随机FitzHugh-Nagumo系统的随机吸引子的分形维数
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摘要
在给定的Hilbert空间中研究了具可加白噪音的非自治FitzHugh-Nagumo系统的解的渐近行为.首先,证明经变换后相等价的动力系统的随机吸引子的存在性.然后,在可分的Banach空间上,提出了估计随机动力系统的随机不变集的分形维数上界的方法.最后,利用随机变量的期望和上述条件,证明了具可加白噪音的随机FitzHugh-Nagumo系统的随机吸引子的分形维数的有限性.
We consider the asymptotic behavior of solutions for non-autonomous FitzHugh-Nagumo system driven by additive white noise in Hilbert space.Firstly,we investigate the existence of random attractor of the random dynamical system generated by the solutions of considered system.Secondly,we present criterion for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable Banach space.Finally,we apply expectation of some random variables and these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic FitzHugh-Nagumo system driven by additive white noise.
We consider the asymptotic behavior of solutions for non-autonomous FitzHugh-Nagumo system driven by additive white noise in Hilbert space.Firstly,we investigate the existence of random attractor of the random dynamical system generated by the solutions of considered system.Secondly,we present criterion for estimating an upper bound of the fractal dimension of a random invariant set of a random dynamical system on a separable Banach space.Finally,we apply expectation of some random variables and these conditions to prove the finiteness of fractal dimension of the random attractors for stochastic FitzHugh-Nagumo system driven by additive white noise.
引文
[1]WANG B X.Random attractors for the stochastic Fitz Hugh-Nagumo systems on unbounded domains[J].Nonliear Anal,2009,71:2 811-2 828.
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[3]ADILI A,WANG B X.Random attractors for stochastic Fitz Hugh-Nagumo systems driven by deterministic non-autonomous forcing[J].Discrete&Continuous Dynamical Systems-Series B,2013,18(3):643-666.
[4]刘林芳,尹福其,徐明亮.无界区域上具可乘白噪音的Fitzhngh-Nagumo方程的渐近行为[J].湘潭大学自然科学学报,2014,36(1):8-15.LIU L F,YIN F Q,XU M L.The asymptotic behavior of Fitzhugh-Nagumo systems with multiplicative white noise on unbounded domains[J].Natural Science Journal of Xiangtan University,2014,36(1):8-15.
[5]WANG Z J,ZHOU S F,GU A.Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains[J].Nonlinear Analysis:Real World Applications,2011,12(6):3 468-3 482.
[6]ZHOU S F,TIAN Y X,WANG Z J.Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations[J].Applied Mathematics and Computation,2016,276:80-95.
[7]王磊,党金宝,林国广.分数次非线性Schrodinger方程的整体吸引子及维数估计[J].云南大学学报:自然科学版,2010,32(2):130-135.WANG L,DANG J B,LIN G G.The global attractor of the fractal nonlinear Schrodinger equation and the estimate of its dimension[J].Journal of Yunnan University:Natural Sciences Edition,2010,32(2):130-135.
[8]乔丽华,吴志敏,林国广.随机Cahn-Hilliard方程的随机吸引子及其Hausdorff维数[J].云南大学学报:自然科学版,2012,34(3):249-257.QIAO L H,WU Z M,LIN G G.Random attractor and Hausdorff dimension for stochastic Cahn-Hilliard equation[J].Journal of Yunnan University:Natural Sciences Edition,2012,34(3):249-257.
[9]WANG G,TANG Y B.Fractal dimension of a random invariant set and applications[J].Appl Math,2013,2013:415764.DOI:10.1155/2013/415764.
[10]LANGA J A,ROBINSON J C.Fractal dimension of a random invariant set[J].Math Pures Appl,2006,85(2):269-294.
[11]ZHOU S F,ZHAO M.Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation[J].Nonlinear Analysis,2016,133(2):292-318.
[12]CRAUEL H,DEBUSSCHE A,FLANDOLI F.Random attractors[J].J Dyn Differ Equ,1997,9(2):307-341.
[13]CHUESHOV I.Monotone random systems theory and applications[M].Berlin Heidlberg:Springer-Verlag,2002.
[14]WANG B X.Sufficient and necessary criteria for existence of pullback attarctors for non-compact random dynamical systems[J].J Differ Equ,2012,253(5):1 544-1 583.
[15]FAN X M.Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise[J].Stoch Anal Appl,2006,24(4):767-793.
[16]尹福其,周盛凡,李红艳.具强阻尼的随机sine-Gordon方程的随机吸引子存在性[J].上海大学学报:自然科学版,2006(3):260-265.YIN F Q,ZHOU S F,LI H Y.Existence of random attractor for strongly damped stochastic sine-Gordon equation[J].Journal of Shanghai University:Natural Science Edition,2006(3):260-265.
[17]尹福其,刘林芳,肖翠辉.无界区域非自治随机sine-Gordon方程的D-周期吸引子[J].数学学报:中文版,2014,57(6):1 127-1 140.YIN F Q,LIU L F,XIAO C H.Periodic D-pullback attractor for stochastic sine-Gordon equation on unbounded domains[J].Acta Mathematica Sinica:Chinese Series,2014,57(6):1 127-1 140.
[18]YIN F Q,LI X L.Fractal dimensions of random attractors for stochastic Benjamin-Bona-Mahony equation driven by white noise[J].Computers&Mathematics with Applications,Available online 8 December 2017,In press,https://doi.org/10.1016/j.camwa.2017.11.025.
[2]ADILI A,WANG B X.Random attractors for non-autonomous stochastic Fitz Hugh-Nagumo systems with multiplicative noise[J].Discrete and Contin Dyn Syst Series A,2013,34(1):269-300.
[3]ADILI A,WANG B X.Random attractors for stochastic Fitz Hugh-Nagumo systems driven by deterministic non-autonomous forcing[J].Discrete&Continuous Dynamical Systems-Series B,2013,18(3):643-666.
[4]刘林芳,尹福其,徐明亮.无界区域上具可乘白噪音的Fitzhngh-Nagumo方程的渐近行为[J].湘潭大学自然科学学报,2014,36(1):8-15.LIU L F,YIN F Q,XU M L.The asymptotic behavior of Fitzhugh-Nagumo systems with multiplicative white noise on unbounded domains[J].Natural Science Journal of Xiangtan University,2014,36(1):8-15.
[5]WANG Z J,ZHOU S F,GU A.Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains[J].Nonlinear Analysis:Real World Applications,2011,12(6):3 468-3 482.
[6]ZHOU S F,TIAN Y X,WANG Z J.Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations[J].Applied Mathematics and Computation,2016,276:80-95.
[7]王磊,党金宝,林国广.分数次非线性Schrodinger方程的整体吸引子及维数估计[J].云南大学学报:自然科学版,2010,32(2):130-135.WANG L,DANG J B,LIN G G.The global attractor of the fractal nonlinear Schrodinger equation and the estimate of its dimension[J].Journal of Yunnan University:Natural Sciences Edition,2010,32(2):130-135.
[8]乔丽华,吴志敏,林国广.随机Cahn-Hilliard方程的随机吸引子及其Hausdorff维数[J].云南大学学报:自然科学版,2012,34(3):249-257.QIAO L H,WU Z M,LIN G G.Random attractor and Hausdorff dimension for stochastic Cahn-Hilliard equation[J].Journal of Yunnan University:Natural Sciences Edition,2012,34(3):249-257.
[9]WANG G,TANG Y B.Fractal dimension of a random invariant set and applications[J].Appl Math,2013,2013:415764.DOI:10.1155/2013/415764.
[10]LANGA J A,ROBINSON J C.Fractal dimension of a random invariant set[J].Math Pures Appl,2006,85(2):269-294.
[11]ZHOU S F,ZHAO M.Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation[J].Nonlinear Analysis,2016,133(2):292-318.
[12]CRAUEL H,DEBUSSCHE A,FLANDOLI F.Random attractors[J].J Dyn Differ Equ,1997,9(2):307-341.
[13]CHUESHOV I.Monotone random systems theory and applications[M].Berlin Heidlberg:Springer-Verlag,2002.
[14]WANG B X.Sufficient and necessary criteria for existence of pullback attarctors for non-compact random dynamical systems[J].J Differ Equ,2012,253(5):1 544-1 583.
[15]FAN X M.Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise[J].Stoch Anal Appl,2006,24(4):767-793.
[16]尹福其,周盛凡,李红艳.具强阻尼的随机sine-Gordon方程的随机吸引子存在性[J].上海大学学报:自然科学版,2006(3):260-265.YIN F Q,ZHOU S F,LI H Y.Existence of random attractor for strongly damped stochastic sine-Gordon equation[J].Journal of Shanghai University:Natural Science Edition,2006(3):260-265.
[17]尹福其,刘林芳,肖翠辉.无界区域非自治随机sine-Gordon方程的D-周期吸引子[J].数学学报:中文版,2014,57(6):1 127-1 140.YIN F Q,LIU L F,XIAO C H.Periodic D-pullback attractor for stochastic sine-Gordon equation on unbounded domains[J].Acta Mathematica Sinica:Chinese Series,2014,57(6):1 127-1 140.
[18]YIN F Q,LI X L.Fractal dimensions of random attractors for stochastic Benjamin-Bona-Mahony equation driven by white noise[J].Computers&Mathematics with Applications,Available online 8 December 2017,In press,https://doi.org/10.1016/j.camwa.2017.11.025.