Отслеживание конечных псевдотраекторий в неравномерно гиперболических системах
Мы оцениваем вероятность отследить случайную конечную псевдотраекторию точной траекторией. Главным результатом работы является теорема о том, что эта вероятность стремится к единице при совместном стремлении длины псевдотраектории к бесконечности и размера ошибки псевдотраектории на каждом шаге к нулю. В доказательстве используется теорема Крамера о вероятности больших уклонений.
The theory of shadowing of pseudotrajectories is a well-developed area of the fundamental theory of dynamical systems. There are a lot of results that show the connection between the shadowing property and structural stabil- ity. First of all, we would mention, that a diffeomorphism has shadowing property near hyperbolic set [1,2]. It is also well known that if a diffeomor- phism is structurally stable on the whole manifold then it has shadowing property [3–5]. Although shadowing property does not imply structural sta- bility, this implication is true with some additional assumptions [6–8]. At the same time, the results of various numerical experiments show that finite pseudotrajectories can be shadowed effectively even for systems that are not structurally stable. For example, logistic map and H ́enon map were studied in [9,10]. The purpose of this research is to study the stochastic setting of the problem mentioned above. It was shown in [11] that considering infinite pseudotrajectories in stochastic setting gives results equivalent to the clas- sical shadowing theory. We introduce a natural way of generating a finite random pseudotrajectory and estimate the probability of it to be shadowed by an exact one.
It was shown in the papers [8,12,13] that the parameters of the shadowing property for finite pseudotrajectories can be obtained using the asymptotics of the growth rate of the solution of the corresponding inhomogeneous linear system. In this work we study two nonuniformly hyperbolic systems with one- dimensional center direction and nonzero Lyapunov exponent. In chapter 1 we state a theorem about a probabilistic shadowing property for a skew product of a Bernoulli shift and a linear map on a real line. We prove this theorem in chapter 2. This result can be considered a partial generalisation of the one shown in [14], where a special case of a linear skew product over Bernoulli shift is studied. In chapter 3 we prove an analogous result for a linear skew product over doubling map of a circle.
The technique we present uses the Cram ́er’s large deviations theorem 2
for i.i.d. random variables, who’s analogs are known to be true for dynam- ical systems with exponential mixing [15]. The last fact suggests that our approach might be used for establishing the shadowing property in a more general setting.
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[10] Hammel, S., Yorke, J., Grebogi, C., (1988). Numerical orbits of chaotic processes represent true orbits. Bulletin of the American Mathematical Society 19, 465-469.
[11] Tikhomirov, S. (2016). On absolute nonshadowability of transitive maps. Differential Equations and Control Processes, N3 57-65.
[12] Tikhomirov, S., (2015). Ho ̈lder shadowing on finite intervals. Ergodic Theory and Dynamical Systems, 35(6), 2000-2016. DOI:10.1017/etds.2014.7
[13] Monakov, G., (2019). Shadowing in not uniformly hyperbolic systems. Bachelor thesis.
[14] Tikhomirov, S., (2015). Shadowing in linear skew products. Zapiski Nauchnykh Seminarov POMI, Vol. 432 pp. 261–273.
[15] Duarte, P., Klein, S., (2016). Lyapunov Exponents of Linear Cocycles: Continuity Via Large Deviations. Atlantis Studies in Dynamical Sys- tems, v. 3.
[16] Shiryaev, A., (2007), Probability (Fourth edition), MCCME.
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