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Is every approximate trajectory of some process near an exact trajectory of a nearby process?

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Abstract

This paper deals with the problem “Can a noisy orbit be tracked by a real orbit?” In particular, we will study the one-parameter family of tent maps and the one-parameter family of quadratic maps. We writeg μ for eitherf μ orF μ withf μ (x)=μx forx≦1/2 andf μ (x)=μ(1−x) forx≧1/2, andF μ (x)=μx(1−x). For a given μ we will say:g μ permits increased parameter shadowing if for each δ x >0 there exists someδ μ >0 and some δ f >0 such that every δ f -pseudog μ -orbit starting in some invariant interval can be δ x -shadowed by a realg α -orbit with α=μ+δ μ . We show thatg μ typically permits increased parameter shadowing.

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Authors and Affiliations

  1. Rijksuniversiteit Groningen, Faculteit der Economische Wetenschappen, Vakgroep Econometrie, Postbus 800, NL-9700 AV, Groningen, The Netherlands

    Helena E. Nusse

  2. Institute for Physical Science and Technology, and Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    James A. Yorke

Authors
  1. Helena E. Nusse
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  2. James A. Yorke
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Additional information

Communicated by J. N. Mather

Research supported in part by the Netherlands organization for the advancement of pure research (Z.W.O.) and under grant AFOSR-81-0217

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Nusse, H.E., Yorke, J.A. Is every approximate trajectory of some process near an exact trajectory of a nearby process?. Commun.Math. Phys. 114, 363–379 (1988). https://doi.org/10.1007/BF01242136

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  • DOI: https://doi.org/10.1007/BF01242136

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