American Institute of Mathematical Sciences

Advanced
July  2001, 7(3): 487-515. doi: 10.3934/dcds.2001.7.487

Moment attractivity, stability and contractivity exponents of stochastic dynamical systems

Henri Schurz 1,

1. 

Department of Mathematics, University of Minnesota, School of Mathematics, Minneapolis, MN 55455, United States

Received  August 1999 Revised  December 2000 Published  April 2001

Nonlinear stochastic dynamical systems as ordinary stochastic differential equations and stochastic difference equations are in the center of this presentation in view of the asymptotic behavior of their moments. We study the exponential p-th mean growth behavior of their solutions as integration time tends to infinity. For this purpose, the concepts of attractivity, stability and contractivity exponents for moments are introduced as generalizations of well-known moment Lyapunov exponents of linear systems. Under appropriate monotonicity assumptions we gain uniform estimates of these exponents from above and below. Eventually, these concepts are generalized to describe the exponential growth behavior along certain Lyapunov-type functionals.
Keywords: (ordinary) stochastic differential equations, moment contractivity exponents, asymptotic behavior, moment stability exponents, Stochastic dynamical systems, sample Lyapunov exponents, moment Lyapunov exponents, explicit-implicit methods., moment attractivity exponents, stochastic numerical methods, characteristic exponents.
Mathematics Subject Classification: Primary 37H10, 34F05, 60H10, 65C30, 93E1.
Citation: Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487
[1]

Nguyen Dinh Cong, Nguyen Thi Thuy Quynh. Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term. Conference Publications, 2011, 2011 (Special) : 332-342. doi: 10.3934/proc.2011.2011.332
[2]

Igor G. Vladimirov. The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 575-600. doi: 10.3934/dcdsb.2013.18.575
[3]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697
[4]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187
[5]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[6]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[7]

Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102
[8]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[9]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[10]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098
[11]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[12]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
[13]

Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807
[14]

Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166
[15]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[16]

Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092
[17]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009
[18]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[19]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[20]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences