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On statistical attractors and the convergence of time averages

Published online by Cambridge University Press:  12 January 2011

ÖZKAN KARABACAK  and
PETER ASHWIN
ÖZKAN KARABACAK
Affiliation:
Mathematics Research Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF. e-mail: [email protected] and [email protected]
PETER ASHWIN
Affiliation:
Mathematics Research Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF. e-mail: [email protected] and [email protected]
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Abstract

There are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 353 - 365
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Arnold, V. I., Afrajmovich, V. S., Il′yashenko, Yu. S. and Shil′nikov, L. P.Bifurcation Theory and Catastrophe Theory (Springer-Verlag, 1999).Google Scholar
[2]Ashwin, P., Aston, P. J. and Nicol, M.On the unfolding of a blowout bifurcation. Phys. D 111 (1–4) (1998), 81–95.CrossRefGoogle Scholar
[3]Ashwin, P. and Field, M.Heteroclinic networks in coupled cell systems. Arch. Ration. Mech. Anal. 148 (2) (1999), 107–143.CrossRefGoogle Scholar
[4]Ashwin, P. and Terry, J.R.On riddling and weak attractors. Phys. D 142 (2000), 87–100.CrossRefGoogle Scholar
[5]Billingsley, PConvergence of Probability Measures (John Wiley & Sons Inc., 1968).Google Scholar
[6]Blank, M. and Bunimovich, L.Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity 16 (1) (2003), 387–401.CrossRefGoogle Scholar
[7]Busse, F. H. and Clever, R. M. Nonstationary convection in a rotating system. In Müller, U., Roesner, K. G. and Schmidt, B., eds., Recent Developments in Theoretical and Experimental Fluid Dynamics, pages 376–385 (Springer Verlag, 1979).CrossRefGoogle Scholar
[8]Chueshov, I. D.Introduction to the theory of infinite-dimensional dissipative systems (http://www.emis.de/monographs/Chueshov/). (ACTA, Kharkov, 2002).Google Scholar
[9]Gaunersdorfer, A.Time averages for heteroclinic attractors. SIAM J. Appl. Math. 52 (5) (1992), 1476–1489.CrossRefGoogle Scholar
[10]Guckenheimer, J. and Holmes, P.Structurally stable heteroclinic cycles. Math. Proc. Camb. Phil. Soc. 103 (1988), 189–192.CrossRefGoogle Scholar
[11]Ilyashenko, Yu. Minimal attractors. In EQUADIFF 2003, pages 421–428 (World Scientific Publishing, 2005).Google Scholar
[12]Ilyashenko, Yu. and Negut, A.Invisible parts of attractors. Nonlinearity 23 (5) (2010), 1199.CrossRefGoogle Scholar
[13]Jordan, T., Naudot, V. and Young, T.Higher order Birkhoff averages. Dyn. Syst. 24 (3) (2009), 299–313.CrossRefGoogle Scholar
[14]Karabacak, Ö. and Ashwin, P.Heteroclinic ratchets in networks of coupled oscillators. J. Nonlinear Sci. 20 (2010), 105–129.CrossRefGoogle Scholar
[15]Katok, A. and Hasselblatt, B.Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, 1995).CrossRefGoogle Scholar
[16]Kleptsyn, V.An example of non-coincidence of minimal and statistical attractors. Ergodic Theory Dynam. Systems 26 (3) (2006), 759–768.CrossRefGoogle Scholar
[17]Milnor, J.On the concept of attractor. Commun. Math. Phys. 99 (1985), 177–195.CrossRefGoogle Scholar
[18]Ott, W. and Yorke, J. A.When Lyapunov exponents fail to exist. Phys. Rev. E 78 (5, Part 2) (2008).CrossRefGoogle ScholarPubMed
[19]Ruelle, D. Historical behaviour in smooth dynamical systems. In Global analysis of dynamical systems, pages 63–66 (Inst. Phys., Bristol, 2001).Google Scholar
[20]Sigmund, K.Time averages for unpredictable orbits of deterministic systems. Ann. Oper. Res. 37 (1–4) (1992), 217–228.CrossRefGoogle Scholar
[21]Takens, F.Heteroclinic attractors: time averages and moduli of topological conjugacy. Bol. Soc. Brasil. Mat. (N.S.) 25 (1) (1994), 107–120.CrossRefGoogle Scholar
[22]Takens, F.Orbits with historic behaviour, or non-existence of averages. Nonlinearity 21 (3) (2008), 33–36.CrossRefGoogle Scholar