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Abstract
Self-stochasticity theory, established for certain classes of deterministic finite-dimensional dynamical systems, is extended to the continuous time difference equations with
and f being continuous interval map. For such equations, self-stochasticity lies in the existence of eventually unpredictable solutions (as smooth as desired) whose long-term behaviour is described with some stochastic processes.
Acknowledgements
This research was partially supported by the Scientific program of National Academy of Sciences of Ukraine, project No 0107 U 002333. The author expresses her gratitude to Professor A.N. Sharkovsky, who has advanced the general ideas on self-stochasticity, for many helpful comments and suggestions. The author also thanks Professor A.A. Dorogovtsev for insightful comments relating to the probabilistic aspects of the work.
Notes
1. The title ‘separator’ takes its name from the fact that in most cases the points of separate the basins of attraction of the attracting cycles of f.
2. That is, a measure absolutely continuous w.r.t. Lebesgue measure.
3. Closed intervals are called a period-p cycle of intervals for a map f if these intervals are cyclically rearranged by f and in pairs do not have common interior points; if, in addition, the union
contains an everywhere dense trajectory of f, then the cycle of intervals is referred to as transitive.
4. The intervals ,
,…,
, named the components of the support of
, need not be among the connected components of the set
since it is possible that certain of them have a common end-point.
5. Generally, f can have several measures meeting the SIM-conditions. If and
are such measures, then: (a)
if
; (b)
and
are mutually singular otherwise (see Ref. [Citation17]).
6. The value of the distribution (7) – some number from [0,1] – is referred to as the probability that ,
,…,
.
7. The requirement of being nonsingular is essential. A trivial example: for the nonsingular functions
that converge to the singular function
, one may readily check that
.
8. See Ref. [Citation2] for the definition and properties of the weak convergence of probability measures.
9. In parallel with the term ‘periodic process’ there is used the term ‘cyclostationary process’.
10. Of course, the extent to which and
are perceived as being arbitrary is limited by the requirement that the left-hand sides of (Equation26
) and (Equation27
) be defined.
11. A random process is called a process with independent values, if for any finite set {
,…,
}, the random values
,…,
are mutually independent, i.e.
.
12. Note that (Equation33) is not a direct consequence of Theorem G, which use a metric weaker than
; however, a close look at the proof of Theorem G has shown that when replacing the metric employed with the metric
, the theorem remains valid and therefore (Equation33
) do holds.
13. Proposition 5, that underlies all the further considerations, is actually a reformulation of Theorem G, which is the main result of Ref. [Citation17].