On transverse stability of random dynamical system

Previous Article
On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains
DCDS Home
This Issue
Next Article
Solitary waves of the rotation-generalized Benjamin-Ono equation

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
Related Papers

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 and Continuous Dynamical Systems, 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), 795-813. 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), 703-737. 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), 563-577. 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), 2615-2634. doi: 10.1098/rspa.1999.0419.
[6]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862. 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), 698-704. 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), 1-12. doi: 10.1214/aop/1176993370.
[9]	Y. Fang, "Stability Analysis of Linear Control Systems with Uncertain Parameters," Ph. D thesis, Case Western Reserve University, 1994.
[10]	S. Floyd and V. Jacobson, The synchronization of periodic routing messages, IEEE Trans. Netw, 2 (1994), 122-136.
[11]	V. Gazi and K. Passino, Stability analysis of social foraging swarms, IEEE Trans. SMCS-Part B: Cybernetics, 34 (2004), 539-556.
[12]	H. Huang, W. Daniel and Y. Qu, Robust stability of stochastic delayed additive neural networks with Markovian switching, Neural Networks, 20 (2007), 799-809. 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), 1231-1259. 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), 399-406. 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), 73-90. doi: 10.2307/3318634.
[17]	X. Mao, Exponetial stability of stochasitic delay interval systems with Markovian swithching, IEEE Trans. on Autom. Control, 47 (2002), 1604-1612.
[18]	G. Tang and L. Guo, Convergence of a class of multi-agent systems in probabilistic framework, Jrl Syst Sci & Complexity, 20 (2007), 173-197. 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), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.
[20]	M. Spivak, "A Comprehensive Introduction to Differential Geometry," Houston, TX: Publish or Perish, 1970.
[21]	G. Ochs, "Weak Random Attractors. Technical Report," Report 449, Institut für Dynamische Systeme, Universität Bremen, 1999.
[22]	D. B. Johnson and D. A. Maltz, Dynamic source routing in ad hoc wireless networks, Mobile Computing, 353 (1996), 153-181. 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), 795-813. 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), 703-737. 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), 563-577. 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), 2615-2634. doi: 10.1098/rspa.1999.0419.
[6]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862. 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), 698-704. 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), 1-12. doi: 10.1214/aop/1176993370.
[9]	Y. Fang, "Stability Analysis of Linear Control Systems with Uncertain Parameters," Ph. D thesis, Case Western Reserve University, 1994.
[10]	S. Floyd and V. Jacobson, The synchronization of periodic routing messages, IEEE Trans. Netw, 2 (1994), 122-136.
[11]	V. Gazi and K. Passino, Stability analysis of social foraging swarms, IEEE Trans. SMCS-Part B: Cybernetics, 34 (2004), 539-556.
[12]	H. Huang, W. Daniel and Y. Qu, Robust stability of stochastic delayed additive neural networks with Markovian switching, Neural Networks, 20 (2007), 799-809. 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), 1231-1259. 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), 399-406. 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), 73-90. doi: 10.2307/3318634.
[17]	X. Mao, Exponetial stability of stochasitic delay interval systems with Markovian swithching, IEEE Trans. on Autom. Control, 47 (2002), 1604-1612.
[18]	G. Tang and L. Guo, Convergence of a class of multi-agent systems in probabilistic framework, Jrl Syst Sci & Complexity, 20 (2007), 173-197. 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), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.
[20]	M. Spivak, "A Comprehensive Introduction to Differential Geometry," Houston, TX: Publish or Perish, 1970.
[21]	G. Ochs, "Weak Random Attractors. Technical Report," Report 449, Institut für Dynamische Systeme, Universität Bremen, 1999.
[22]	D. B. Johnson and D. A. Maltz, Dynamic source routing in ad hoc wireless networks, Mobile Computing, 353 (1996), 153-181. doi: 10.1007/978-0-585-29603-6_5.

[1]	Fumihiko Nakamura, Yushi Nakano, Hisayoshi Toyokawa. Lyapunov exponents for random maps. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022058
[2]	Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[3]	Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433
[4]	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
[5]	Xiaojin Huang, Hongfu Yang, Jianhua Huang. Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 601-610. doi: 10.3934/naco.2021024
[6]	Wenxiong Chen, Congming Li. Harmonic maps on complete manifolds. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 799-804. doi: 10.3934/dcds.1999.5.799
[7]	Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[8]	Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[9]	Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102
[10]	Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[11]	Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[12]	Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697
[13]	Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487
[14]	Woojoo Shim. On the generic complete synchronization of the discrete Kuramoto model. Kinetic and Related Models, 2020, 13 (5) : 979-1005. doi: 10.3934/krm.2020034
[15]	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
[16]	Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
[17]	Peter Giesl, Sigurdur Freyr Hafstein, Stefan Suhr. Existence of complete Lyapunov functions with prescribed orbital derivative. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022027
[18]	Pierre Guiraud, Etienne Tanré. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5183-5201. doi: 10.3934/dcdsb.2019056
[19]	Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[20]	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

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Xiangnan He Wenlian Lu Tianping Chen
on Google Scholar
Xiangnan He Wenlian Lu Tianping Chen

[Back to Top]