| dc.description.abstract | We consider the dynamical system defined by the pair ([a,b],f)([a, b], f)([a,b],f), where [a,b][a, b][a,b] is a compact interval of R\mathbb{R}R and f:[a,b]?[a,b]f : [a, b] \rightarrow [a, b]f:[a,b]?[a,b] is a continuous function or map. The time evolution is realized by the successive iteration of the map fff on an initial condition belonging to [a,b][a, b][a,b]. When fff is an onto map, for an x?[a,b]x \in [a, b]x?[a,b] we associate a bi-directional sequence
O(x)={x?2,x?1,x0,x1,x2,�}\mathcal{O}(x) = \{x_{-2}, x_{-1}, x_0, x_1, x_2, \dots\}O(x)={x?2?,x?1?,x0?,x1?,x2?,厎
such that f(xn)=xn+1,?n?Zf(x_n) = x_{n+1}, \, n \in \mathbb{Z}f(xn?)=xn+1?,n?Z, and call it an orbit of xxx.
Definition. Denote fnf^nfn to be the nnn-fold composition of the map fff with itself. A map f:[a,b]?[a,b]f : [a, b] \rightarrow [a, b]f:[a,b]?[a,b] is topologically transitive if for any given pair UUU and VVV of non?empty open subsets of [a,b][a, b][a,b], there exists a k>0k > 0k>0 such that fk(U)?V??f^k(U) \cap V \neq \emptysetfk(U)?V?=?.
Definition. An orbit O(x)\mathcal{O}(x)O(x) of a surjective map fff is said to be a separating orbit with an instability constant ?>0\tau > 0?>0 if for every pair of distinct points xi,xj?O(x)x_i, x_j \in \mathcal{O}(x)xi?,xj??O(x), there exists an integer n>0n > 0n>0 such that
?fn(xi)?fn(xj)?>?.|f^n(x_i) - f^n(x_j)| > \tau.?fn(xi?)?fn(xj?)?>?.
Topological transitivity is one of the fundamental irreducibility conditions imposed on a dynamical system, and on [a,b][a, b][a,b] it is equivalent to the map being chaotic in the sense of Devaney. A feature of topologically transitive maps on compact intervals is that they possess plenty of dense orbits. In the first part of the work, we prove that all dense orbits of any topologically transitive map on [a,b][a, b][a,b] are separating orbits and can be endowed with a uniform instability constant.
In the second part of the work, we consider two signal estimation problems.
The model used in the first problem is described as follows:
The noise {xi}\{x_i\}{xi?} is modeled as a chaotic time series; more precisely, it is a separating orbit
O(x)={�,x?2,x?1,x0,x1,x2,�}\mathcal{O}(x) = \{\dots, x_{-2}, x_{-1}, x_0, x_1, x_2, \dots \}O(x)={�,x?2?,x?1?,x0?,x1?,x2?,厎
obtained from a transitive interval map with an instability constant ?\tau?. The observed time series {yi}\{y_i\}{yi?} is given by
yi=xi+si,i?Z,y_i = x_i + s_i,\quad i \in \mathbb{Z},yi?=xi?+si?,i?Z,
where {si}\{s_i\}{si?} is a real sequence such that si=0s_i = 0si?=0 for all i<0i < 0i<0, ?si??C<?/4|s_i| \leq C < \tau/4?si???C<?/4 for all i>0i > 0i>0, and si?0s_i \to 0si??0 as i??i \to \inftyi??. The task is to estimate each element of the sequence {s0,s1,�}\{s_0, s_1, \dots\}{s0?,s1?,厎 given only the sequence {yi}\{y_i\}{yi?}. This model is motivated by results in the literature referring to ambient ocean noise and 搊cean clutter拻 being chaotic in nature.
With k?1k \geq 1k?1 as a parameter, for each sis_isi? (0?i?n0 \leq i \leq n0?i?n), we create an estimate si,n,ks_{i,n,k}si,n,k?, which is a function of the observations y?k,�,yny_{-k}, \dots, y_ny?k?,�,yn?. Under a mild recurrence condition on the orbit {xi}\{x_i\}{xi?}, we prove that
lim?k???lim?sup?n????si?si,n,k?=0.\lim_{k\to\infty} \, \limsup_{n\to\infty} \, |s_i - s_{i,n,k}| = 0.k??lim?n??limsup??si??si,n,k??=0.
In the second estimation problem, {si}\{s_i\}{si?} is treated as a sequence of independent bounded real random variables with zero mean and identical variance. We provide sufficient conditions for different limiting behaviours of {si}\{s_i\}{si?} under which a similar convergence result of the estimation error holds with probability 1. In both problems, the topological properties of the sequence {xi}\{x_i\}{xi?} are exploited. Although the main motivation is to handle dense orbits of interval maps, our results in general apply to any separating orbit on a compact subspace of R\mathbb{R}R. | |
