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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

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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

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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

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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

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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

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