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27/02/2023
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Phan, T. H. (2022). NONAUTONOMOUS ATTRACTORS FOR YOUNG DIFFERENTIAL EQUATIONS DRIVEN BY UNBOUNDED VARIATION PATHS. Tạp chí Khoa học Thăng Long: Toán và các Khoa học liên quan, 1(4), 117-134. https://science.thanglong.edu.vn/index.php/volc/article/view/53
NONAUTONOMOUS ATTRACTORS FOR YOUNG DIFFERENTIAL EQUATIONS DRIVEN BY UNBOUNDED VARIATION PATHS
Nội dung chính của bài viết
Tóm tắt
We prove the existence of the pullback attractor of the generated ow by a dissipative nonautonomous diferential equations driven by unbounded variation paths under the condition of smallness of nonlinear term. In case perturbed term is linear we prove that the attractor is singleton and also is forward one.
Chi tiết bài viết
Từ khóa
stochastic di erential equations (SDE), Young integral, rough path theory, rough differential equations, exponential stability
Tài liệu tham khảo
[1] L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg New York, 1998. Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, (1990).
[2] N. D. Cong, L. H. Duc, P. T. Hong. Young differential equations revisited. J. Dyn. Di . Equat., Vol. 30, Iss. 4, (2018), 1921-1943.
[3] N. D. Cong, L. H. Duc, P. T. Hong. Lyapunov spectrum of nonautonomous linear Young differential equations. J. Dyn. Di . Equat, (2020), 1749-1777.
[4] L. H. Duc, P. T. Hong, N. D. Cong. Asymptotic stability for stochastic dissipative systems with a Holder noise. Preprint. ArXiv: 1812.04556
[5] N. D. Cong, L. H. Duc, P. T. Hong. Pullback Attractors for Stochastic Young Differential Delay Equations. J. Dyn. Di. Equat., 34(2022), 605- 636.
[6] L. H. Duc, P. T. Hong, N. D. Cong. Asymptotic stability for stochastic dissipative systems with a Holder noise. SIAM J. Control Optim.,Vol. 57, No. 4, pp. 3046 - 3071, (2019)
[7] H. Crauel, P. Kloeden, Nonautonomous and random attractors. Jahresber Dtsch. Math-Ver. 117 (2015), 173-206.
[8] H. Cui, P. E.Kloeden Invariant forward attractors of non-autonomous random dynamical systems. J. Di . Equat., 265, (2018), 6166-6186.
[9] H. Crauel, A. Debussche, F. Flandoli. Random attractors J. Dyn. Di . Equat. 9, (1997), 307-341.
[10] L. H. Duc, M. J. Garrido-Atienza, A. Neuenkirch, B. Schmalfuß. Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2; 1). J. Di . Equat., 264, Iss. 2, (2018), 1119-1145.
[11] L. H. Duc, P. T. Hong. Asymptotic stability of controlled differential equations. J. Dyn. Di . Equat., to appear.
[12] P. Friz, N. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge Unversity Press, Cambridge, 2010.
[13] M. Garrido-Atienza, B. Maslowski, B. Schmalfuß. Random attractors for stochastic equations driven by a fractional Brownian motion. International Journal of Bifurcation and Chaos, Vol. 20, No. 9 (2010) 27612782.
[14] J. A Langa, J. C. Robinson, A. Suarez. Stability, instability, and bifurcation phenomena in non-autonomous differential equations
[15] B. Mandelbrot, J. van Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review, 4, No. 10, (1968), 422-437.
[16] V. V. Nemytskii, V. V. Stepanov. Qualitative theory of differential equations. GITTL, Moscow-Leningrad. (1949). English translation, Princeton University Press, (1960).
[17] D. Nualart, A.R ascanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53, No. 1, (2002), 55-81.
[18] G. Sell. Nonautonomous differential equations and topological dynamics. I. The Basic theory Transactions of the American Mathematical Society, Vol. 127, No. 2 (May, 1967), pp. 241-262.
[19] L.C. Young. An integration of Holder type, connected with Stieltjes integration. Acta Math. 67, (1936), 251-282.
[20] M. Zahle. Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields, 111 (1998), 333-374.
[2] N. D. Cong, L. H. Duc, P. T. Hong. Young differential equations revisited. J. Dyn. Di . Equat., Vol. 30, Iss. 4, (2018), 1921-1943.
[3] N. D. Cong, L. H. Duc, P. T. Hong. Lyapunov spectrum of nonautonomous linear Young differential equations. J. Dyn. Di . Equat, (2020), 1749-1777.
[4] L. H. Duc, P. T. Hong, N. D. Cong. Asymptotic stability for stochastic dissipative systems with a Holder noise. Preprint. ArXiv: 1812.04556
[5] N. D. Cong, L. H. Duc, P. T. Hong. Pullback Attractors for Stochastic Young Differential Delay Equations. J. Dyn. Di. Equat., 34(2022), 605- 636.
[6] L. H. Duc, P. T. Hong, N. D. Cong. Asymptotic stability for stochastic dissipative systems with a Holder noise. SIAM J. Control Optim.,Vol. 57, No. 4, pp. 3046 - 3071, (2019)
[7] H. Crauel, P. Kloeden, Nonautonomous and random attractors. Jahresber Dtsch. Math-Ver. 117 (2015), 173-206.
[8] H. Cui, P. E.Kloeden Invariant forward attractors of non-autonomous random dynamical systems. J. Di . Equat., 265, (2018), 6166-6186.
[9] H. Crauel, A. Debussche, F. Flandoli. Random attractors J. Dyn. Di . Equat. 9, (1997), 307-341.
[10] L. H. Duc, M. J. Garrido-Atienza, A. Neuenkirch, B. Schmalfuß. Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2; 1). J. Di . Equat., 264, Iss. 2, (2018), 1119-1145.
[11] L. H. Duc, P. T. Hong. Asymptotic stability of controlled differential equations. J. Dyn. Di . Equat., to appear.
[12] P. Friz, N. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge Unversity Press, Cambridge, 2010.
[13] M. Garrido-Atienza, B. Maslowski, B. Schmalfuß. Random attractors for stochastic equations driven by a fractional Brownian motion. International Journal of Bifurcation and Chaos, Vol. 20, No. 9 (2010) 27612782.
[14] J. A Langa, J. C. Robinson, A. Suarez. Stability, instability, and bifurcation phenomena in non-autonomous differential equations
[15] B. Mandelbrot, J. van Ness. Fractional Brownian motion, fractional noises and applications. SIAM Review, 4, No. 10, (1968), 422-437.
[16] V. V. Nemytskii, V. V. Stepanov. Qualitative theory of differential equations. GITTL, Moscow-Leningrad. (1949). English translation, Princeton University Press, (1960).
[17] D. Nualart, A.R ascanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53, No. 1, (2002), 55-81.
[18] G. Sell. Nonautonomous differential equations and topological dynamics. I. The Basic theory Transactions of the American Mathematical Society, Vol. 127, No. 2 (May, 1967), pp. 241-262.
[19] L.C. Young. An integration of Holder type, connected with Stieltjes integration. Acta Math. 67, (1936), 251-282.
[20] M. Zahle. Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields, 111 (1998), 333-374.