NEW SCENARIO FOR TRANSITION TO SLOW 3-D TURBULENCE PART I.SLOW 1-D TURBULENCE IN NIKOLAEVSKII SYSTEM.

International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015
DOI : 10.14810/ijrap.2015.4102 21
NEW SCENARIO FOR TRANSITION TO SLOW
3-D TURBULENCE PART I.SLOW 1-D
TURBULENCE IN NIKOLAEVSKII SYSTEM.
J. Foukzon
Israel Institute of Technology,Haifa, Israel
Abstract:
Analyticalnon-perturbative study of thethree-dimensional nonlinear stochastic partialdifferential equation
with additive thermal noise, analogous to thatproposed by V.N.Nikolaevskii [1]-[5] to describelongitudinal
seismic waves, ispresented. Theequation has a threshold of short-waveinstability and symmetry, providing
longwavedynamics.New mechanism of quantum chaos generating in nonlineardynamical systemswith
infinite number of degrees of freedom is proposed. The hypothesis is said,that physical turbulence could be
identifiedwith quantum chaos of considered type. It is shown that the additive thermal noise destabilizes
dramatically the ground state of theNikolaevskii system thus causing it to make a direct transition from a
spatially uniform to a turbulent state.
Keywords:
3D turbulence, chaos, quantum chaos,additive thermal noise,Nikolaevskii system.
1.Introduction
In the present work a non-perturbative analyticalapproach to the studying of problemof quantum chaos in
dynamical systems withinfinite number of degrees of freedom isproposed.Statistical descriptions of
dynamical chaos and investigations of noise effects on chaoticregimes arestudied.Proposed approach also
allows estimate the influence of additive (thermal)fluctuations on the processes of formation ofdeveloped
turbulence modes in essentially nonlinearprocesses like electro-convection andother. A principal rolethe
influence ofthermalfluctuations on thedynamics of some types of dissipative systems inthe approximate
environs of derivation rapid of ashort-wave instability was ascertained. Impotentphysicalresults follows
from Theorem 2, is illustrated by example of 3D stochastic model system:
߲‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
߲‫ݐ‬
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
+ ቈߜଵ
߲‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
߲‫ݔ‬ଵ
+ ߜଶ
߲‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
߲‫ݔ‬ଶ
+ ߜଷ
߲‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
߲‫ݔ‬ଷ
቉ ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ +
+݂ሺ‫,ݔ‬ ‫ݐ‬ሻ − ඥߟ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ = 0, ‫ݔ‬ ∈ ℝଷ
(1.1)
‫ݑ‬ఎሺ‫,ݔ‬ 0, ߝሻ = 0, ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ =
డరௐሺ௫,௧ሻ
డ௫భడ௫మడ௫యడ௧
, ߟ ≪ 1,0 < ߜ௝, ݆ = 1,2,3, (1.2)

22
which was obtained from thenon-stochastic3‫ܦ‬Nikolaevskiimodel:
డ௨ሺ௫,௧,ఌሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ + ቂߜଵ
డ௫భ
+ ߜଶ
డ௫మ
+ ߜଷ
డ௫య
ቃ ‫ݑ‬ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ + ݂ሺ‫,ݔ‬ ‫ݐ‬ሻ (1.3)
which is perturbed by additive “small” white noise ඥߟ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ. And analytical result also illustrated by
exampleof1‫ܦ‬stochasticmodel system
డ௨ആሺ௫,௧,ఌሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ + ߜ
డ௨ആሺ௫,௧,ఌሻ
డ௫భ
‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ + ݂ሺ‫,ݔ‬ ‫ݐ‬ሻ − ඥߟ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ = 0, ‫ݔ‬ ∈ ℝ (1.4)
‫ݑ‬ఎሺ‫,ݔ‬ 0, ߝሻ = 0, ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ =
డమௐሺ௫,௧ሻ
డ௫డ௧
, ߟ ≪ 1,0 < ߜ, (1.5)
which was obtained from thenon-stochastic 1DNikolaevskiismodel:
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ + ߜ
డ௫భ
‫ݑ‬ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ + ݂ሺ‫,ݔ‬ ‫ݐ‬ሻ = 0, ‫ݑ‬ሺ‫,ݔ‬ 0, ߝሻ = 0, ‫ݔ‬ ∈ ℝ, (1.6)
‫ݑ‬ሺ‫,ݔ‬ 0, ߝሻ = 0.(1.7)
Systematic study of a different type of chaos at onset ‘‘soft-mode turbulence’’based onnumerical
integration of the simplest 1DNikolaevskii model (1.7)has been executed by many authors [2]-[7].There is
an erroneous belief that such numerical integration gives a powerful analysisismeans of the processes of
turbulence conception, based on the classical theory ofchaos of the finite-dimensional classical systems
[8]-[11].
Remark1.1.However, as it well known, such approximations correct only in a phase ofturbulence
conception, when relatively small number of the degrees of freedom excites. In general case, when a very
large number of the degrees of freedom excites, well known phenomena of thenumerically induced chaos,
can to spoils in the uncontrollable wayany numerical integration[12]-[15]
Remark1.2.Other non trivial problem stays from noise roundoff error in computer computation using
floatingpoint arithmetic [16]-[20].In any computer simulation the numerical solution is fraught with
truncation by roundoff errors introduced by finite-precision calculation of trajectories of dynamical systems,
where roundoff errors or other noise can introduce new behavior and this problem is a very more
pronounced in the case of chaotic dynamical systems, because the trajectories of such systems
exhibitextensivedependence on initial conditions. As a result, a small random truncation or roundoff error,
made computational error at any step of computation will tend to be large magnified by future
computationalof the system[17].
Remark1.3.As it well known, if the digitized or rounded quantity is allowed to occupy the nearest of a
large number of levels whose smallest separation is ‫ܧ‬଴, then, provided that the
original quantity is large compared to ‫ܧ‬଴ and is reasonably well behaved, theeffect of the quantization or
rounding may betreated as additive random noise [18].Bennett has shown that such additive noise is nearly
white, withmean squared value of ‫ܧ‬଴
ଶ
/12[19].However the complete uniform white-noise model to be valid
in the sense of weak convergence of probabilistic measures as the lattice step tends to zero if the matrices
of realization of the system in the state space satisfy certain nonresonance conditions and the
finite-dimensional distributions of the input signal are absolutely continuous[19].
The method deprived of these essential lacks in general case has been offered by the author in papers

23
[23]-[27].
Remark1.4.Thus from consideration above it is clear thatnumerical integration procedure ofthe1D
Nikolaevskii model (1.6)-(1.7) executed in papers [2]-[7]in fact dealing withstochastic model
(1.4)-(1.5).There is an erroneous the point of view,that a white noise with enough small intensity does not
bringanysignificant contributions in turbulent modes, see for example [3]. By this wrong assumptions the
results of the numerical integration procedure ofthe1D Nikolaevskii model (1.6)-(1.7) were mistakenly
considered and interpretedas a veryexact modeling the slow turbulence within purely non stochastic
Nikolaevskii model (1.6)-(1.7).Accordingly wrongconclusionsabout that temperature noisesdoes not
influence slowturbulence have been proposed in [3].However in [27] has shown non-perturbativelythat that
a white noise with enough small intensity can to bring significant contributions in turbulent modes and
even to change this modes dramatically.
At the present time it is generally recognized thatturbulence in its developed phase has essentiallysingular
spatially-temporal structure. Such asingular conduct is impossible to describeadequately by the means of
some model system of equations of a finite dimensionality. In thispoint a classical theory of chaos is able
todescribe only small part of turbulencephenomenon in liquid and another analogous sof dynamical
systems. The results of non-perturbative modeling ofsuper-chaotic modes, obtained in the present paper
allow us to put out a quite probablehypothesis: developed turbulence in the realphysical systems with
infinite number of degreesof freedom is a quantum super-chaos, at that the quantitative characteristics of
this super-chaos, iscompletely determined by non-perturbativecontribution of additive (thermal)
fluctuations inthe corresponding classical system dynamics [18]-[20].
2.Main Theoretical Results
We study the stochastic ‫-ݎ‬dimensionaldifferential equation analogous proposed byNikolaevskii [1] to
describe longitudinal seismicwaves:
డ௨ആሺ௫,௧,ఌ,ఠሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ + ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ ∑ ߜ௜
డ௨ആሺ௫,௧,ఌ,ఠሻ
డ௫೔
௥
௝ୀଵ + ݂ሺ‫,ݔ‬ ‫ݐ‬ሻ −
ඥߟ‫ݓ‬ሺ‫,ݔ‬ ‫,ݐ‬ ߱ሻ = 0, (2.1)
‫ݔ‬ ∈ ℝ௥
, ‫ݑ‬ఎሺ‫,ݔ‬ 0, ߝ, ߱ሻ = 0, ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ =
డೝశభௐሺ௫,௧ሻ
డ௫భడ௫మ…డ௫ೝడ௧
, 0 < ߜ௝, ݆ = 1, … , ‫)2.2(.ݎ‬
The main difficulty with the stochasticNikolaevskii equationis that the solutions do not take values in an
function space but in generalized functionspace. Thus it is necessary to give meaning to the non-linear
terms ߲௫ೕ
‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ, ݆ = 1, … , ‫ݎ‬ because the usual product makes no sense for arbitrary distributions.
We deal with product of distributions via regularizations, i.e., we approximate the distributions by
appropriate way and pass to the limit.In this paper we use the approximation of the distributions by
approach of Colombeaugeneralized functions [28].
Notation 2.1.We denote byࣞሺℝ௥
× ℝାሻthe space of the infinitely differentiable functionswith compact
supportin ℝ௥
× ℝାandbyࣞ′ሺℝ௥
× ℝାሻ its dual space.Let ℭ = ሺΩ, Σ, µሻbe a probability space. We denote
by۲the space of allfunctionsܶ: Ω → ࣞ′ሺℝ௥
× ℝାሻ such that 〈ܶ, ߮〉is a random variable for all߮ ∈
ࣞሺℝ௥
× ℝାሻ.Theelements of۲are called random generalized functions.
Definition 2.1.[29].We say that a random field ሼℜሺ‫,ݔ‬ ‫ݐ‬ሻ|‫ݐ‬ ∈ ℝା, ‫ݔ‬ ∈ ℝ௥ሽ isa spatiallydependent
semimartingale if for each ‫ݔ‬ ∈ ℝ௥
, ሼℜሺ‫,ݔ‬ ‫ݐ‬ሻ|‫ݐ‬ ∈ ℝାሽ is asemimartingale in relation to the same
filtration ሼℱ௧|‫ݐ‬ ∈ ℝାሽ. If ℜሺ‫,ݔ‬ ‫ݐ‬ሻ is a ‫ܥ‬∞
-function of ‫ݔ‬ and continuous inalmost everywhere,it is called
a‫ܥ‬∞
-semimartingale.

24
Definition 2.2.We say that that ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ ∈ ۲ is a strong generalized solution(SGS) of the
Eq.(2.1)-(2.2) if there exists asequence of‫ܥ‬∞
-semimartingales‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ߳ ∈ ሺ0,1ሿ such that there
exists
ሺiሻ ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ =ୢୣ୤ limఢ→଴‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻinࣞ′ሺℝ௥
× ℝାሻ almost surely for ߱ ∈ Ω,
ሺiiሻ ߲௫ೕ
‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ =ୢୣ୤ limఢ→଴߲௫ೕ
‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ݆ = 1, … , ‫ݎ‬almost surely for ߱ ∈ Ω,
ሺiiiሻfor all ߮ ∈ ࣞሺℝ௥
× ℝାሻ
ሺivሻ〈߲௧‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ, ߮〉 − 〈∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ, ߮〉 − ෍
ߜ௜
2
〈߲௫ೕ
‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ, ߮〉
௥
௝ୀଵ
− 〈݂ሺ‫,ݔ‬ ‫ݐ‬ሻ, ߮〉 +
+ඥߟ ‫׬‬ ݀‫ݔ‬ℝೝ ‫׬‬ ߮ሺ‫,ݐ‬ ‫ݔ‬ሻ
∞
଴
ܹ݀௧ሺ‫,ݔ‬ ‫ݐ‬ሻ = 0, ‫ݐ‬ ∈ ℝାalmost surely for ߱ ∈ Ω,
and where ܹ௧ሺ‫,ݔ‬ ‫ݐ‬ሻ =
డೝௐሺ௫,௧ሻ
డ௫భడ௫మ…డ௫ೝ
,
ሺvሻ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ = 0almost surely for ߱ ∈ Ω.
However in this paper we use the solutionsofstochastic Nikolaevskii equation only in the sense of
Colombeaugeneralized functions [30].
Remark2.1.Note that from Definition 2.2it is clear that any strong generalized solution‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ of
the Eq.(2.1)-(2.2) one can to recognized as Colombeaugeneralized function such that
‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ =ୢୣ୤ ቀ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻቁ
ఢ
ሺ#ሻ
By formula ሺ#ሻone can todefine appropriate generalized solutionof the Eq.(2.1)-(2.2) even if a strong
generalized solutionof the Eq.(2.1)-(2.2) does not exist.
Definition 2.3.Assumethata strong generalized solution of the Eq.(2.1)-(2.2) does not exist.We shall say
that:
(I)Colombeaugeneralized stochastic process ቀ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻቁ
ఢ
is a weak generalized solution (WGS) of
the Eq.(2.1)-(2.2) orColombeausolutionof the Eq.(2.1)-(2.2) iffor all ߮ ∈ ࣞሺℝ௥
× ℝାሻandfor all߳ ∈ ሺ0,1ሿ
ሺiሻ〈‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ߲௧߮〉 − 〈∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ߮〉 − ෍
ߜ௜
2
〈߲௫ೕ
‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ߮〉
௥
௝ୀଵ
+
+ 〈݂ሺ‫,ݔ‬ ‫ݐ‬ሻ, ߮〉 + ඥߟ ‫׬‬ ݀‫ݔ‬ℝೝ ‫׬‬ ߮ሺ‫,ݐ‬ ‫ݔ‬ሻ
∞
଴
ܹ݀௧ሺ‫,ݔ‬ ‫ݐ‬ሻ = 0, ‫ݐ‬ ∈ ℝାalmost surely for ߱ ∈ Ω,
ሺiiሻ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ = 0almost surely for ߱ ∈ Ω.

25
(II)Colombeaugeneralized stochastic process ቀ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻቁ
ఢ
is aColombeau-Ito’ssolutionof the
Eq.(2.1)-(2.2) if for all ߮ ∈ ࣞሺℝ௥ሻand for all߳ ∈ ሺ0,1ሿ
ሺiሻ〈߲௧‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ߮〉 + 〈∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ߮〉 ෍
ߜ௜
2
〈߲௫ೕ
‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ, ߮〉
௥
௝ୀଵ
−
−〈݂ሺ‫,ݔ‬ ‫ݐ‬ሻ, ߮〉 − ඥߟ ‫׬‬ ߮ሺ‫ݔ‬ሻℝೝ ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ݀‫ݔ‬ = 0, ‫ݐ‬ ∈ ℝାalmost surely for ߱ ∈ Ω,
ሺiiሻ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߳, ߱ሻ = 0almost surely for ߱ ∈ Ω.
Notation 2.2.[30]. Thealgebra of moderate element we denote byℰெሾℝ௥ሿ.The Colombeau algebra of the
Colombeau generalized functionwe denote by ࣡ሺℝ௥ሻ.
Notation 2.3.[30].We shall use the following designations. If ܷ ∈ ࣡ሺℝ௥ሻit representatives will be
denoted byܴ௎, their values on ߮ = ൫߮ఢሺ‫ݔ‬ሻ൯,߳ ∈ ሺ0,1ሿwill be denoted by ܴ௎ሺ߮ሻ and it pointvalues at
‫ݔ‬ ∈ ℝ௥
will be denoted ܴ௎ሺ߮, ‫ݔ‬ሻ.
Definition 2.4.[30].Let ‫ܣ‬଴ = ‫ܣ‬଴ሺℝ௥ሻ be the set of all ߮ ∈ ‫ܦ‬ሺℝ௥ሻ such that‫׬‬ ߮ሺ‫ݔ‬ሻ ݀‫ݔ‬ = 1.
Let ℭ = ሺΩ, Σ, µሻbe a probability space.Colombeau random generalized function this is a
map ܷ: Ω → ࣡ሺℝ௥ሻ such that there is representing functionܴ௎: ‫ܣ‬଴ × ℝ௥
× Ω with the properties:
(i) for fixed߮ ∈ ‫ܣ‬଴ሺℝ௥ሻ the function ሺ‫,ݔ‬ ߱ሻ → ܴ௎ሺ߮, ‫,ݔ‬ ߱ሻ is a jointly measurable on ℝ௥
× Ω;
(ii) almost surely in ߱ ∈ Ω,the function ߮ → ܴ௎ሺ߮, . , ߱ሻbelongs to ℰெሾℝ௥ሿ and is a
representative of ܷ;
Notation 2.3.[30]. TheColombeaualgebra ofColombeau random generalized functionis denoted
by࣡Ωሺℝ௥ሻ.
Definition 2.5.Let ℭ = ሺΩ, Σ, µሻbe a probability space. Classically, a generalizedstochastic process on
ℝ௥
is a weakly measurable mapܸ: Ω → ‫ܦ‬′ሺℝ௥ሻ denoted by ܸ ∈ ‫ܦ‬Ω
′
ሺℝ௥ሻ. If ߮ ∈ ‫ܣ‬଴ሺℝ௥ሻ, then
(iii) ܸሺ߱ሻ ∗ ߮ሺ‫ݔ‬ሻ = 〈ܸሺ߱ሻ, ߮ሺ‫.−ݔ‬ ሻ〉 is a measurable with respect to߱ ∈ Ω and
(iv) smooth with respect to‫ݔ‬ ∈ ℝ௥
and hence jointly measurable.
(v) Also ൫ܸሺ߱ሻ ∗ ߮ሺ‫ݔ‬ሻ൯ ∈ ℰெሾℝ௥ሿ.
(vi) Therefore ܴ௏ሺ߮, ‫,ݔ‬ ߱ሻ = ܸሺ߱ሻ ∗ ߮ሺ‫ݔ‬ሻ qualifies as an representing function for an element
of࣡Ωሺℝ௥ሻ.
(vii) In this way we have an imbedding ࣞ′ሺℝ௥ሻ → ࣡Ωሺℝ௥ሻ.
Definition 2.6.Denote by ܵሺܶሻ = ܵሺℝ௥ାଵሻ ↾ ܶ the space of rapidly decreasing smooth functions on
ܶ = ℝ௥
× ሾ0, ∞ሻ. Letℭ = ሺΩ, Σ, µሻwith (i) Ω = ܵ′ሺܶሻ, ሺiiሻ Σ- the Borelߪ-algebra generated by the weak
topology. Therefore there is unique probability measure ߤ on ሺΩ, Σሻ such that
න ݀ߤሺ߱ሻexp ሾ݅〈߱, ߮〉ሿ = exp ൬−
1
2
‖φ‖௅మሺ்ሻ
ଶ
൰
for all ߮ ∈ ܵሺܶሻ. White noise‫ݓ‬ሺ߱ሻ with the support in ܶ is the generalized process ‫ݓ‬ሺ߱ሻ: Ω →
ࣞ′ሺℝ௥ାଵሻ such that: (i) ‫ݓ‬ሺ߮ሻ = 〈‫ݓ‬ሺ߱ሻ, ߮〉 = 〈߱, ߮ ↾ ܶ〉 (ii) ۳ሾ‫ݓ‬ሺ߮ሻሿ = 0, (iii) ۳ሾ‫ݓ‬ଶሺ߮ሻሿ =
‖φ‖௅మሺ்ሻ
ଶ
.Viewed as a Colombeau random generalized function, it has a representative (denoting on
variables in ℝ௥ାଵ
by ሺ‫,ݔ‬ ‫ݐ‬ሻ): ܴ௪ሺ߮, ‫,ݔ‬ ‫,ݐ‬ ߱ሻ = 〈߱, ߮ሺ‫,−ݔ‬ ‫ݐ‬ −ሻ ↾ ܶ〉, which vanishes if ‫ݐ‬ is less than
minus the diameter of the support of߮.Therefore‫ݓ‬ is a zero on ℝ௥
× ሺ−∞, 0ሻ in ࣡Ωሺℝ௥ାଵሻ. Note that its
variance is the Colombeau constant:۳ሾܴ௪
ଶ ሺ߮, ‫,ݔ‬ ‫,ݐ‬ ߱ሻሿ = ‫׬‬ ݀‫ݕ‬ ‫׬‬ |߮ሺ‫ݔ‬ − ‫,ݕ‬ ‫ݐ‬ − ‫ݏ‬ሻ|ଶ
݀‫ݏ‬
∞
଴ℝೝ .

26
Definition 2.7.Smoothedwith respect to ℝ௥
white noise ൫‫ݓ‬ఢሺ‫,ݔ‬ ‫ݐ‬ሻ൯ఢ
the
representativeܴ௪ሺ߮, ‫,ݔ‬ ‫,ݐ‬ ߱ሻwith߮ ∈ ‫ܣ‬଴ሺℝ௥ሻ × ࣞ′ሺℝ௥ሻ,such that߮ = ൫߮ఢሺ‫ݔ‬ሻ൯,߳ ∈ ሺ0,1ሿ and߮ఢ =
߳ି௥
߮ ቀ
௫
ఢ
ቁ ߜሺ‫ݐ‬ሻ.
Theorem 2.1.[25]. (Strong Large Deviation Principle fo SPDE)(I) Let൫‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ൯ఢ
,߳ ∈
ሺ0,1ሿbe solution of theColombeau-Ito’sSPDE [26]:
డ൫௨ചሺ௫,௧,ఌ,ఎ,ఠሻ൯
ച
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ൫‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ൯ఢ
+
ቀቀ‫ܨ‬ఢ൫‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ൯ቁ
ఢ
ቁ ∑ ߜ௜ ቆ‫ܨ‬ఢ ቀ
డ௨ചሺ௫,௧,ఌ,ఎ,ఠሻ
డ௫೔
ቁቇ
ఢ
௥
௜ୀଵ + ൫݂ఢሺ‫,ݔ‬ ‫ݐ‬ሻ൯ఢ
− ඥߟ൫‫ݓ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߱ሻ൯ఢ
= 0, (2.3)
‫ݔ‬ ∈ ℝ௥
, ‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ ≡ 0, ߜ௝ > 0, ݆ = 1, … , ‫)4.2(.ݎ‬
Here: (1)൫‫ܨ‬ఢሺ‫ݖ‬ሻ൯
ఢ
∈ ‫ܩ‬ሺℝሻ, ‫ܩ‬ሺℝሻthe Colombeau algebra of Colombeau generalized functions and ‫ܨ‬଴ሺ‫ݖ‬ሻ =
‫.ݖ‬
(2) ൫݂ఢሺ‫,ݔ‬ ‫ݐ‬ሻ൯ఢ
∈ ࣡ሺℝ௥ାଵሻ.
(3)‫ݓ‬ఢሺ‫,ݔ‬ ‫ݐ‬ሻis a smoothedwith respect to ℝ௥
white noise.
(II)Letቀ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻቁ
ఢ
,߳ ∈ ሺ0,1ሿ be solution of the Colombeau-Ito’s SDE[26]:
ௗቀ௨ച,࢔ሺ௫࢔,௧,ఌ,ఎ,ఠሻቁ
ച
ௗ௧
+ ∆࢔ሾߝ − ሺ1 + ∆࢔ሻଶሿ ቀ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻቁ
ఢ
+
൬൬‫ܨ‬ఢ ቀ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻቁ൰
ఢ
൰ ∑ ߜ௜ ቆ‫ܨ‬ఢ ቀ
௨ച,࢔శభ,೔൫௫࢔శ૚,೔,௧,ఌ,ఎ,ఠ൯ି௨ച,࢔൫௫࢔,೔,௧,ఌ,ఎ,ఠ൯
௛ಿ
ቁቇ
ఢ
௥
௜ୀଵ + ൫݂ఢሺ‫ݔ‬࢔, ‫ݐ‬ሻ൯ఢ
−
ඥߟ൫‫ݓ‬ఢሺ‫ݔ‬࢔, ‫,ݐ‬ ߱ሻ൯ఢ
= 0, (2.5)
‫ݔ‬࢔ ∈ દ ⊂ ℎே ∙ ℤ௥
, ࢔ = ሺ݊ଵ, … , ݊௥ሻ ∈ ℤ௥
, |࢔| = ∑ ݊௝
௥
௜ୀଵ , ‫ݔ‬࢔ାଵ,௜ = ൫‫ݔ‬௡భ
, … , ‫ݔ‬௡೔
+ ℎே, … , ‫ݔ‬௡ೝ
൯, −ܰ ≤ ݊௜ ≤
ܰ,(2.6)
‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, 0, ߝ, ߟ, ߱ሻ ≡ 0, ‫ݔ‬࢔ ∈ દ.(2.7)
Here Eq.(2.5)-Eq.(2.7) is obtained from Eq.(2.3)-Eq.(2.4) byspatialdiscretizationon finite latticeદ
ℎே → 0ifܰ → ∞ and ∆࢔-is a latticed Laplacian[31]-[33].
(III)Assume that Colombeau-Ito’s SDE (2.5)-(2.7)is a strongly dissipative.[26].
(IV) Letℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ be the solutions of thelinear PDE:
డℜሺ௫,௧,ఌ,ఒሻ
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ + ߣ ∑ ߜ௜
డ௫೔
௥
௜ୀଵ − ݂ሺ‫,ݔ‬ ‫ݐ‬ሻ = 0, ߣ ∈ ℝ,(2.8)
ℜሺ‫,ݔ‬ 0, ߝ, ߣሻ = 0. (2.9)
Then
liminfఢ→଴۳ሾ|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶሿ ≤ ℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ. (2.10)

27
Proof.The proof based on Strong large deviations principles(SLDP-Theorem) for Colombeau-Ito’ssolution
of the Colombeau-Ito’s SDE, see [26],theorem 6.BySLDP-Theorem one obtain directlythe differential
master equation (see [26],Eq.(90)) for Colombeau-Ito’s SDE(2.5)-(2.7):
݀ ቀܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝሻቁ
ఢ
݀‫ݐ‬
+ ∆࢔ሾߝ − ሺ1 + ∆࢔ሻଶሿ ቀܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝሻቁ
ఢ
+
+ߣ࢔ ∑ ߜ௜ ቆ‫ܨ‬ఢ ቀ
௎ച,࢔శభ,೔൫௫࢔శ૚,೔,௧,ఌ൯ି௎ച,࢔൫௫࢔,೔,௧,ఌ൯ା
௛ಿ
ቁቇ
ఢ
௥
௜ୀଵ + ൫݂ఢሺ‫ݔ‬࢔, ‫ݐ‬ሻ൯ఢ
+ ‫݋‬ሺ߳ሻ = 0,(2.11)
ܷఢ,࢔ሺ‫ݔ‬࢔, 0, ߝሻ = −ߣ࢔. (2.12)
We set now ߣ࢔ ≡ ߣ ∈ ℝ. Then from Eq.(2.13)-Eq.(2.14) we obtain
݀ ቀܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߣሻቁ
ఢ
݀‫ݐ‬
+ ∆࢔ሾߝ − ሺ1 + ∆࢔ሻଶሿ ቀܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߣሻቁ
ఢ
+
+ߣ ∑ ߜ௜ ቆ‫ܨ‬ఢ ቀ
௎ച,࢔శభ,೔൫௫࢔శ૚,೔,௧,ఌ,ఒ൯ି௎ച,࢔൫௫࢔,೔,௧,ఌ,ఒ൯
௛ಿ
ቁቇ
ఢ
௥
௜ୀଵ + ൫݂ఢሺ‫ݔ‬࢔, ‫ݐ‬ሻ൯ఢ
+ ܱሺ߳ሻ = 0,(2.13)
ܷఢ,࢔ሺ‫ݔ‬࢔, 0, ߝ, ߣሻ = −ߣ. (2.14)
From Eq.(2.5)-Eq.(2.7) and Eq.(2.13)-Eq.(2.14) by SLDP-Theorem (see see[26], inequality(89)) we obtain
the inequality
liminfఢ→଴۳ ቂห‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣห
ଶ
ቃ ≤ ܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߣሻ, ߳ ∈ ሺ0,1ሿ.(2.15)
Let us consider now the identity
|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
= หൣ‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻ൧ + ൣ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ൧ห
ଶ
. (2.16)
Fromtheidentity (2.16) bythetriangle inequality we obtaintheinequality
|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ ห‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻห
ଶ
+ ห‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣห
ଶ
. (2.17)
From theidentity (2.17) by integration we obtain theinequality
۳|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤
≤ ۳ห‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻห
ଶ
+ ห‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣห
ଶ
. (2.18)
From theidentity (2.18) by theidentity (2.15) for all ߳ ∈ ሺ0,1ሿwe obtain theinequality
۳|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤
≤ ۳ห‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻห
ଶ
+ ܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߣሻ.(2.19)

28
In the limit ܰ → ∞ fromthe inequality we obtain the inequality
۳|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤
≤ limsupே→∞۳ห‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻห
ଶ
+ limsupே→∞ܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߣሻ. (2.20)
Wenote that
limsupே→∞۳ห‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ‫ݑ‬ఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߟ, ߱ሻห
ଶ
= 0. (2.21)
Therefore from (2.20) and (2.21) we obtain the inequality
۳|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ limsupே→∞ܷఢ,࢔ሺ‫ݔ‬࢔, ‫,ݐ‬ ߝ, ߣሻ(2.22)
In the limit ܰ → ∞ from Eq.(2.13)-Eq.(2.14) for any fixed ߳ ≠ 0, ߳ ≪ 1, we obtain the differential master
equationforColombeau-Ito’s SPDE (2.3)-(2.4)
݀൫ܷఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ൯ఢ
݀‫ݐ‬
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ൫ܷఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ൯ఢ
+
+ߣ ∑ ߜ௜ ቆ‫ܨ‬ఢ ቀ
డ௎ചሺ௫,௧,ఌ,ఒሻ
డ௫೔
ቁቇ
ఢ
௥
௜ୀଵ + ൫݂ఢሺ‫,ݔ‬ ‫ݐ‬ሻ൯ఢ
+ ܱሺ߳ሻ = 0,(2.23)
ܷఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ = −ߣ. (2.24)
Therefore from the inequality (2.22) followsthe inequality
۳|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ ܷఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ. (2.25)
In the limit ߳ → 0from differential equation (2.23)-(2.24)we obtain the differential equation (2.8)-(2.9)and
it is easy to see that
limఢ→଴ܷఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ = ℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ. (2.26)
From the inequality (2.25) one obtainthe inequality
liminfఢ→଴۳|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ limఢ→଴ܷఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ = ℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ.(2.27)
From the inequality (2.27) and Eq.(2.26) finally we obtainthe inequality
liminfఢ→଴۳|‫ݑ‬ఢሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߟ, ߱ሻ − ߣ|ଶ
≤ ℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ. (2.28)
The inequality (2.28) finalized the proof.
Definition 2.7.(TheDifferential Master Equation)The linear PDE:
డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ + ߣ ∑ ߜ௜
డ௫೔
௥
௜ୀଵ − ݂ሺ‫,ݔ‬ ‫ݐ‬ሻ = 0, ߣ ∈ ℝ,(2.29)
ℜሺ‫,ݔ‬ 0, ߝ, ߣሻ = 0 (2.30),
We will call asthe differential master equation.

29
Definition 2.8.(TheTranscendental Master Equation)Thetranscendental equation
ℜ൫‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ൯ = 0, (2.31)
wewill call asthe transcendentalmaster equation.
Remark2.2.We note that concrete structure of the Nikolaevskii chaos is determined by the
solutionsߣሺ‫,ݔ‬ ‫,ݐ‬ ߝሻvariety bytranscendentalmaster equation(2.31).Master equation (2.31)is determines by the
only way some many-valued functionߣሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ which is the main constructive object, determining
thecharacteristics of quantum chaos in the corresponding model of Euclidian quantum fieldtheory.
3.Criterion of the existence quantum chaos in Euclidian quantum
N-model.
Definition3.1.Let ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻbe the solution of the Eq.(2.1). Assume that for almost all pointsሺ‫,ݔ‬ ‫ݐ‬ሻ ∈
ℝ௥
× ℝା(in the sense of Lebesgue–measureonℝ௥
× ℝା), there exist a function ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ such that
limఎ→଴۳ ൤ቀ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ − ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻቁ
ଶ
൨ = 0. (3.1)
Then we will say that afunction ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻis a quasi-determined solution (QD-solution of the Eq.(2.).
Definition3.2. Assume that there exist a setℌ ⊂ ℝ௥
× ℝାthat is positive
Lebesgue–measure, i.e.,ߤሺℌሻ > 0 and
∀ሺ‫,ݔ‬ ‫ݐ‬ሻ൛ሺ‫,ݔ‬ ‫ݐ‬ሻ ∈ ℌ → ¬∃limఎ→଴۳ൣ‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ൧ൟ,(3.2)
i.e., ሺ‫,ݔ‬ ‫ݐ‬ሻ ∈ ℌ imply that the limit: limఎ→଴۳ൣ‫ݑ‬ఎ
ଶሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ൧does not exist.
Then we will say thatEuclidian quantum N-model has thequasi-determined Euclidian quantum chaos
(QD-quantum chaos).
Definition3.3.For each pointሺ‫,ݔ‬ ‫ݐ‬ሻ ∈ ℝ௥
× ℝାwe define a set ൛ℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻൟ ⊂ ℝ by the condition:
∀ߣൣߣ ∈ ൛ℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻൟ ⟺ ℜሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ = 0൧.(3.3)
Definition3.4.Assume that Euclidian quantum N-model(2.1) has the Euclidian QD-quantum chaos.
For each point ሺ‫,ݔ‬ ‫ݐ‬ሻ ∈ ℝ௥
× ℝା we define a set-valued functionℜ෩ሺ‫,ݔ‬ ‫ݐ‬ሻ: ℝ௥
× ℝା → 2ℝ
by the condition:
ℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = ൛ℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻൟ(3.4)
We will say thattheset -valued functionℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻis a quasi-determinedchaotic solution(QD-chaotic
solution)of the quantum N-model.

Pic.3.1.Evolution of
at point‫ݔ‬
Pic.3.2.The spatial structure of
instant
Theorem3.1.Assume that݂ሺ‫,ݔ‬ ‫ݐ‬ሻ
such that ‫ݎ‬ ∈ Գ, ߜ௝ ∈ ℝା, ݆ = 1, …
QD-chaotic solutions.
Definition3.5.For each point ሺ‫,ݔ‬
(i) ‫ݑ‬ାሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = limsupఎ→଴
(ii) ‫ݑ‬ିሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = liminfఎ→଴۳
(iii) ‫ݑ‬௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = ‫ݑ‬ାሺ‫,ݔ‬ ‫,ݐ‬ ߝ,
Definition3.7.
(i) Function ‫ݑ‬ାሺ‫,ݔ‬ ‫,ݐ‬ ߝሻis called
Evolution of QD-chaotic solutionℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻin time‫ݐ‬ ∈ ሾ0,10ሿ
‫ݔ‬ = 3. ‫ݐ‬ ∈ ሾ0,10ሿ, ߝ = −10ିଶ
, ߪ = 10ଷ
, ‫݌‬ = 1.1.
spatial structure ofQD-chaoticsolutionℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ at
instant ‫ݐ‬ = 3, ߝ = −10ିଶ
, ߪ = 10ଷ
, ‫݌‬ = 1.1.
ሺ ሻ = ߪ sinሺ‫݌‬ ∙ ‫ݔ‬ሻThen for all values of parameters‫,ݎ‬ ߝ, ߪ
… , ‫,ݎ‬ ߝ ∈ ሾ−1,1ሿ, ‫݌‬ ∈ ℝ௥
, ߪ ≠ 0, quantum N-model (2.1) has the
ሺ ‫ݐ‬ሻ ∈ ℝ௥
× ℝା we define the functions such that:
଴۳ሾ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻሿ,
۳ሾ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻሿ,
߱ሻ − ‫ݑ‬ିሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻ.
is calledupper boundof the QD-quantum chaosat point ሺ‫,ݔ‬ ‫ݐ‬ሻ
30
ሿ
ߪ, ߜ௝, ݆ = 1, … , ‫ݎ‬
model (2.1) has the
ሺ ሻ.

(ii) Function ‫ݑ‬ିሺ‫,ݔ‬ ‫,ݐ‬ ߝሻis called lower bound of the
(iii) Function ‫ݑ‬௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻis called
Definition3.8. Assume now that
limsup௧→∞‫ݑ‬௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = ‫ݑ‬௪ሺ‫,ݔ‬ ߝሻ ൏
Then we will say thatEuclidian quantum N
finitewidthat point‫ݔ‬ ∈ ℝ௥
.
Definition3.9.Assume now that
limsup௧→∞‫ݑ‬௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = ‫ݑ‬௪ሺ‫,ݔ‬ ߝሻ =
Then we will say that Euclidian quantum N
width at point ‫ݔ‬ ∈ ℝ௥
.
Pic.3.3.TheQD-quantum chaos of the asymptotically infinite width at point
is called lower bound of the QD-quantum chaosat point ሺ‫,ݔ‬ ‫ݐ‬
ሻis called width oftheQD-quantum chaosat pointሺ‫,ݔ‬ ‫ݐ‬ሻ.
that
ሻ ൏ ∞.(3.5)
Then we will say thatEuclidian quantum N-model has QD-quantum chaos of the asymptotically
ሻ = ∞. (3.6)
Then we will say that Euclidian quantum N-model has QD-quantum chaos of the asymptotically
quantum chaos of the asymptotically infinite width at point ‫ݔ‬ = 3. ߝ = 0.1
10ଷ
, ‫݌‬ = 1.
31
ሺ ‫ݐ‬ሻ.
quantum chaos of the asymptotically
quantum chaos of the asymptotically infinite
1, ߜ = 10, ߪ =

32
Pic.3.4.The fine structure of the QD-quantum chaos of the asymptotically infinite width at point ‫ݔ‬ =
3, ߝ = 0, ߜ = 10, ߪ = 10ସ
, ‫݌‬ = 1, ‫ݐ‬ ∈ ሾ10ସ
, 10ସ
+ 10ିଵሿ, ߣ ∈ ሾ−0.676,0.676ሿ.
Definition3.10. For each point ሺ‫,ݔ‬ ‫ݐ‬ሻ ∈ ℝ௥
× ℝା we define the functions such that:
(i) ℜ෩ାሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = sup൛ℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻൟ,
(ii) ℜ෩ିሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = inf൛ℜ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻൟ,
(iii) ℜ෩௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ = ℜ෩ାሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ − ℜ෩ିሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ.
Theorem3.2. For each point ሺ‫,ݔ‬ ‫ݐ‬ሻ ∈ ℝ௥
× ℝାis satisfiedtheinequality
ℜ෩௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ ≤ ‫ݑ‬௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ.(3.7)
Proof. Immediately follows by Theorem2.1 and Definitions 3.5, 3.10.
Theorem3.3. (Criterion of QD-quantum chaos in Euclidian quantum N-model)
Assume that
mes൛ሺ‫,ݔ‬ ‫ݐ‬ሻ|ℜ෩௪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ > 0ൟ > 0.(3.8)
Then Euclidian quantum N-model has QD-quantum chaos.
Proof. Immediately follows bytheinequality(3.7)and Definition3.2.
4. Quasi-determined quantum chaos and physical turbulencenature.
In generally accepted at the present time hypothesiswhatphysical turbulencein the dynamical systems with
an infinite number of degrees of freedom really is, thephysical turbulence is associated with a
strangeattractors, on which the phase trajectories of dynamical system reveal the knownproperties of
stochasticity: a very high dependence on the initial conditions, whichis associated with exponential
dispersion of the initially close trajectories and bringsto their non-reproduction; everywhere the density on
the attractor almost of all thetrajectories a very fast decreaseoflocal auto-correlation function[2]-[9]
Φሺx, τሻ = 〈‫ݑ‬෤ሺ‫,ݔ‬ ‫ݐ‬ሻ‫ݑ‬෤ሺ‫,ݔ‬ ߬ + ‫ݐ‬ሻ〉,(4.1)
Here
‫ݑ‬෤ሺ‫,ݔ‬ ‫ݐ‬ሻ = ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ − 〈‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ〉, 〈݂ሺ‫ݐ‬ሻ〉 = lim்→∞〈݂ሺ‫ݐ‬ሻ〉், 〈݂ሺ‫ݐ‬ሻ〉் =
1
ܶ
න ݂
்
଴
ሺ‫ݐ‬ሻ.
In contrast with canonical numerical simulation, by using Theorem2.1 it is possible to study
non-perturbativelythe influence of thermal additive fluctuationson classical dynamics, which in the
consideredcase is described by equation (4.1).
The physicalnature of quasi-determined chaosis simple andmathematically is associated
withdiscontinuously of the trajectories of the stochastic process‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߱ሻon parameter ߟ.
In order to obtain thecharacteristics of this turbulence, which is a very similarlytolocal auto-correlation
function (3.1) we define bellowsomeappropriatefunctions.

33
Definition 4.1.The numbering functionܰሺ‫,ݐ‬ ‫ݔ‬ሻ of quantum chaos in Euclidian quantum N-modelis
defined by
ܰሺ‫,ݔ‬ ‫ݐ‬ሻ = card൛ℜ෩ሺ‫,ݔ‬ ‫ݐ‬ሻൟ.(4.2)
Here by cardሼܺሽ we denote the cardinality of a finite setܺ,i.e., the number of its elements.
Definition 4.2.Assume now that a set ൛ℜ෩ሺ‫,ݔ‬ ‫ݐ‬ሻൟis ordered be increase of its elements. We introduce the
functionℜ෩௜ሺ‫,ݔ‬ ‫ݐ‬ሻ, ݅ = 1, … , ܰሺ‫,ݔ‬ ‫ݐ‬ሻwhich value at pointሺ‫,ݔ‬ ‫ݐ‬ሻ, equals the ݅-th element ofa set ൛ℜ෩ሺ‫,ݔ‬ ‫ݐ‬ሻൟ.
Definition 3.3.The mean value function‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ ofthe chaotic solution ℜ෩ሺ‫,ݔ‬ ‫ݐ‬ሻat point ሺ‫,ݔ‬ ‫ݐ‬ሻis defined by
‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ = ൫ܰሺ‫,ݔ‬ ‫ݐ‬ሻ൯
ିଵ
∑ ℜ෩௜ሺ‫,ݔ‬ ‫ݐ‬ሻேሺ௫,௧ሻ
௜ୀଵ .(4.3)
Definition 3.4.The turbulent pulsations function ‫ݑ‬∗ሺ‫,ݔ‬ ‫ݐ‬ሻ of the chaotic solution ℜ෩ሺ‫,ݔ‬ ‫ݐ‬ሻat point ሺ‫,ݔ‬ ‫ݐ‬ሻis
defined by
‫ݑ‬∗ሺ‫,ݔ‬ ‫ݐ‬ሻ = ට൫ܰሺ‫,ݔ‬ ‫ݐ‬ሻ൯
ିଵ
∑ หℜ෩௜ሺ‫,ݔ‬ ‫ݐ‬ሻ − ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻหேሺ௫,௧ሻ
௜ୀଵ .(4.4)
Definition3.5.Thelocal auto-correlation function is definedby
Φሺ‫,ݔ‬ ߬ሻ = lim்→∞〈‫ݑ‬෤ሺ‫,ݔ‬ ߬ሻ‫ݑ‬෤ሺ‫,ݔ‬ ߬ + ‫ݐ‬ሻ〉் = lim்→∞
ଵ
்
‫׬‬ ‫ݑ‬෤
்
଴
ሺ‫,ݔ‬ ‫ݐ‬ሻ‫ݑ‬෤ሺ‫,ݔ‬ ߬ + ‫ݐ‬ሻ݀‫)5.4(,ݐ‬
‫ݑ‬෤ሺ‫,ݔ‬ ‫ݐ‬ሻ = ‫ݑ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ − ‫ݑ‬ුሺ‫ݔ‬ሻ, ‫ݑ‬ුሺ‫ݔ‬ሻ = lim்→∞
ଵ
்
‫׬‬ ‫ݑ‬
்
଴
ሺ‫,ݔ‬ ‫ݐ‬ሻ݀‫)6.4(.ݐ‬
Definition 3.5.Thenormalized local auto-correlation function is defined by
Φ୬ሺ‫,ݔ‬ ߬ሻ =
Φሺ௫,ఛሻ
Φሺ୶,଴ሻ
.(4.7)
Let us consider now 1DEuclidian quantum N-model corresponding to classical dynamics
డమ
డ௫మ ൤ߝ − ቀ1 +
డమ
డ௫మቁ
ଶ
൨ ‫ݑ‬ሺ‫,ݔ‬ ߝሻ + ߜ
డ௨ሺ௫,ఌሻ
డ௫
‫ݑ‬ሺ‫,ݔ‬ ߝሻ − ߪ sinሺ‫݌‬ ∙ ‫ݔ‬ሻ = 0,(4.8)
Corresponding Langevin equation are [34]-[35]:
߲‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
߲‫ݐ‬
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿ‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ + ߜ
߲‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
߲‫ݔ‬
‫ݑ‬ఎሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ −
−ߪ sinሺ‫ݔ݌‬ሻ = ඥߟ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ, ߜ > 0∆=
డమ
డ௫మ,(4.9)
‫ݑ‬ఎሺ‫,ݔ‬ 0, ߝሻ = 0, ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ =
డమௐሺ௫,௧ሻ
డ௫డ௧
.(4.10)
Corresponding differential master equation are

డ௧
+ ∆ሾߝ − ሺ1 + ∆ሻଶሿℜሺ‫,ݔ‬ ‫,ݐ‬
ℜሺ‫,ݔ‬ 0, ߝ, ߣሻ = −ߣ.(4.12)
Corresponding transcendental master equation
ሼୡ୭ୱሺ௣∙௫ሻିୣ୶୮ሾ௧∙ఞሺ௣ሻሿୡ୭ୱሾ௣ሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఒ∙ఋ∙
ఞమሺ௣ሻାఒమ∙ఋమ∙௣మ
߯ሺ‫݌‬ሻ = ‫݌‬ଶሾߝ − ሺ‫݌‬ଶ
− 1ሻଶሿ.(4.14)
We assume now that ߯ሺ‫݌‬ሻ = 0.Then from Eq.(4.13) for a
ሼୡ୭ୱሺ௣∙௫ሻିୡ୭ୱሾ௣ሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఒ∙ఋ∙௣
ఒమ∙ఋమ∙௣మ +
ఒ
ఙ
= 0
ሼcosሺ‫݌‬ ∙ ‫ݔ‬ሻ − cosሾ‫݌‬ሺ‫ݔ‬ − ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ
The result of calculation using transcendental
is presented by Pic.4.1 and Pic.4.2.
Pic.4.1.Evolution of QD-chaotic solution
ሺ , ߝ, ߣሻ + ߣߜ
డ௫
− ߪ sinሺ‫ݔ݌‬ሻ = 0,(4.11)
Corresponding transcendental master equation (2.29)-(2.30) are
∙௣
+
ሼୱ୧୬ሺ௣∙௫ሻିୣ୶୮ሾ௧∙ఞሺ௣ሻሿୱ୧୬ሾ௣ሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఞሺ௣ሻ
ఞమሺ௣ሻାఒమ∙ఋమ∙௣మ +
ఒ
ఙ
= 0,(4.13)
Then from Eq.(4.13) for all‫ݐ‬ ∈ ሾ0, ∞ሻ we obtain
0,or(4.14)
ሻሿሽ ∙ ߪ ∙ ߜିଵ
∙ ‫݌‬ିଵ
൅ ߣଶ
ൌ 0.(4.15)
transcendental master equation (4.15) the corresponding function
is presented by Pic.4.1 and Pic.4.2.
chaotic solutionԸ෩ሺ10ଷ
, ‫,ݐ‬ ߝሻin time‫ݐ‬ ∈ ሾ0, 10ଷሿ, ∆‫ݐ‬ ൌ 0.1, ߝ ൌ
ߪ ൌ 10ଶ
, ߜ ൌ 1, , ∆ߣ ൌ 0.01
34
master equation (4.15) the corresponding function Ը෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ
ൌ 0, ‫݌‬ ൌ 1,

Pic.4.2.The spatial structure ofQD
The result of calculation using master equation(4.
presented by Pic.4.3 and Pic.4.4
Pic.4.3.The
EuclidianquantumN-model
Pic.4.4.The development
Euclidian quantum N-model at point
Let us calculate now corresponding
of calculation using Eq.(4.7)-Eq.(4.7)
QD-chaoticsolutionԸ෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ at instant ‫ݐ‬ ൌ 10ଷ
, ߝ ൌ 0, ‫݌‬ ൌ
ߜ ൌ 1, ∆‫ݔ‬ ൌ 0.1, ∆ߣ ൌ 0.01.
calculation using master equation(4.13) the correspondingfunction
4.
Thedevelopment of temporal chaotic regime of1D
model at point ‫ݔ‬ ൌ 1, ‫ݐ‬ ∈ ሾ0, 10ଶሿ. ߝ ൌ 10ି଻
, ߪ ൌ 10ଶ
, ߜ ൌ 1,
The developmentof temporal chaotic regimeof 1D
model at point‫ݔ‬ ൌ 1, ‫ݐ‬ ∈ ሾ0, 10ଶሿ, ߝ ൌ 10ି଻
, ߪ ൌ 5 ∙ 10ହ
, ߜ ൌ 1
correspondingnormalized local auto-correlation functionΦ୬ሺ‫ݔ‬
Eq.(4.7) is presented by Pic.4.5 and Pic.4.6.
35
= 1, ߪ ൌ 10ଶ
,
unction Ը෪ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻis
, ‫݌‬ ൌ 1.
1, ‫݌‬ ൌ 1.
ሺ‫,ݔ‬ ߬ሻ.The result

Pic.4.5. Normalized local auto
‫ݐ‬ ∈
Pic.4.6.Normalized local auto
‫ݐ‬ ∈ ሾ0
Inpaper [7]the mechanism of the onset of chaos and its relationship to the characteristics of the spiral
attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg
equation(4.14). Numerical data are compared with experimental results.
డ௔ሺ௫,௧ሻ
డ௧
ൌ ݅߱ሺ‫ݔ‬ሻܽሺ‫,ݔ‬ ‫ݐ‬ሻ ൅
ଵ
ଶ
ሺ1 െ |ܽሺ
߲ܽሺ
߲‫ݔ‬
Normalized local auto-correlation function Φ୬ሺ1, ߬ሻ
∈ ሾ0,50ሿ, ߝ ൌ 10ି଻
, ߪ ൌ 10ଶ
, ߜ ൌ 1, ‫݌‬ ൌ 1.
Normalized local auto-correlation functionΦ୬ሺ1, ߬ሻ.
ሾ0,100ሿ, ߝ ൌ 10ି଻
, ߪ ൌ 5 ∙ 10ହ
, ߜ ൌ 1, ‫݌‬ ൌ 1.
the mechanism of the onset of chaos and its relationship to the characteristics of the spiral
attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg
. Numerical data are compared with experimental results.
ሺ‫,ݔ‬ ‫ݐ‬ሻ|ଶሻܽሺ‫,ݔ‬ ‫ݐ‬ሻ ൅ ݃
డమ௔ሺ௫,௧ሻ
డ௫
, (4.14)
ሺ0, ‫ݐ‬ሻ
߲‫ݔ‬
ൌ 0,
߲ܽሺ݈, ‫ݐ‬ሻ
߲‫ݔ‬
ൌ 0, ‫ݔ‬ ∈ ሾ0, ݈ሿ, ݈ ൌ 50.
36
the mechanism of the onset of chaos and its relationship to the characteristics of the spiral
attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg– Landau

Pic.4.7. Normalized local auto
However as pointed out above (see
for stochastic model
డ௔ሺ௫,௧ሻ
డ௧
ൌ ݅߱ሺ‫ݔ‬ሻܽሺ‫,ݔ‬ ‫ݐ‬ሻ ൅
ଵ
ଶ
ሺ1 െ |ܽሺ
߲ܽሺ
߲‫ݔ‬
5.The order of the phase transition
turbulent state at instant
In order to obtain the character of the phase transition
a spatially uniform to a turbulent state
Ը൫‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ൯ ൌ 0. (5.1)
Bydifferentiation the Eq.(5.1) one obtain
ௗԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
ௗఌ
ൌ
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఒ
ௗఒሺ௫
ௗఌ
FromEq.(5.2) one obtain
ௗఒሺ௫,௧,ఌሻ
ௗఌ
ൌ െ ቀ
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఌ
ቁ ∙ ቀ
డԸ൫
Let us consider now 1DEuclidian quantum N
transcendental master equation Eq.(4.13)
ߣone obtain
Normalized local auto-correlation functionΦ୬ሺ25, ߬ሻ[7].
However as pointed out above (see Remark1.1-1.4 ) such numerical simulation in factgives
ሺ‫,ݔ‬ ‫ݐ‬ሻ|ଶሻܽሺ‫,ݔ‬ ‫ݐ‬ሻ ൅ ݃
డమ௔ሺ௫,௧ሻ
డ௫
൅ √ߝ‫ݓ‬ሺ‫,ݔ‬ ‫ݐ‬ሻ, ߝ ≪ 1, (4.15)
ሺ0, ‫ݐ‬ሻ
߲‫ݔ‬
ൌ 0,
߲ܽሺ݈, ‫ݐ‬ሻ
߲‫ݔ‬
ൌ 0, ‫ݔ‬ ∈ ሾ0, ݈ሿ, ݈ ൌ 50.
phase transitionfrom a spatially uniformstate
at instant ࢚ ൎ ૙.
In order to obtain the character of the phase transition (first-order or second-order on parameter
a spatially uniform to a turbulent stateat instant ‫ݐ‬ ൎ 0one can to use the master equation () of the form
one obtain
൯ ሺ௫,௧,ఌሻ
ௗఌ
൅
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఌ
ൌ 0. (5.2)
ቀ
൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఒ
ቁ
ିଵ
.(5.3)
Euclidian quantum N-model given byEq. (4.9)-Eq. (4.10). From corresponding
transcendental master equation Eq.(4.13)by differentiation the equation Eq.(4.13)with respect to
37
givesnumerical data
state to a
on parameters ߝ, ‫)݌‬ from
one can to use the master equation () of the form
Eq. (4.10). From corresponding
with respect to variable

38
߲Ը൫‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ൯
߲ߣ

ൌ
ሼcosሺ‫݌‬ ∙ ‫ݔ‬ሻ െ expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿcosሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ ∙ ߜ ∙ ‫݌‬
߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶ
൅
ሼcosሺ‫݌‬ ∙ ‫ݔ‬ሻ െ expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿsinሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ ∙ ߣ ∙ ‫ݐ‬ ∙ ߜଶ
∙ ‫݌‬ଶ
߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶ
െ
െ
ሼcosሺ‫݌‬ ∙ ‫ݔ‬ሻ െ expሾ‫ݐ‬ ∙ ߯ሺpሻሿcosሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ ∙ 2 ∙ ߣ ∙ ߜଷ
∙ ‫݌‬ଷ
ሾ߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶሿଶ
െ
ሼsinሺ‫݌‬ ∙ ‫ݔ‬ሻ െ expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿcosሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ ∙ ߜ ∙ ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻ
߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶ
െ
െ
ሼୱ୧୬ሺ௣∙௫ሻିୣ୶୮ሾ௧∙ఞሺ௣ሻሿୱ୧୬ሾ௣ሺ௫ିఒ∙ఋ∙௧ሻሿሽଶ∙ఒ∙ఞሺ௣ሻ∙ఋమ∙௣మ
ሾఞమሺ௣ሻାఒమ∙ఋమ∙௣మሿమ ൅
ଵ
ఙ
. (5.4)
From Eq.(5.4)for a sufficiently small ‫ݐ‬ ൎ 0 one obtain
ቂ
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఒ
ቃ
௧ൎ଴
ൌ
ଵ
ఙ
.(5.5)
From master equation Eq.(4.13) one obtainby differentiation the equation Eq.(4.13)with respect to variable
ߝone obtain
߲Ըሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ
߲ߝ
ൌ
ቄെ‫ݐ‬ ∙
ௗఞሺ௣ሻ
ௗఌ
expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿcosሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿቅ ∙ ߣ ∙ ߜ ∙ ‫݌‬
߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶ
െ
ሼcosሺ‫݌‬ ∙ ‫ݔ‬ሻ െ expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿcosሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ ∙ 2 ∙
ௗఞሺ௣ሻ
ௗఌ
∙ ߣ ∙ ߜ ∙ ‫݌‬
ሾ߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶሿଶ
൅
൅
ቄെ‫ݐ‬ ∙
ௗఞሺ௣ሻ
ௗఌ
expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿsinሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿቅ ∙ ߯ሺ‫݌‬ሻ
߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶ
൅
ሼsinሺ‫݌‬ ∙ ‫ݔ‬ሻ െ expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿsinሾ‫݌‬ሺ‫ݔ‬ െ ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ ∙
ௗఞሺ௣ሻ
ௗఌ
߯ଶሺ‫݌‬ሻ ൅ ߣଶ ∙ ߜଶ ∙ ‫݌‬ଶ
െ
െ
ሼୱ୧୬ሺ௣∙௫ሻିୣ୶୮ሾ௧∙ఞሺ௣ሻሿୱ୧୬ሾ௣ሺ௫ିఒ∙ఋ∙௧ሻሿሽଶ∙ఞమሺ௣ሻ∙
೏ഖሺ೛ሻ
೏ഄ
ሾఞమሺ௣ሻାఒమ∙ఋమ∙௣మሿమ (5.6)
From Eq.(5.6)for a sufficiently small ‫ݐ‬ ൎ 0one obtain
ቂ
డԸ൫௫,௧,ఌ,ఒሺ௫,௧,ఌሻ൯
డఌ
ቃ
௧ൎ଴
ൌ െ ቀ‫ݐ‬ ∙
ௗఞሺ௣ሻ
ௗఌ
ቁ
ఞሺ௣ሻ∙௣∙ୱ୧୬ሺ௣∙௫ሻ
ఞమሺ௣ሻାఒమ∙ఋమ∙௣మ ൌ ‫݌ݐ‬ଶሺ‫݌‬ଶ
െ 1ሻଶ ఞሺ௣ሻୱ୧୬ሺ௣∙௫ሻ
ఞమሺ௣ሻାఒమ∙ఋమ∙௣మ,(5.7)
Therefore from Eq.(5.3),Eq. (5.5) and Eq.(5.7) one obtain

39
ሾℑሺߝ, ‫,ݔ‬ ‫ݐ‬ሻሿ௧ൎ଴ ൌ ቂ
ଵ
௧
ௗఌ
ቃ
௧ൎ଴
ൎ ߪ‫݌‬ଶሺ‫݌‬ଶ
െ 1ሻଶ ୱ୧୬ሺ௣∙௫ሻ
ఞሺ௣ሻ
signሺ‫݌‬ ∙ ‫ݔ‬ሻ (5.8)
In the limit ‫ݐ‬ → 0 from Eq. (5.8) one obtain
ሺߝ, ‫ݔ‬ሻ = lim௧→଴
௧ௗఌ
= ߪ‫݌‬ଶሺ‫݌‬ଶ
− 1ሻଶ ୱ୧୬ሺ௣∙௫ሻ
ఞሺ௣ሻ
signሺ‫݌‬ ∙ ‫ݔ‬ሻ,(5.9)
and where ߯ሺ‫݌‬ሻ = ‫݌‬ଶሾߝ − ሺ‫݌‬ଶ
− 1ሻଶሿ = ‫݌‬ଶ
ߩሺߝሻ, ߩሺߝሻ = ߝ − ሺ‫݌‬ଶ
− 1ሻଶ
.
FromEq. (5.9) follows that
limఘሺఌሻ→଴శ
ሺߝ, ‫ݔ‬ሻ = +∞,(5.10)
limఘሺఌሻ→଴ష
ሺߝ, ‫ݔ‬ሻ = −∞.(5.11)
FromEq. (5.10)-(5.11) follows second orderdiscontinuity of the quantity ሺߝ, ‫,ݔ‬ ‫ݐ‬ሻ at instant ‫ݐ‬ = 0.
Therefore the system causing it to make a direct transitionfrom a spatially uniformstate‫ݑ‬ఎ≈଴ሺ‫,ݔ‬ 0, ߝሻ = 0
to a turbulent statein an analogous fashion to the second-order phase transition inquasi-equilibrium
systems.
6.Chaotic regime generatedby periodical multi-modes external
perturbation.
Assume nowthat external periodical force݂ሺ‫ݔ‬ሻhas the followingmulti-modes form
݂ሺ‫ݔ‬ሻ = − ∑ ߪ௞
௠
௞ୀଵ sinሺ‫݌‬௞‫ݔ‬ሻ.(6.1)
Corresponding transcendental master equation are
ሺ‫,ݔ‬ ‫,ݐ‬ ߝ, ߣሻ = ෍ ߪ௞
ሼcosሺ‫݌‬௞ ∙ ‫ݔ‬ሻ − expሾ‫ݐ‬ ∙ ߯ሺ‫݌‬ሻሿcosሾ‫݌‬௞ሺ‫ݔ‬ − ߣ ∙ ߜ ∙ ‫ݐ‬ሻሿሽ ∙ ߣ ∙ ߜ ∙ ‫݌‬௞
߯ଶሺ‫݌‬௞ሻ + ߣଶ ∙ ߜଶ ∙ ‫݌‬௞
ଶ
௠
௞ୀଵ
+
+ ∑ ߪ௞
ሼୱ୧୬ሺ௣ೖ∙௫ሻିୣ୶୮ሾ௧∙ఞሺ௣ೖሻሿୱ୧୬ሾ௣ೖሺ௫ିఒ∙ఋ∙௧ሻሿሽ∙ఞሺ௣ೖሻ
ఞమሺ௣ೖሻାఒమ∙ఋమ∙௣ೖ
మ
௠
௞ୀଵ + ߣ = 0, ߯ሺ‫݌‬ሻ = ‫݌‬ଶሾߝ − ሺ‫݌‬ଶ
− 1ሻଶሿ.(6.2)
Let us consider the examples of QD-chaotic solutions with a periodical force:
݂ሺ‫ݔ‬ሻ = −ߪ ∑ sin ቀ
௞௫
௡
ቁ௠
௞ୀଵ .(6.3)

40
Pic.6.1.Evolution of QD-chaotic solution෩ሺ10ଷ
, ‫,ݐ‬ ߝሻin time‫ݐ‬ ∈ ሾ7 ∙ 10ଷ
, 10ସሿ,
∆‫ݐ‬ = 0.1, ݉ = 1, ݊ = 100, ߝ = −1, ‫݌‬ = 1, ߪ = 10ଶ
, ߜ = 1, ∆ߣ = 0.01.
Pic.6.2.The spatial structure ofQD-chaoticsolution෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ at instant ‫ݐ‬ = 10ଷ
,
‫ݔ‬ ∈ ሾ1.4 ∙ 10ଷ
, 2.5 ∙ 10ଷሿ, ݉ = 1, ݊ = 100, ߝ = −1, ‫݌‬ = 1, ߪ = 10ଶ
, ߜ = 1, ∆‫ݔ‬ = 0.1, ∆ߣ = 0.01.
Pic.6.2.The spatial structure ofQD-chaoticsolution෩ሺ‫,ݔ‬ ‫,ݐ‬ ߝሻ at instant ‫ݐ‬ = 5 ∙ 10ଷ
,
‫ݔ‬ ∈ ሾ1.4 ∙ 10ଷ
, 2.5 ∙ 10ଷሿ, ݉ = 1, ݊ = 100, ߝ = −1, ‫݌‬ = 1, ߪ = 10ଶ
, ߜ = 1,
∆‫ݔ‬ = 0.1, ∆ߣ = 0.01.
7.Conclusion
A non-perturbative analytical approach to the studying of problemof quantum chaos in dynamical systems
withinfinite number of degrees of freedom isproposed and developed successfully.It is shown that the
additive thermal noise destabilizes dramatically the ground state of the system thus causing it to make a
direct transition from a spatially uniform to a turbulent state.

41
8.Acknowledgments
A reviewer provided important clarifications.
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