On transverse stability of random dynamical system

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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

Abstract
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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.
[2]	L. Arnold, "Random Dynamical Systems,", Springer-Verlag Berlin Heidelberg, (1998).
[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.
[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.
[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.
[6]	F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Autom. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842.
[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.
[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.
[9]	Y. Fang, "Stability Analysis of Linear Control Systems with Uncertain Parameters,", Ph. D thesis, (1994).
[10]	S. Floyd and V. Jacobson, The synchronization of periodic routing messages,, IEEE Trans. Netw, 2 (1994), 122.
[11]	V. Gazi and K. Passino, Stability analysis of social foraging swarms,, IEEE Trans. SMCS-Part B: Cybernetics, 34 (2004), 539.
[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.
[13]	A. Lotka, "Elements of Physical Biology,", Williams & Wilkins Company, (1925).
[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.
[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.
[16]	X. Mao, A. Matasov and A. Piunovskiy, Stochastic differential delay equations with Markovian switching,, Bernoulli, 6 (2000), 73. doi: 10.2307/3318634.
[17]	X. Mao, Exponetial stability of stochasitic delay interval systems with Markovian swithching,, IEEE Trans. on Autom. Control, 47 (2002), 1604.
[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.
[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.
[20]	M. Spivak, "A Comprehensive Introduction to Differential Geometry,", Houston, (1970).
[21]	G. Ochs, "Weak Random Attractors. Technical Report,", Report 449, (1999).
[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.

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.
[2]	L. Arnold, "Random Dynamical Systems,", Springer-Verlag Berlin Heidelberg, (1998).
[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.
[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.
[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.
[6]	F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Autom. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842.
[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.
[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.
[9]	Y. Fang, "Stability Analysis of Linear Control Systems with Uncertain Parameters,", Ph. D thesis, (1994).
[10]	S. Floyd and V. Jacobson, The synchronization of periodic routing messages,, IEEE Trans. Netw, 2 (1994), 122.
[11]	V. Gazi and K. Passino, Stability analysis of social foraging swarms,, IEEE Trans. SMCS-Part B: Cybernetics, 34 (2004), 539.
[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.
[13]	A. Lotka, "Elements of Physical Biology,", Williams & Wilkins Company, (1925).
[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.
[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.
[16]	X. Mao, A. Matasov and A. Piunovskiy, Stochastic differential delay equations with Markovian switching,, Bernoulli, 6 (2000), 73. doi: 10.2307/3318634.
[17]	X. Mao, Exponetial stability of stochasitic delay interval systems with Markovian swithching,, IEEE Trans. on Autom. Control, 47 (2002), 1604.
[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.
[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.
[20]	M. Spivak, "A Comprehensive Introduction to Differential Geometry,", Houston, (1970).
[21]	G. Ochs, "Weak Random Attractors. Technical Report,", Report 449, (1999).
[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.

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