American Institute of Mathematical Sciences

Advanced
February  2013, 33(2): 701-721. doi: 10.3934/dcds.2013.33.701

On transverse stability of random dynamical system

Xiangnan He 1, , Wenlian Lu 2, and  Tianping Chen 3,

1. 

Center for Computational Systems Biology, Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University，Shanghai, 200433, China
2. 

Center for Computational Systems Biology, Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433
3. 

Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433

Received  July 2011 Revised  April 2012 Published  September 2012

In this paper, we study the transverse stability of random dynamical systems (RDS). Suppose a RDS on a Riemann manifold possesses a non-random invariant submanifold, what conditions can guarantee that a random attractor of the RDS restrained on the invariant submanifold is a random attractor with respect to the whole manifold? By the linearization technique, we prove that if all the normal Lyapunov exponents with respect to the tangent space of the submanifold are negative, then the attractor on the submanifold is also a random attractor of the whole manifold. This result extends the idea of the transverse stability analysis of deterministic dynamical systems in [1,3]. As an explicit example, we discuss the complete synchronization in network of coupled maps with both stochastic topologies and maps, which extends the well-known master stability function (MSF) approach for deterministic cases to stochastic cases.
Keywords: RDS, stochastic topologies and maps, Lyapunov exponents, complete synchronization., transverse stability.
Mathematics Subject Classification: Primary: 37H15, 37D10; Secondary: 60F1.
Citation: Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701
References:
[1]

J. C. Alexander, I. Kan, J. A. Yorke and Z. You, Riddled basins,, Int. J. Bifurcation Chaos, 2 (1992), 795.  doi: 10.1142/S0218127492000446.  Google Scholar

[2]

L. Arnold, "Random Dynamical Systems,", Springer-Verlag Berlin Heidelberg, (1998).   Google Scholar

[3]

P. Ashwin, J. Buescu and I. Stewart, From attractor to chaotic saddle: a tale of transverse instability,, Nonlinearity, 9 (1996), 703.  doi: 10.1088/0951-7715/9/3/006.  Google Scholar

[4]

P. Ashwin, E. Covas and R. Tavakol, Transverse instability for non-normal parameters,, Nonlinearity, 12 (1999), 563.  doi: 10.1088/0951-7715/12/3/009.  Google Scholar

[5]

P. Ashwin, Minimal attractors and bifurcations of random dynamical systems,, Proc. Rhys. Soc. Lond. A, 455 (1999), 2615.  doi: 10.1098/rspa.1999.0419.  Google Scholar

[6]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Autom. Control, 52 (2007), 852.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[7]

D. Cui, X. Liu, Y. Wan and X. Li, Estimation of genuine and random synchronization in multivariate neural series,, Neural Networks, 23 (2010), 698.  doi: 10.1016/j.neunet.2010.04.003.  Google Scholar

[8]

R. S. Ellis, Large deviation for a general class of random vectors,, The Annals of Probability, 12 (1984), 1.  doi: 10.1214/aop/1176993370.  Google Scholar

[9]

Y. Fang, "Stability Analysis of Linear Control Systems with Uncertain Parameters,", Ph. D thesis, (1994).   Google Scholar

[10]

S. Floyd and V. Jacobson, The synchronization of periodic routing messages,, IEEE Trans. Netw, 2 (1994), 122.   Google Scholar

[11]

V. Gazi and K. Passino, Stability analysis of social foraging swarms,, IEEE Trans. SMCS-Part B: Cybernetics, 34 (2004), 539.   Google Scholar

[12]

H. Huang, W. Daniel and Y. Qu, Robust stability of stochastic delayed additive neural networks with Markovian switching,, Neural Networks, 20 (2007), 799.  doi: 10.1016/j.neunet.2007.07.003.  Google Scholar

[13]

A. Lotka, "Elements of Physical Biology,", Williams & Wilkins Company, (1925).   Google Scholar

[14]

W. Lu, M. F. Atay and J. Jost, Synchronization of discrete-time dynamical networks with time-varying couplings,, SIAM Journals on Mathematical Analysis, 39 (2007), 1231.  doi: 10.1137/060657935.  Google Scholar

[15]

W. Lu, M. F. Atay and J. Jost, Chaos synchronization in networks of coupled map with time-varying topologies,, Eur. Phys. J. B, 63 (2008), 399.  doi: 10.1140/epjb/e2008-00023-3.  Google Scholar

[16]

X. Mao, A. Matasov and A. Piunovskiy, Stochastic differential delay equations with Markovian switching,, Bernoulli, 6 (2000), 73.  doi: 10.2307/3318634.  Google Scholar

[17]

X. Mao, Exponetial stability of stochasitic delay interval systems with Markovian swithching,, IEEE Trans. on Autom. Control, 47 (2002), 1604.   Google Scholar

[18]

G. Tang and L. Guo, Convergence of a class of multi-agent systems in probabilistic framework,, Jrl Syst Sci & Complexity, 20 (2007), 173.  doi: 10.1007/s11424-007-9016-3.  Google Scholar

[19]

L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems,, Phys. Rev. Lett, 80 (1998), 2109.  doi: 10.1103/PhysRevLett.80.2109.  Google Scholar

[20]

M. Spivak, "A Comprehensive Introduction to Differential Geometry,", Houston, (1970).   Google Scholar

[21]

G. Ochs, "Weak Random Attractors. Technical Report,", Report 449, (1999).   Google Scholar

[22]

D. B. Johnson and D. A. Maltz, Dynamic source routing in ad hoc wireless networks,, Mobile Computing, 353 (1996), 153.  doi: 10.1007/978-0-585-29603-6_5.  Google Scholar

show all references

References:
[1]

J. C. Alexander, I. Kan, J. A. Yorke and Z. You, Riddled basins,, Int. J. Bifurcation Chaos, 2 (1992), 795.  doi: 10.1142/S0218127492000446.  Google Scholar

[2]

L. Arnold, "Random Dynamical Systems,", Springer-Verlag Berlin Heidelberg, (1998).   Google Scholar

[3]

P. Ashwin, J. Buescu and I. Stewart, From attractor to chaotic saddle: a tale of transverse instability,, Nonlinearity, 9 (1996), 703.  doi: 10.1088/0951-7715/9/3/006.  Google Scholar

[4]

P. Ashwin, E. Covas and R. Tavakol, Transverse instability for non-normal parameters,, Nonlinearity, 12 (1999), 563.  doi: 10.1088/0951-7715/12/3/009.  Google Scholar

[5]

P. Ashwin, Minimal attractors and bifurcations of random dynamical systems,, Proc. Rhys. Soc. Lond. A, 455 (1999), 2615.  doi: 10.1098/rspa.1999.0419.  Google Scholar

[6]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Autom. Control, 52 (2007), 852.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[7]

D. Cui, X. Liu, Y. Wan and X. Li, Estimation of genuine and random synchronization in multivariate neural series,, Neural Networks, 23 (2010), 698.  doi: 10.1016/j.neunet.2010.04.003.  Google Scholar

[8]

R. S. Ellis, Large deviation for a general class of random vectors,, The Annals of Probability, 12 (1984), 1.  doi: 10.1214/aop/1176993370.  Google Scholar

[9]

Y. Fang, "Stability Analysis of Linear Control Systems with Uncertain Parameters,", Ph. D thesis, (1994).   Google Scholar

[10]

S. Floyd and V. Jacobson, The synchronization of periodic routing messages,, IEEE Trans. Netw, 2 (1994), 122.   Google Scholar

[11]

V. Gazi and K. Passino, Stability analysis of social foraging swarms,, IEEE Trans. SMCS-Part B: Cybernetics, 34 (2004), 539.   Google Scholar

[12]

H. Huang, W. Daniel and Y. Qu, Robust stability of stochastic delayed additive neural networks with Markovian switching,, Neural Networks, 20 (2007), 799.  doi: 10.1016/j.neunet.2007.07.003.  Google Scholar

[13]

A. Lotka, "Elements of Physical Biology,", Williams & Wilkins Company, (1925).   Google Scholar

[14]

W. Lu, M. F. Atay and J. Jost, Synchronization of discrete-time dynamical networks with time-varying couplings,, SIAM Journals on Mathematical Analysis, 39 (2007), 1231.  doi: 10.1137/060657935.  Google Scholar

[15]

W. Lu, M. F. Atay and J. Jost, Chaos synchronization in networks of coupled map with time-varying topologies,, Eur. Phys. J. B, 63 (2008), 399.  doi: 10.1140/epjb/e2008-00023-3.  Google Scholar

[16]

X. Mao, A. Matasov and A. Piunovskiy, Stochastic differential delay equations with Markovian switching,, Bernoulli, 6 (2000), 73.  doi: 10.2307/3318634.  Google Scholar

[17]

X. Mao, Exponetial stability of stochasitic delay interval systems with Markovian swithching,, IEEE Trans. on Autom. Control, 47 (2002), 1604.   Google Scholar

[18]

G. Tang and L. Guo, Convergence of a class of multi-agent systems in probabilistic framework,, Jrl Syst Sci & Complexity, 20 (2007), 173.  doi: 10.1007/s11424-007-9016-3.  Google Scholar

[19]

L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems,, Phys. Rev. Lett, 80 (1998), 2109.  doi: 10.1103/PhysRevLett.80.2109.  Google Scholar

[20]

M. Spivak, "A Comprehensive Introduction to Differential Geometry,", Houston, (1970).   Google Scholar

[21]

G. Ochs, "Weak Random Attractors. Technical Report,", Report 449, (1999).   Google Scholar

[22]

D. B. Johnson and D. A. Maltz, Dynamic source routing in ad hoc wireless networks,, Mobile Computing, 353 (1996), 153.  doi: 10.1007/978-0-585-29603-6_5.  Google Scholar

[1]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[2]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[3]

Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021001
[4]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006
[5]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378
[6]

Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124
[7]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104
[8]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044
[9]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[10]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309
[11]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045
[12]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228
[13]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317
[14]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[15]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[16]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012
[17]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[18]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
[19]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[20]

Jing Qin, Shuang Li, Deanna Needell, Anna Ma, Rachel Grotheer, Chenxi Huang, Natalie Durgin. Stochastic greedy algorithms for multiple measurement vectors. Inverse Problems & Imaging, 2021, 15 (1) : 79-107. doi: 10.3934/ipi.2020066

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences