Self-stochasticity in deterministic boundary value problems
This paper presents the experience of applying dynamical systems theory to an investigation into nonlinear boundary value problems for partial differential equations (PDE for short) in the case that their solutions become chaotic with time. To describe the long time behavior of such solutions, the c...
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Інститут прикладної математики і механіки НАН України
1999
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| Cite this: | Self-stochasticity in deterministic boundary value problems / E.Yu. Romanenko, A.N. Sharkovsky, M.B. Vereikina // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 174-184. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1692902020-06-10T01:26:08Z Self-stochasticity in deterministic boundary value problems Romanenko, E.Yu. Sharkovsky, A.N. Vereikina, M.B. This paper presents the experience of applying dynamical systems theory to an investigation into nonlinear boundary value problems for partial differential equations (PDE for short) in the case that their solutions become chaotic with time. To describe the long time behavior of such solutions, the concept of self-stochasticity had been suggested. The results reported in this work are concerned linear systems of PDE with nonlinear boundary conditions; general ideas on the manner in which chaotic solutions may be described are set forth by the example of several simplest boundary value problems. 1999 Article Self-stochasticity in deterministic boundary value problems / E.Yu. Romanenko, A.N. Sharkovsky, M.B. Vereikina // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 174-184. — Бібліогр.: 14 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169290 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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This paper presents the experience of applying dynamical systems theory to an investigation into nonlinear boundary value problems for partial differential equations (PDE for short) in the case that their solutions become chaotic with time. To describe the long time behavior of such solutions, the concept of self-stochasticity had been suggested. The results reported in this work are concerned linear systems of PDE with nonlinear boundary conditions; general ideas on the manner in which chaotic solutions may be described are set forth by the example of several simplest boundary value problems. |
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Romanenko, E.Yu. Sharkovsky, A.N. Vereikina, M.B. |
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Romanenko, E.Yu. Sharkovsky, A.N. Vereikina, M.B. Self-stochasticity in deterministic boundary value problems Нелинейные граничные задачи |
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Romanenko, E.Yu. Sharkovsky, A.N. Vereikina, M.B. |
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Romanenko, E.Yu. |
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Self-stochasticity in deterministic boundary value problems |
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Self-stochasticity in deterministic boundary value problems |
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Self-stochasticity in deterministic boundary value problems |
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Self-stochasticity in deterministic boundary value problems |
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Self-stochasticity in deterministic boundary value problems |
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self-stochasticity in deterministic boundary value problems |
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Інститут прикладної математики і механіки НАН України |
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Self-stochasticity in deterministic boundary value problems / E.Yu. Romanenko, A.N. Sharkovsky, M.B. Vereikina // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 174-184. — Бібліогр.: 14 назв. — англ. |
| series |
Нелинейные граничные задачи |
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AT romanenkoeyu selfstochasticityindeterministicboundaryvalueproblems AT sharkovskyan selfstochasticityindeterministicboundaryvalueproblems AT vereikinamb selfstochasticityindeterministicboundaryvalueproblems |
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2025-07-15T04:02:48Z |
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1837684143022931968 |
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Self-stochasticity in deterministic
boundary value problems
c© E.Yu.Romanenko, A.N.Sharkovsky and M.B.Vereikina
1 Introduction
This paper presents the experience of applying dynamical systems theory to an investiga-
tion into nonlinear boundary value problems for partial differential equations (PDE for
short) in the case that their solutions become chaotic with time. To describe the long time
behavior of such solutions, the concept of self-stochasticity had been suggested (10–12).
The results reported in this work are concerned linear systems of PDE with nonlinear
boundary conditions; general ideas on the manner in which chaotic solutions may be
described are set forth by the example of several simplest boundary value problems. A
fuller treatment can be found in (5–14). It takes considerable efforts to extend these
results for nonlinear systems of PDE.
Definition 1. By self-stochasticity in deterministic dynamical systems we
mean the fact that: for a deterministic system there exists a completion of its phase space
with random functions such that the system has a “massive” set of trajectories whose
ω-limit sets contain random functions. Each such a trajectory is referred to as self-
stochastic.
It is obvious that self-stochasticity phenomenon may occur only if the starting phase
space of a dynamical systems is noncompact. In this case, the cardinal problem is to find
a suitable metric which allows to compare deterministic functions with random ones.
By the term “massive set” is implied either a set of positive measure or a set of full
measure (“almost all (in measure)”) and either an everywhere dense set or a set of second
category (“almost all”) or whatever, as required by the problem. Thereby we necessitate a
system to have a great deal of self-stochastic trajectories (for instance, if there were a few
self-stochastic trajectories among all the trajectories, then almost surely (with probability
1) these self-stochastic trajectories should escape detection by a computer).
To reveal that self-stochasticity phenomenon is actually feasible and to do the concept
of self-stochasticity more clear, we discuss some dynamical systems induced by nonlinear
boundary value problems for PDE.
Among nonlinear boundary value problems which serve as mathematical models for
chaotic processes, one can separate two fundamental classes:
1. problems for PDE of parabolic type (such as Navier–Stokes equation much used in
hydrodynamics);
2. problems for PDE of hyperbolic type (which are characteristic of electrodynamics).
In many cases boundary value problems from both these classes induce infinite dimen-
sional dynamical systems of the form
{
Ck(D,E), T, St
}
(1)
where Ck = Ck(D, E) is the space of Ck-functions ϕ : D → E, D ⊂ Rl, E ⊂ Rm
(k, l, m ≥ 1), T = R+ or Z+, and St is the shift operator along solutions.
The phase space Ck equipped a priori with the Ck-metric is noncompact. It may
therefore occur that for some (and even for almost all) ϕ ∈ Ck the corresponding trajectory
St[ϕ] is noncompact. As a consequence, its ω-limit set ω[ϕ] is found to be either empty
or, if not, noncompact and hence this ω-limit set characterizes the asymptotic behavior
of St[ϕ] incompletely. For boundary value problems from the first class, trajectories
are generally compact and, in contrast, for those of the second class, trajectories are in
many cases noncompact. Every so often it is the noncompactness of trajectories that is
responsible for initiation of chaos in problems from the second class.
Let us consider the simplest nonlinear problem
∂u
∂t
=
∂u
∂x
, x ∈ [0, 1], t ∈ R+, (2)
u |x=1 = f(u) |x=0, (3)
u |t=0 = ϕ(x), (4)
with f ∈ C1(I, I) being an irreversible map of a closed interval I into itself and ϕ ∈
C1
(
[0, 1], I
)
complying with the consistency relations ϕ(1) = f
(
ϕ(0)
)
and ϕ̇(1) =
ḟ
(
ϕ(0)
)
ϕ̇(0) (which insure the bounded solutions of the problem to be C1-smooth).
Pr. (2)-(4) is the simplest nonlinear problem that one can envision. We have here
taken it as a model problem for representation of self-stochasticity phenomenon because
in this case, explanations are quite simple.
Pr. (2)–(4) is reduced to a difference equation. Indeed, on substituting the general
solution of (2), which has the form u(x, t) = w(x + t), into (3), we arrive at
w(τ + 1) = f(w(τ)), τ ∈ R+. (5)
Then from (4) and the consistency relations it follows that the solution uϕ of Pr. (2)–(4)
can be presented in the form
uϕ(x, t) = wϕ(x + t), (6)
where wϕ is a solution of Eq. (5) such that
wϕ(τ) = ϕ(τ) for τ ∈ [0, 1]. (7)
Consequently,
uϕ(x, t) = ϕ(x + t), if 0 ≤ x + t < 1,
uϕ(x, t) = (f ◦ ϕ)(x + t− 1), if 1 ≤ x + t < 2,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
uϕ(x, t) = (fn ◦ ϕ)(x + t− n), if n ≤ x + t < n + 1,
and hence uϕ(x, t) can be written in the form
uϕ(x, t) = f [t+x]
(
ϕ({t + x})
)
, t ∈ R+, x ∈ [0, 1], (8)
with fn standing for the n-th iteration of f (i. e., fn = f ◦ fn−1 and f 0 = id ), [·] and {·}
standing for the integral and fractional parts of a number.
Thus Pr. (2) and (3) generates the dynamical system
{
Ck ([0, 1], I) , T, St
}
(9)
with
St[ϕ](x) =
(
f [t+x] ◦ ϕ
)
({t + x}) . (10)
The formula (10) indicates that the long time behavior of a trajectory St[ϕ] is dictated
by the asymptotic (as n →∞) properties of the iteration fn of the map f , which specifies
the nonlinearity in the boundary condition.
What consequences may the noncompactness of St[ϕ] bring about ? The
situations prove to be possible that although in theory every trajectory St[ϕ] is determined
uniquely by ϕ, it is not practical to predict the values of the function St[ϕ](x)
for t large enough.
To make clear why the aforesaid is valid, we consider Pr. (2)–(4) with the nonlinearity
in the boundary condition being given by the map
f : z 7−→ 4z(1− z), z ∈ [0, 1], (11)
which is often referred to as chaotic parabola. The map (11) has the following property:
for any open interval J ⊂ [0, 1] there exists n∗ > 0 such that fn(J) = [0, 1] if n > n∗. As
for Pr. (2)–(4) and (11), this implies that for any x ∈ [0, 1] and any ε > 0 there exists
t∗ > 0 such that
St[ϕ]
(
Vε(x)
)
= [0, 1] for t > t∗, (12)
where Vε(·) is for the ε-neighborhood of a point. If ε is not over a computer precision,
then the computer recognizes the neighborhood Vε(x) as a single point and by virtue of
(12) the value of St[ϕ] calculated at a point x for t > t∗ might appear to be equal to
anyone of numbers from [0, 1] according to which implementation of f in code is used. In
this case, St[ϕ] is said to be out of predictability horizon (at least at given x for
t > t∗).
Let us summarize. There exist boundary value problems such that the pre-
dictability of their solutions sooner or later breaks down and, in this sense,
nonrandom solutions behave much like random processes (this fact is a powerful
argument in support of that self-stochasticity phenomenon is really a possibility). In such
situations, it is reasonable to turn a probabilistic description: rather than ask “what is
the value of a solution at a point (x, t) ?”, we should ask “what is the probability of
the value of a solution at a point (x, t) falling into a certain set ?” If the answer
to the latter question can be provided (of course, where a solution becomes unpredictable
with time), we have called such a situation self-stochasticity , in contrast to chaoticity
as a behavior with no regularity at all.
2 Construction of a Metric
Let us consider the general dynamical system (1) and assume that it possesses a wealth
of noncompact trajectories with unpredictable long time behavior. To describe Syst. (1),
we need complete its phase space Ck via an appropriate metric so as the trajectories to
be compact in a new phase space.
As we have seen just now, the ordinary sup-metric, which effects pointwise comparison,
is unfit for Syst. (1). The sought-for metric must allow for the values of functions ϕ1, ϕ2
not only at a point x but also in some neighborhood of x (this approach is conformable
to ideas developed in statistical physics [1, 3, 4]).
In other words, the sought-for metric must involve the distributions of the
values of ϕ1 and ϕ2 near a point x or more specifically, the ensemble-averaged
distributions
F ε
ϕi
(x, z) =
1
mes Vε(x)
∫
Vε(x)
Fϕi
(y, z)dy, i = 1, 2, (13)
where Fϕ(x, z) is the distribution function ϕ ∈ Ck(X,Z) with Z being bounded. Inasmuch
as ϕ is deterministic,
Fϕ(x, z) = χ(−∞,z)
(
ϕ(x)
)
(14)
with χA(·) being the indicator function of a set A.
Thus we arrive at the following metric
%(ϕ1, ϕ2) = sup
ε>0
min
{
ε,
1
mes Z
sup
x∈X
∫
Z
∣∣∣F ε
ϕ1
(x, z)− F ε
ϕ2
(x, z)
∣∣∣ dz
}
. (15)
One can extend the formula for % to a set of functions that is essentially wider then
Ck(X, Z), namely, to the set of random processes ψ : X → Z that are specified by all
their finite-dimensional distributions. Write <(X,Z) for this set.
It is clear that % is a semimetric on <(X,Z), whereas it is a metric on Ck(X, Z). To
turn % into a metric on <(X,Z), it is necessary to involve all finite-dimensional distribu-
tions into (15). Thus we are led to the metric
%#(ψ1, ψ2) = sup
ε>0
min
{
ε, (16)
∞∑
r=1
1
2rmes Zr
sup
(x1,...,xr)∈Xr
∫
Zr
∣∣∣F ε
ψ1
(x1, . . . , zr)− F ε
ψ2
(x1, . . . , zr)
∣∣∣ dz1 . . . dzr
}
,
where
• F ε
ψ(x1 . . . zr) =
1
m ε(x1 . . . xr)
∫
Vε(x1...xr)
Fψ (y1 . . . zr)dy1 . . . dyr;
• given (x1, . . . , xr) ∈ Xr and (z1, . . . , zr) ∈ Zr, the collections
(x1, . . . , xr; z1, . . . , zr) is written (x1 . . . zr);
• Fψ(x1 . . . zr) is the r-dimensional distribution of a function
ψ : X → Z;
• Vε(x1 . . . xr) is the ε-neighborhood of (x1, . . . , xr) ∈ Xr;
• m ε(x1 . . . xr) = mes Vε(x1 . . . xr) for (x1, . . . , xr) ∈ Xr.
The metric %# is applicable both to deterministic functions and to random
functions. A little though reveals that %# is well-defined on the subset of those functions
ψ ∈ <(X, Z) such that
F ε
ψ(x1 . . . zr) −→ Fψ(x1 . . . zr) as ε → 0 (17)
for all points (x1, . . . , zr) ∈ Xr × Zr outside of a set of Lebesgue measure zero.
3 Some Results
Let C# = C#(D, E) be the completion of the phase space Ck(D, E) via the metric %#
with random and deterministic functions having the property (17). The space C# is not
compact. But there exist classes of boundary value problems (among which is Pr. (2)-
(4) ) such that, under certain conditions, the corresponding dynamical system possesses
a massive set of trajectories that are compact in C#.
As for Pr. (2)-(4), this conditions, which we will refer to as (IM), are as follows:
• the map f has an invari-
ant measure µ with the
support being a cycle of
intervals J1, J2, . . ., Jp
⇐⇒ supp µ =
p⋃
i=1
Ji,
where f(Ji) = Ji+1(mod p),
and µ(f−1(A)) = µ(A);
• the measure µ is equiv-
alent to Lebesgue mea-
sure on supp µ
⇐⇒ mes A = 0 ⇐⇒ µ(A) = 0;
• the map f p has the
property of intermixing⇐⇒ µ(A ∩ f−pj(B)) → µ(A) · µ(B)
as j →∞,
A,B ⊂ Ji, 1 ≤ i ≤ p;
• the map f is nonsin-
gular (with respect to
Lebesgue measure)
⇐⇒ mes A = 0 =⇒ mes f−1(A) =
0,
mes f(A) = 0.
These conditions are, in particular, fulfilled if f is a unimodal map with negative
Schwarzian derivative Sf := f ′′′/f ′ − 3/2 (f ′′/f ′)2
and satisfies Collet-Eckmann’s condi-
tions lim
n→∞ inf
1
n
log
∣∣∣∣∣
d
dt
fn(c)
∣∣∣∣∣ > 0, where c is the (unique) extreme point of f . For ease of
subsequent formulations, we assume f to be such a map.
Syst. (9) and (10), generated by Pr. (2) and (3), being uniformly continuous with
respect to %#, induces a dynamical system on the extended space C#, namely,
{
C# (D, E) , T, St
}
(18)
with
St[ψ](x) =
(
f [t+x] ◦ ψ
) (
{t + x}
)
, (19)
where by the superposition g ◦ ψ of deterministic and random functions we mean the
random function specified by the infinite-dimensional distributions
Fg◦ψ(x1, . . . , xr; z1, . . . , zr) :=
∫
g−1
(
(−∞,z1)
)
. . .
∫
g−1
(
(−∞,zr)
)
∂rFψ (x1, . . . , xr; y1, . . . , yr)
∂y1 . . . ∂yr
dy1 . . . dyr, r = 1, 2, . . . .
With Syst. (18) and (19) in hand, we are in a position to characterize the ω-limit set of the
trajectory St[ϕ] of Syst. (9) and (10), which corresponds to the solution uϕ of Pr. (2)-(4).
Theorem 1. For every nonsingular ϕ ∈ C1, the ω-limit set of the trajectory St[ϕ]
consists of random functions that combine into a cycle of Syst. (18) and (19) with period p,
more precisely,
ω[ϕ] =
⋃
t∈[0,p)
St
[
f# ◦ ϕ
]
,
where f# is the purely random process specified by the distribution function
Ff#(u, z) := p · µ (Ji ∩ (−∞, z)) for u ∈ Ĵi
with Ĵi =
⋃
j≥0
∫
f−jp(Ji), 1 ≤ i ≤ p,
and by the superposition f# ◦ ϕ is meant the random process specified by the distributions
Ff#◦ϕ(x1, . . . , xr; z1, . . . , zr) := Ff#(ϕ(x1), . . . , ϕ(xr); z1, . . . , zr),
r = 1, 2, . . . .
(Recall that a stochastic process Y (x) is said to be purely random, if for any x1 and
x2 random variables Y (x1) and Y (x2) are mutually independent. In this case, Y (x) is
completely determinated by its distribution function FY .) From now on we assume ϕ to
be nonsingular (without specifying this fact when no confusion can arise).
As Th. 1 implies, if p > 1 trajectories St[ϕ1] and St[ϕ2] with ϕ1 6≡ ϕ2 have, in general,
distinct ω-limit sets, and if p = 1 all the trajectories St[ϕ] with ϕ being nonsingular are
attracted by the single fixed point f# ◦ id of Syst. (18) and (19).
As an illustration of Th. 1 we refer to the dynamical system generated by Pr. (2), (3)
and (11). The map f : z 7→ 4z(1 − z), z ∈ [0, 1], satisfies the above conditions, in
particular, f has the invariant measure
µf (dz) =
1
π
dz√
z(1− z)
and supp µf = [0, 1] is of period 1. Every trajectory St[ϕ] tends to the single fixed point
f# ◦ id , which is the random function given by the distribution
Ff#(x, z) =
1
π
z∫
0
dy√
y(1− y)
=
2
π
arcsin
√
z.
What information about the solutions of Pr. (2)-(4) does Th. 1 provide ?
If we want to go from a trajectory St[ϕ] to the solutions uϕ(x, t), we should associate with
uϕ the random process
Pϕ(x, t) =
(
f [t+x] ◦ f# ◦ ϕ
) (
{t + x}
)
, (20)
which gives a statistical description for uϕ in the following sense.
Theorem 2. For any σ > 0, there are ε1 = ε1(σ) > 0, ε2 = ε2(σ) > 0 and T =
T (σ) > 0 such that
for any r ≥ 1 and all points (x1, t1, . . . , xr, tr; z1, . . . , zr) outside of a set of Lebesgue measure
zero,
∣∣∣F ε
uϕ
(x1, t1, . . . , xr, tr; z1, . . . , zr)− FPϕ (x1, t1, . . . , xr, tr; z1, . . . , zr)
∣∣∣ < σ
if ε ∈ (ε1, ε2) and ti > T , i = 1, 2, . . . r.
Th. 2 shows that statistical properties of a deterministic solution uϕ are asymptotical
exactly reproduced by the random process Pϕ. We can thus say that Pϕ is the approx-
imate statistical image of uϕ.
Properties of the approximate statistical image Pϕ:
• Pϕ is a continuous random process;
• Pϕ(x, t) is p-periodic if p > 1 and stationary if p = 1;
• Pϕ(x, t) is autocorrelated only at those points (x1, t1) and (x2, t2) such that
{t1 + x1} = {t2 + x2}.
The expressions for the distribution densities and autocorrelation function of Pϕ are rather
unwieldy and we here omit them.
Unlike the case considered, it may occur that self-stochastic trajectories are generate
only by part of initial functions ϕ. An example of such is given by Eq. (2) with the
boundary condition
∂u
∂t
|x=1= h(u)
∂u
∂t
|x=0 (21)
and the initial condition (4). The condition (21) integrates to the expression
u |x=1= f(u) |x=0 +λ,
where f is an antiderivative of h and λ is an arbitrary constant.
Thus we arrive at a problem with parameter, which is given by Eq. (2) with the initial
condition (4) and the boundary conditions
u |x=1= fλ(u) |x=0, where fλ : z 7−→ f(z) + λ, λ ∈ R. (22)
For reasons of continuity, with each ϕ ∈ C1 ([0, 1],R) there is, in general, associated only
one of the conditions (22), namely, the condition with
λ = γ[ϕ] := ϕ(1)− f(ϕ(0)). (23)
Thus Pr. (2) and (22) presets a measure ν on C1, namely, for any open set Φ ⊂ C1
ν(Φ) = mes {λ : λ = γ[ϕ], ϕ ∈ Φ} .
Let us assume h to satisfy the conditions
• h is a C2-smooth function with |h′(z)| > L > 0,
• (z − z0)h
′′(z) ≥ 0 for z ∈ R and some z0 ∈ R.
Under these conditions, for any h, there exists a (semiopen) interval Λ(h) such that with
λ ∈ Λ(h), the map fλ has bounded nontrivial invariant intervals, the largest of which,
denoted by Iλ, contains all these intervals. Moreover there exist a bounded (generally
speaking, noninvariant) interval I = I(h) such that Iλ ⊂ I for any λ ∈ Λ(h). Therefore
the bounded nonconstant solutions of Pr. (2) and (22) arise from the set
B1(h) =
{
ϕ ∈ C1([0, 1], I) : γ[ϕ] ∈ Λ(h) and ϕ(x) ∈ Iγ[ϕ] for x ∈ [0, 1]
}
.
Pr. (2) and (22) induces the dynamical system
{
B1(h), T, St
}
(24)
with
St[ϕ](x) =
(
f
[t+x]
γ[ϕ] ◦ ϕ
) (
{t + x}
)
, (25)
which in its turn induces the extended dynamical system
{
C#
1 (h), T, St
}
, (26)
where C#
1 (h) is the completion of the phase space B1(h) with functions from <([0.1], I)
via the metric %# and St is given by (25).
A peculiarity of this type of problems is that the action St depends on ϕ. This
fact results in that the trajectories can differ radically in their long time behavior. In
particular, if for some ϕ ∈ B1(h), the map fγ[ϕ] satiefies the conditions (IM), then the
corresponding trajectory St[ϕ] is self-stochastic. A precise formulation is given by the
following theorem.
Theorem 3. There exist two sets Φd, Φr ⊂ B1(h), of positive ν-measure each, such
that the ω-limit set ω[ϕ] of a trajectory St[ϕ] of Syst. (24) and (25) consists of
• deterministic functions if ϕ ∈ Φd,
• random functions if ϕ ∈ Φr.
Moreover, for any integer n ≥ 1, there exist ϕ′(n) ∈ Φd and ϕ′′(n) ∈ Φr such that ω[ϕ′(n)] and
ω[ϕ′′(n)] are cycles of period n (of the extended system).
For ϕ ∈ Φd, functions ψ ∈ ω[ϕ] are, in general, step functions; the set of discontinuities
Dψ of ψ may be infinite but mesDψ = 0.
For ϕ ∈ Φr, functions ψ ∈ ω[ϕ] are described similar to that in Th. 1, namely, in terms
of the invariant measure of the map fγ[ϕ] : z 7→ f(z) + γ[ϕ].
It is not improbable for a self–stochastic trajectory that “points” of its ω-limit set
ω[ϕ] ⊂ C#(D, E), as functions D → E, are random over one subdomain of D and
deterministic over another. An example of such is provided by the problem
∂u1
∂t
=
∂u1
∂x1
+
∂u1
∂x2
, (27)
∂u2
∂t
= −∂u2
∂x1
− ∂u2
∂x2
, x = (x1, x2) ∈ R× [0, 1], t ∈ R+;
u1 = u2 |x2=0,
∂u1
∂t
= h(u2)
∂u2
∂t
|x2=1 . (28)
u1 |t=0= ϕ1(x), u2 |t=0= ϕ2(x). (29)
As well as in the above case, one can introduce
γ[ϕ](x1) = ϕ1(x1, 1)− f(ϕ2(x1, 1)), x1 ∈ R,
and find B2(h) ⊂ C1(R × [0, 1],R2) such that the bounded nonconstant solutions are
generated by those, and only those, ϕ such that ϕ ∈ B2(h) and construct the completion
C#
2 (h) of B2(h) with functions from <. Corresponding to Pr. (27) and (28) is the dynam-
ical system
{
B2(h), T, St
}
and the extended dynamical system
{
C#
2 (h), T, St
}
, where for
both these systems, St is given by (10) with f being replaced with fγ[ϕ](x1).
Theorem 4. There exist two open (in C1) sets Φd, Φr ⊂ B2(h) such that the ω-limit
set ω[ϕ] of a trajectory St[ϕ] of the dynamical system{
B2(h), T, St
}
consists of
• deterministic functions (which combine into a cycle of the extended system) if ϕ ∈ Φd;
• random functions (which combine into a family of almost periodic trajectories of the
extended system) if ϕ ∈ Φr.
Moreover, for ϕ ∈ Φr the domain R × [0, 1] falls into subdomains D1(ϕ) and D2(ϕ), of
positive Lebesgue measure each, such that any ψ ∈ ω[ϕ] is a random function on D1 and a
deterministic function on D2.
It should be noted that self-stochasticity phenomenon is not exotic because
mes Λacim > 0, where Λacim(h) = {λ : fλ has a smooth invariant measure} [2].
4 Universal Properties
The theorems 3-4 can essentially be widened through the use of a number of properties of
1-D dynamical systems. Here we present several statements for Pr. (2) and (21), which
involve, in particular, Feigenbaum’s constants δ = 4.6992 . . . and α = 2.502 . . . .
Let λ(n) be for the lower bound of those λ such that fλ has a cycle of period n; β(n)
be for the lower bound of those λ such that fλ has a smooth invariant measure with the
support consisting of n intervals.
As known, with ordering for natural numbers
1 ≺ 2 ≺ 22 ≺ . . . ≺ 5 · 22 ≺ 3 · 22 ≺ . . . ≺ 5 · 2 ≺ 3 · 2 ≺ . . . ≺ 9 ≺ 7 ≺ 5 ≺ 3,
the following relations hold:
λ(n) < λ(n′) for n ≺ n′,
λ(n) < β(n) < λ(n′) for n ≺ n′, if n 6= 2i, and
λ(6n) < β(n) < λ((2s + 1)n) for any s ≥ 1, if n = 2i, i = 0, 1, 2, . . . .
Moreover,
λ (2is)− λ (2i−1s)
λ (2i+1s)− λ (2is)
→ δ as i →∞, whatever s ≥ 1.
Let ϕξ, ξ ∈ (ξ1, ξ2), be a family of functions from B1(h) that depend on the parameter
ξ continuously, and let λj = γ[ϕξj
], j = 1, 2.
Theorem 5. Let n1 ≺ n ≺ n2. If the map fλ1 has no cycles of period n1 and the map
fλ2 has a cycle of period n2, then
• there exists an interval Ξn ⊂ (ξ1, ξ2) such that for ξ ∈ Ξn the ω-limit set ω[ϕξ] is a
cycle of period n;
• if n1 6= 2i there exists ξ′ ∈ Ξn such that the ω-limit set ω[ϕξ′ ] is a cycle of period n,
whose points are random functions.
Set
Pi =
{
ϕ ∈ B1(h) : λ(2i) < γ[ϕ] < λ(2i+1)
}
,
Qi =
{
ϕ ∈ B1(h) : λ(2i) < γ[ϕ] < β(2i−1)
}
,
Ri = {ϕ ∈ Qi : γ[ϕ] ∈ Λacim(h)} , i = 0, 1, 2, . . . .
Theorem 6. The ω-limit set of a trajectory St[ϕ] of Syst. (24), (25) is
• a cycle of the period equal to 2i, which consists of deterministic functions, for ϕ ∈ Pi;
• a cycle of the period divisible by 2i, which consists of deterministic functions, for almost
all ϕ ∈ Qi \ Ri;
• a cycle of the period divisible by 2i, which consists of random functions, for almost all
ϕ ∈ Ri.
Furthermore, the following universal relations hold:
lim
i→∞
ν(Pi)
ν(Pi+1)
= lim
i→∞
ν(Qi)
ν(Qi+1)
= lim
i→∞
ν(Ri)
ν(Ri+1)
= δ.
Theorem 7. There exists a constant C > 0 such that for any ε > 0 and any ϕ ∈ Qn
one can find a 2n-periodic step function qϕ with the property that for the solution uϕ of
Pr. (2) and (21) there is a θ(ϕ, ε) > 0 such that
mes
{
(x, t) : |uϕ(x, t)− qϕ(x, t)| > Cα−n for θ < t < θ + 1
}
< ε
for θ > θ(ϕ, ε).
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