[7] Kushner, H. J.: On the stability of processes defined by stochastic difference-differential equations, J. differential equations 4, 424-443 (1968) · Zbl 0169.11601 · doi:10.1016/0022-0396(68)90028-4
[8] Mao, X.: Stochastic stabilisation and destabilisation, Systems control lett. 23, 279-290 (1994) · Zbl 0820.93071
[9] Meerkov, S.: Condition of vibrational stabilizability for a class of non-linear systems, IEEE trans. Automat. control 27, 485-487 (1982) · Zbl 0491.93034 · doi:10.1109/TAC.1982.1102897
[10] Scheutzow, M.: Stabilisation and destabilisation by noise in the plane, Stochastic anal. Appl. 11, No. 1, 97-113 (1993) · Zbl 0766.60072 · doi:10.1080/07362999308809304
[11] Zhabko, A. P.; Kharitonov, V. L.: Problem of vibrational stabilisation of linear systems, Automat. i. Telemekh. 2, 31-34 (1980) · Zbl 0553.93047
[12] Mao, X.; Marion, G.; Renshaw, E.: Environmental noise suppresses explosion in population dynamics, Stochastic process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[13] Bahar, A.; Mao, X.: Stochastic delay Lotka--Volterra model, J. math. Anal. appl. 292, 364-380 (2004) · Zbl 1043.92034 · doi:10.1016/j.jmaa.2003.12.004
[14] Jiang, D. Q.; Shi, N. Z.: A note on nonautonomous logistic equation with random perturbation, J. math. Anal. appl. 303, 977-986 (2005) · Zbl 1076.34062 · doi:10.1016/j.jmaa.2004.08.027
[15] Deng, F.; Luo, Q.; Mao, X.; Pang, S.: Noise suppresses or expresses exponential growth, Systems control lett. 57, 262-270 (2008) · Zbl 1157.93515 · doi:10.1016/j.sysconle.2007.09.002
[16] Mao, X.; Yuan, C.: Stochastic differential equations with Markovian switching, (2006) · Zbl 1109.60043 · doi:10.1155/JAMSA/2006/59032
[17] Skorohod, A. V.: Asymptotic methods in the theory of stochastic differential equations, (1989)
[18] Mao, X.: Stochastic differential equations and their applications, (2007) · Zbl 1138.60005