jmathphys95

8/8/2019 JMathPhys95

1/18

Variational representation of the projection dynamicsand random motion of highly dissipative systems

V. V. GafiychukInstitute for Applied Problems of Mechanics and Mathematics,

Ukrainian Academy of Sciences, 3b, Naukova strz Lviv, 290601, UkraineI. A. LubashevskiiInstitute of General Physics, Academy of Science of Russian Federation,Vavilovo St,: 46~92, Moscow 117333, Russia

(Received 21 July 1994; accepted for publication 4 May 1995)

We propose the method of investigation of highly dissipative systems, which arebased on the approximation of the attractor by some manifold. The projectiondynamic equations for the general form of such manifold of the dissipative dynamicsystem are obtained. Variational principles for the projection dynamics are consid-ered. On the basis of the projection dynamic equations we investigate the influenceof the random forces on the behaviour of the system. 0 I995 American Institute ofPhysics.

I. INTRODUCTION

Evolution of dissipative distributed systems is conventionally described in terms of the fol-lowing nonlinear equations (see, e.g., Refs. 1 and 2):

or in the vector form

g=F{$,A}.

(l.la)

Here i= 1,2,3 , . . . , N where N is a given integer number, { Gli( ,t)} are certain fields specifying astate of this system and are regarded as real functions of the time f and the spatial coordinatesr, {Fi} are the components of a nonlinear evolution operatorF which depends on both the fields{ +i} and the external parametersA = (A t ,...,AM) (M is also an integer number). Due to dissipa-tion, the system tends to a certain state in the space 1I of the functions {pi} as t-w. This stateis conventionally treated as a certain set CX* called the attractor of dissipative system. Thereforeone of the main problems in the description of dissipative systems is analysis of the attractorgeometry and the system motion in the vicinity of the corresponding attractor. In the following weshall confine ourselves to this problem.

For certain systems there are a large number of experimental data and results obtained bynumerical modeling that allow one to approximately imagine the general form of the attractorsbeforehand. In more exact terms, these results show the general form of the fields { fii( r,t)} beingthe asymptotic solution of equations (l.la) as t-+m and, thereby, enable one to construct somemanifold n in the space I that characterizes such solutions. Therefore, for this system we canspecify its attractor by paths in the space T that go in a small neighborhood of the manifoldR. The given manifold may be described symbolically in terms of

~={$=4(w)l, (1.2)

0022-24a8/95/36( 10)15735/18/$6.00J. Math. Phys. 36 (lo), October 1995 0 1995 American Institute of Physi cs 5735

Copyright 2001. All Rights Reserved.


2/18

5736 V. V. Gafiychuk and 1. A. Lubashevskii: Variational representation of the projection dynami cs

where +=(a I,..., @N)T. Here @i(r,ut ,..., uP,ul(r ) ,...,uJr )) are certain functions of the spa-tial coordinates r, the collection of real variables (u , ,..., up) and real functions ut(r ) ,..., uq(r );o are the generalized coordinates of the manifold a.

The basic idea of the approach to be developed in the present paper is to reduce the system ofequations (l.la) to some evolution equations that contain solely the variables (U t ,...,u,) and may

be ui(r ),..., u4( r ), whose time dependence characterizes the motion of the system along theattractor. For some systems solving such evolution equations can be found to be more simple thansolving the system of equations (l.la). In describing the evolution of a quasiconservative systema similar idea has already been considered (see, e.g., Ref. 3).

The plan of the article is as follows. Section II presents the main idea of the method proposedin this article and is called the projection dynamics. In Sec. III we develop a regular procedure thatallows one to obtain the equations, governing the motion of the parameters characterizing approxi-mate forms of the attractors, to a given accuracy. Section IV is devoted to various variationalrepresentations of the projection dynamics equations. In Sec. V we also consider the motion of thesystem under random forces in terms of the projection dynamics.

Although in Sets. III-V we use the formal operator language, we omit mathematical rigor andobtain a result at physical rigorous level.

II. ESSENTIALS

In the following we shall consider high dissipative systems. The characteristic feature of suchsystems is that a high dissipative system for a small time reaches a small nei ghborhood of itsattractor fix and then slowly goes along it. According to the assumption adopted in Sec. I theattractor O* is localized in the vicinity of the manifold CI. In other words, the motion of thedissipative system along the attractor O*, i.e., the asymptotic solution of equation (l.lb) may berepresented as

(2.1)where A(t)=(Ar(r,t),...,AN(r,t)) r is practically constant on temporal scales during which thesystem travels along the attractor CI*. Thus on such temporal scales

where c$: is the Frechet derivative of the operator 4.For a high dissipative system the evolution operator F practically involves two different

terms. The first one describes the fast motion towards the attractor R* and determines its geom-etry or the geometry of the manifold CI whose small neighborhood contains the attractor. The

second one causes the slow motion of the system along the attractor a*. Therefore if we couldcalculate the values of the functionals {Fi} at the attractor and project the obtained vector onto theattractor then we would be able to represent the system dynamics in terms of its local motion inthe plane tangent to the manifold a. Thereby we would be able to reduce the system of equations(l.lb) to evolution equation for the generalized coordinates w. In fact the rigorous attractorposition in the space q is unknown, and we may solely suppose that the value A is small and cancalculate the functionals {Fi} at the manifold fl only. However expressions (2.1), (2.2) enable usto linearize equat ions (1. lb) with respect to A in the vicinity of the manifold a. In this way weobtain

(2.3)

J. Math. Phys., Vol. 36, No. 10, October 1995



3/18

V. V. Gafiychuk and I. A. Lubashevskii: Variational representation of the projection dynamics 5737

Here F&l Jr=+ is the Frechet derivative of the operatorF with respect to $ at #= 4. We assumethat there exists the operator G inverse to the Frechet derivative Fb at every point 4 of themanifold n defined as G(J&F;= - 1. In the following this operator will be called the Greenoperator and its kernel will be called the Green function. The operator G allows us to rewriteequation (2.3) as

G & -FIti=+ 1-A. (2.4)In formula (2.1) the vector A has not been uniquely determined because in the general form A canbe practically represented as the sum

A=An+r, (2.5)

where A,, is the vector normal to the manifold fl at the point 4 under consideration and T is vectorlying in the plane tangent to the manifold fl at the same point 4. Let us demand that the vector

Al,=+ be normal to the manifold, i.e. T=O. Since the collection of the vectors { $:6w}, where6w is an arbitrary whatever small increment of the generalized coordinates o, gives the planetangent to the manifold fl at the point $= 4(w) this requirement is equivalent to the followingcondition

(&&lA)=O. (2.6)

Here the symbol (.**I***) denotes the scalar product in the space ?, i.e.,

Substituting (2.4) into (2.6) we get

I)o.

0.7)

It should be pointed out that expression (2.8) contains solely variables defined at the manifolda, i.e., it depends on w and dwldt only. Thereby the set of equations (2.8) associated withpossible different values of SW forms the desired evolution equations of the point 4 on themanifold R. Condition (2.6) or, what is the same, equation (2.8) makes up the evolution equationsfor the projection (i.e., for the point 4) of the dissipative system onto the manifold a. We notethat the latter has motivated the title of the present paper.

In particuIar, when the manifold n is of finite dimension, i.e., w= (ut ,. ..,u,) from (2.8) wecan obtain p ordinary differential equations which give (ti i , . . . tip) as function of u t , . . . , up and inthese terms the attractor is described by the time dependencies u i( t),. . .,u,(t). If the manifoldfi is of infinite dimension, i.e., the general ized coordinates contain certain functionsv,(r,t),..., u,(r,t) from (2.8) we get a set of partial differential equations.

It should be noted that expression (2.8) can be rewritten in the form

4-P _ 4-j--&h Sh (~-FIGI~-F)~~=~=O, (2.9)

where D and w are treated as pairwise independent variables, and a,-, I&J is the functionalderivative with respect to & which acts solely on the left term in brackets (2.9). When the Greenoperator G and consequently the Frechet derivative are Hermitian operators in equation (2.9) the

J. Math. Phys., Vol. 36, No. 10, October 1995Copyright 2001. All Rights Reserved.


4/18

5738 V. V. Gafiychuk and I. A. Lubashevskii: Variational representation of the projection dynamics

derivative SC.-, S&J may be replaced by the conventional functional derivative S/S&J with respectto &. Thereby in this case equation (2.9) amounts to the extreme condition of the functional

D=++i-FIGI&-F)I,++ (2.10)

with respect to the variable ci,. Thus for dissipative systems where the Frechet derivative of theirevolution operators is Hermitian one may make use of the generalized variational Gauss principlewhose kernel coincides with the corresponding Green function.

It should be noted that from the general point of view the finite dimensional manifold R canbe treated as an approximate manifold and the reduction of equati ons (1 . a),( 1. b) to the approxi-mate equations (2.8) is actually a version of the nonlinear Gale&in methods (see Refs. 4-6 andreference therein). Nevertheless, there is discrepancy between the procedure proposed in thepresent article and the methods based on the concept of approximate inertial manifold in that formas it has already been developed. This issue is discussed in detail in Sec. III.

Concluding the given section of the article we note that the procedure developed above allowsus to construct a regular method for obtaining the projection evolution equations to a givenaccuracy. This method is based on expanding the evolution operator F into a power series ofA, and sequential iteration of equation (l.lb). Besides, in some case it is possible to simplifyequation (1.1 b) by replacement of the Green operator by an operator being more simple in struc-ture. In particular, the latter can match the part of the evolution operator that causes the fast motionof the system towards the attractor R* rather than the whole evolution operator. The next sectionof the paper is devoted to these problems.

III. PERTURBATION TECHNIQUE

Let us analyze the motion of a dissipative system where the evolution operatorF=(F , ,...,FN)T involves two parts. The first one (F,) determines the fast motion towards a

manifold R and the second one ( EFJ gives rise to the slow motion along the manifold R. Inmathematical terms the motion of this system is described by the equation

g=Fo{$}+EF&~ (3.1)

where E is a small parameter and the operator Fo{ #} becomes zero at the manifold Q, i.e.,

FdJI)I~=~=O (3.2)

for any point I,& 4 of the manifold 0. The manifold !J is supposed to be specified by the

expression (1.2) in the space 9.Following Sec. II we represent the solution of equation (3.1) as the sume(t) = +( w(t))+ A(t) [see (2.1)], where the first term on the right-hand side describes the motionalong the manifold a. The second one is the small deviation of the system from the manifoldR. This expression enables us to expand the operators Fo{ Ijl), and FP{ Icl) into the Taylor series ofA:

and

J. Math. Phys., Vol . 36, No. 10, October 1995

(3.4)



5/18


Here the term F()(c$J)(A,A ,...,A) denoted also as F(){c$IA} ths e n-th order differential opera-tor which is a homogeneous operator of degree n with respect to A. It should be pointed out thatthe term F( 4) is a linear operator with respect to A and conventionally represented as

F()(+)A=L(+)A, (3.5)

where L( 4) is the Frechet derivative of the operator F{ $} at the point += 4. The term F()X ( 4) (A, A) is a symmetrical bilinear operator wi th respect to A.

The Frechet derivative La{ 4} = dFo ld+I += + of the operator Fo{ +}, given at the manifolda, plays an important role in the perturbation technique. We assume that for any point 4 E fi theeigenvectors { fii( 4)) of the Frechet derivative LO{ c$} form a complete system of linearly inde-pendent vectors in the space 1Ir. In other words, any vector of the space T, in particular, A can beexpended relative to the basis { &,( 4));

A(f)=? -4(+)h(4). (3.6)

As follows from expansion (3.6) in the case under consideration the motion of the system canbe represented as the motion of the point C$ the shadow) on the manifold R and the time variationof the coefficients .A,, . At the present stage the motion of the shadow is not uniquely determinedbecause the motion along the manifold fi is independently described by the motion of the pointC$ nd the time variations of the coefficients h$,(t),+ Since the evolution operator F,,{t,b} has noeffect on the system motion along the manifold 0 in the general case the coefficientsJA,(t) I h =c may increase beyond all bounds. In order to analyze the system dynamics in terms ofthe shadow motion, the distance between the point $, showing the real system position in thespace q, and the point #I must be small. Therefore , it is reasonable to specify the shadow motionin such way that at every instant of time all the coefficients fm,(t) IhCO be equal to zero. Thisprocedure is equivalent to eliminating singular terms in the evolution equation for the coefficientsJ& , obtained by perturbation technique. Keeping the latter in mind we note that the proceduredeveloped in the present work, the perturbation technique for ordinary differential equations,7 theBogolubov-Metropolskii method of averages* as well as the perturbation technique for nonlinearwaves proposed in Ref. 9 are similar in eliminating singular terms.

We now proceed to formal construction of the perturbation technique. Substituting (3.3),(3.4)and (3.6) into (3.1) we get

(3.7)

Here

d&do and deA ldw are the corresponding Frechet derivatives. Equation (3.7) is completed bythe conditions

&lh=O=O. (3.9)

In order to find the explicit expansion of the vector P relative to the basis { $*( 4)) we introducethe linear operator G~(c$} defined by the formula

G~Wd4~-E~)= -E, (3.10)

J. Math. Phys., Vol. 36, No. 10, October 1995Copyright 2001. All Rights Reserved.


6/18


where the regularization parameter 6-+ + 0 and E is the unit operator. The operator G6 possessesthe same set of the eigenvectors { Gx} and its eigenvalues are { - l/(X - s>} respectively. Thesecond type of operators that we need are the projection operator

.9= lim G$ (3.11)&-++o

and the operator, called the Green operator

1F= 61rno Gg- y.[ 1 (3.12)

The operators Yand *Tenable us to divide equati on (3.7) in two parts governing the motion of thesystem towards and along the manifold s2.

5; A=-A+FP(A),

d+-&- kil+y A&$ W=PP. (3.14)

Here the prime on the sums indicates that the terms matching the zero eigenvalue are omitted andA = X .A$ &( 4). We consider such systems called the highly dissipative systems for which thetransient term in equation (3.13) can be treated as a small perturbation. Physically, this meansthat from the view-point of the fast motion towards the manifold fi the system motion along itmay be regarded as quasistationary. In this case equation (3.13) describing evolution of the vectorA can be reduced to the explicit relationship determining the vector A in terms of projectiondynamics, viz.

x[ 3% -g(rP)]+sj ti -g-JP)]+....

In mathematical terms the highly dissipative systems are characterized by the convergence of thelatter series. In order to verify whether a given system belongs to this class one should analyze indetail the spectrum of the Frechet operator. In particular, if there is a finite gap separating the zeroeigenvalue (X =O> from other ones (Re X-CO for Cl) then for an arbitrary 1/1he vector %$4will be

finite. In this case due to the system motion along the manifold fl being caused by the perturbationoperator eFp the time derivative dAldt according to (3.15) may be estimated as (dAldt)- EAand, thus the transient term in (3.13) is small in comparison with the first one on the right-handside. Therefore such a system can be classified as a highly dissipative system.

Finding the derivatives of the operators .Y andF along the manifold as functions of thisoperator determined on the manifold, and taking into account expression (3.19, conditionSF@,, 0 (X # 0) and identity .EF= 0 we get the projection dynamics equation (3.14) in the form

where




7/18


F=,zYA+P (3.17)

is the generalized evolution operator for the system motion along the manifold R. The explicitexpressions for P and A as functionals of q5can be obtained by successive iteration of equations(3.8), (3.15), (3.16), and at lower order in E from (3.8) and (3.15) we get

@I)= P(l)= E&{(6),

A(l)=~p(l)s~~F (4)P

(3.18)

(3.19)

and, thus the shadow motion equation at first order in E takes the form

(3.20)

To the next order in E from (3.8) and (3.16) we find

(3.2 1)

and

(3.22)

Substituting (3.19) and (3.20) into (3.21) and (3.22) we obtain the shadow motion equation tosecond order in E

(3.23)

where

s2=~Fp{~}+~2F~)(~)(~~p{~}~+~2~F~2)(~)(~~p{~};Fp{+))+ ~2F62WWFp~~h FP{41).

In obtaining equation (3.23) we have taken into account the relation

(3.24)

dLo;dWYA=Fg)(c+$YA) '

and substituted (3.20) into the latter equality. It should be noted that the term of second order intz can play an essential part when the operator Fp is degenerate at the manifold 0 and the firstorder approximation is inadequate to give the right results.

Equation (3.16) is actually of the vector form whose components are specified by the coordi-nate system, given initially in the space q. Therefore the formal dimension of equation (3.16)coincides with the dimension of the space q. However, in actual truth, the amount of the inde-pendent equations as well as the independent variables is determined by the dimension of themanifold R


8/18


The plane T+ tangent to the manifold R at the point +!r= 4(w) can be specified by the set ofvectors {(d&do). SW}, where the vector 6w runs all the possible values. Let the collection ofvectors {e,} form a basis in the space {SW}, and, thus, the vector system {(d+ldo)e,} be a basisof the plane T+ in the space !P.

Since equation (3.16) contains solely the vectors lying in the plane T+ it can be equivalently

represented as the system of equations

(3.25)

The convenience of the given equation system is that it contains the complete collection ofindependent equations explicitly describing the system motion in terms of time variations of theparameters w only.

As has been mentioned in Sec. II the manifold R may be regarded as an approximate inertialmanifold, at least,because the attractor CI* is located in its small neighborhood. Nevertheless, theparticular technique with which we reduce the full problem (1.1) to the approximate equations(3.16),(3.25) of the shadow motion differs essentially from the classical methods widely used inobtaining the inertial forms of dissipative system dynamics.

Indeed, (see, e.g., Refs. 6 and 11 and references therein) in the context of the approximateinertial governing equations of the dissipative manifold theory system are also represented in theform (3.1) where, however, the operator Fo{ $} = -A $ is a linear one. The operator i is assumedto be dissipative, and its eigenvectors {qi} meet the conditions imposed on their eigenvalues{Xi}:O


9/18


dDldv,=O (4.2a)

directly lead to equations (3.20). In the symbolical form the system of equations (4.2) may berewritten as

dDla$=O. (4.2b)

In other worlds, the real motion of the system shadow on the manifold 0 matches such a rate& at which functional (4.1) attains the minimum with respect to 4 at given point 4. Following theconventionally used nomenclature we call this property of functional (4.1) the generalized varia-tional Gauss principle.2-5

It should be noted that the specific form of the functional D{$J,~} is of little importancebecause solely the equations governing the system motion make physical sense. Therefore, allfunctionals that lead to the same equations can be regarded as identical. In particular, if infollowing Sec. II we make use of the left-hand side functional derivative a((-) /S(i, with respect to4, then equations (3.20) result from the condition

+,-D*{4,&=0,84

(4.3)

where

D*{~,~}=(~l~-~cb>~F~(~)). (4.4)By virtue of (3.2) in the latter expression EFJ Cp)may be reduced by the total evolution operatorF= Fo( 4) + EFJ 4). If we consider the system motion along the manifold fi approximating themanifold R rather that along the manifold 0 itself this replacement will be also justified and theequations for the shadow motion on the manifold d can be obtained from the correspondingextremum conditions for the functional

D*{&$}=($~~c$-F). (4.5)

Indeed, when the function 4 differs by a small value A+ from (b we may setF{&= ~Fp{4+W~ and, thus 9F{ +} = &Fp{ $} because of 9L0 = 0.

Recalling the definition of the projection operator 9 [see formula (3.1 l)] it is reasonable toreplace 9 in expression (4.5) by the whole Green operator G6 for S # 0 when we deal with acertain manifold d approximating fi rather than the manifold n itself. In this way we establishthe correspondence between the results obtained in Sec. II and the present section.

When the Frechet derivative Lo(+) is Hermitian and, thereby, the projection operator isHermitian too, in expression (4.5) we may replace P by the unit operator. Indeed, in this case

Clearly, for the given functional D*{qb,c$} equation (4.3) is equivalent to the equation (4.2b)where D{ q5,4} = DC{ I,&,cl) is of the form

(4.6)

In literature the latter functional is known as the minimal constraint functional of the conventionalvariational Gauss principle.12-I5

It should be pointed out that for non-Hermitian systems functional (4.6) also may be appliedin cases where the manifold R closely approximates the geometry of the attractor !J*.14*15 The

J. Math. Phys., Vol. 36, No. IO, October 1995Copyright 2001. All Rights Reserved.


10/18


latter follows from the fact that at such a manifold s1: Falo= 0 and the operatorF, mustpractically lie in the plane T, tangent to the manifold a. This allows us to set $Fp= Fp and thenreplace Fp by F.

Summarizing the results obtained in this section we note that the possibility of using suffi-ciently simple variational principles, for example, the conventional Gauss principle, depends on

the specific properties of a dissipative system (the attractor geometry or geometry of the manifoldR, the Frechet derivative, etc). Besides, such simple variational principles practically allow one toobtain the evolution equations being valid solely at lower order in the small parameter E. Whenthe evolution operator eFp is degenerate at the manifold it is necessary to use directly evolutionequations (3.25). In dealing with a nondegenerate Hermitian dissipative system, i.e., such a systemwhere the Frechet derivative Lo{ $} is Hermitian and the evolution operator eFp is nondegenerateat the manifold a, it is convenient to use the functional (4.6) rather than equations (3.20) directlybecause the former is sufficiently simple in structure and contains $ and F{ @} only.

Given some additional restrictions, the conventional Gauss principle can be represented inother equivalent forms. By way of example, let us consider Hermitian fields with weak interaction.By definition, for such fields { ei} the evolution operator F,{ Icl) is of the form { Foi{ s,bi}} where theith component Foi{ qi} depends on the function rcrionly and the derivative dFoi ld *i is Hermitian.In this case the manifold Cl can be specified as { +;= ~i( Wi)} and the total set of its generalizedcoordinates w = { wi} involves N different collections of generalized coordinates wi , each beenassociated with the corresponding function only. Under these conditions equation (4.2a) can berewritten as

(4.7)

where F;( $)= Fio{ @i} + EF~{ (cl)= eFp{ 1/1). fwe introduce the functionals ci{JI) for which

(4.8)

then equation (4.7) may be represented in terms of

(4.9)

We note that the motion equations similar to (4.9) has been formulated in Ref. 16 for investigationof reaction-diffusion systems.

B. The Whitham type variational principle for high dissipative thermodynamical

systemsIn essence, evolution of high dissipative systems being near thermodynamic equilibrium is

conventional ly described in terms of the projection dynamics. Indeed, typically in such systemsthere are two types of state variables that are distingui shed by temporal scales of time variations.The first type variable conventional ly called fast ones, characterize attaining a local quasi equi-librium state by the system within a short time TV.The second type variables, called slow ones,describe evolution of the system as a whole , for a long time rs% rf. Thus on time scales of orderTVmagnitudes of the fast variables can be treated as functions of the slow variables and, therebyevolution of the system in time can be described by the motion of the slow variables.

In view of the fact that equations governing the system motion are given in a microscopicform which contains both the slow and fast variables the problem is to obtain complete motion

equations for the slow variables.




11/18


In terms of the microscopic variables ~={~l(u,t),~2(v,t)~~~} evolution of a high dissipativethermodynamical system can be described by the equationsI

S.% SHs(i;+ sll;=O (4.10)

where the functional H{ fi} is the free energy of the system, and the positive-definite quadraticform 3

is the Rayleigh dissipative functional operator. We note that the form of equat ion (4.10) is invari-ant under one-to-one transformations of variables.

Let us convert functional (4.11) to diagonal form

-WI=; T U4A2 (4.12)

by transition from the variables { ei} to new variables {qr}. Here {A,} are the eigenvalues of thelinear operator A and the sum runs over all the variables {qr}.

The set of eigenvalues {A,} is assumed to involve two types of eigenvalues. The first typeeigenvalues { Xrf} corresponding to the fast variables {q!} are of order one. The second typeeigenval ues {X,,} matching the slow variables (4:) contains the cofactor l/e, i.e.,X,,=X,*,( l/e), where A,*,- 1 and E= rf/rS@ 1. In terms of {qr} equations (4.10) take the form

1 SH

if=-;n;;a

In this way we have reduced equations (4.10) to the equations of form (l.la) or (l.lb).Following the developed procedure we rewrite equations (4.13a), (4.13b) as

3 qf

I/ If 4s= Fo-!cI} + ~Fpbzh

SH

Fo{d =-P s4p

0

0

CM = -x,--. SH &I,

(4.13a)

(4.13b)

(4.14)

(4.15)

(4.16)

and

J. Math. Phys., Vol. 36, No. IO, October 1995



12/18


xf=

qf= 4: ; qs= 4: 7

. . . 0 . . . 0

Arf &,= A,*,0 .. . 0 . . .

During the short time rf the system reaches a small neighborhood of the manifoldfi = (4 = g( w)} determined by the equation:

Fo{g}=O. (4.17)

By virtue of (4.13a), (4.13b) the solution of (4.17) can be represented in terms of

qfdqs) (4.18)

and, thereby, qs may be regarded as the generalized coordinates o of the manifold a, i.e.,o=q,. In addition, in the vicinity of the manifold fl the free energy H(q) takes the form

H{q)=Hdq,>+ ~qf-g(q8)lA(q,)lqf-g(q,)). (4.19)

Here H~{q,}=H{q, ,gf=g(q,)}, the second term is a quadratic form positive-definite with re-gard to the vector qf-g(q,) and A(q,) is a linear operator acting in the space {q,}. The positive-definiteness of the given quadratic form follows from the condition that the all eigenvalues of theFrechet derivative dFo ldq Io must be negative except for the zero eigenvalue associated with theplane, tangent to the manifold fi at the point (qs ,qf=g(q,)).

As follows from (4.15) and (4.19) the Frechet derivative of the evolution operator F. at thepoint (q8 ,qf=g(q,)) of the manifold 0 is at the form

dFo -+A(q,)-=+W$

s4

0 0

From (3.11)-(3.12) for the given Frechet derivative we obtain

(4.20)

(4.2 1)

and

yz l~A-(OU.)~f ,II, (4.22)

where E is the unit operator. According to (3.18) at lower order in E the vector PC)= ~F,lo anddue to (4.16), (4.19) PPp=O. Thus by virtue of (3.21) to the second order in E f12)=eFploalso. Substituting (4.16), (4.18), (4.21) into (3.23) to the second order in E we get




13/18


dg . dg A-Edq qs dqs *s &?ss =E

4s$H

-x*ssq,

The system of equations (4.23) can be rewritten as follows

x,4,+ H=o&s

or in the equivalent form

;Mdh,llJ+ g =o.s s

(4.23)

(4.24a)

(4.24b)

The first term in equation (4.24b) can be replaced by the total dissipative functional given at themanifold Cl, i.e., by the sum

q:(Wdq,)%hfl$ 4ss

Indeed, the first term in the latter expression is small by virtue of EG 1 and, thus, can be ignored.By this way we reduce (4.24b) to the equation of form (4.10) where

(4.25)

(4.26)

are the dissipative functional and the free energy given at the manifold Ck. Due to the form ofequation (4.10) being invariant under one-to-one transformations of the generalized coordinates, inthe general case the equations governing the shadow motion along the manifold R is also of theform

(4.27)

The given equation is actually the content of the Whitham type variational principle for highdissipative systems. We note that the procedure of transition to the slow variables for the distrib-uted Hamiltonian type systems has been developed first by Whitham.8-20 The latter causes thename of the variational principle under consideration. It should pointed out that equation (4.27)can be also represented as

AD{o,O}=O,

where

(4.29)




14/18


V. RANDOM MOTION ON THE MANIFOLD t-l

When the dimension of the manifold 0 is large enough the characteristics of the evolutionoperator Fp{ +} can cause the existence of several attractors lying on the manifold a. In this caseit is reasonable to study possible transitions between the attractors under random forces. Randomforces are inherent in every physical system owing to dissipative processes being random in nature

at microscopic levels. So in this section we shall describe the influence of random forces on thesystem motion along the manifold a. In addition we shall assume that the evolution operatorF,,{ $} is nondegenera ted at the manifold a. The latter allows us to confine our consideration tothe first order evolution equation (3.20) for projection dynamics.

Let us consider a high dissipative system { +( r,r)} evolving according to the stochastic equa-tion:

(5.1)

Here the mean properties of the random fieldf are specified by the conditions

f=o, (5.2)fi(r,t)fi,(r',l')=rii'(r,r'l~}~(r-r'), (5.3)

where the symbol (:) denotes averaging over all possible realizations, S(t t ) is the S - functionand

dr dr&(r)Iiit(r,r)+/i,(r)

is a positive-definite quadratic form. Besides the fieldf is assumed to be Gaussian, and in the

following will be treated as a small operator. The latter allows us to ignore possible random forceinduced renormalization of the regular operators.2 In this case for the random fieldf the prob-ability of a given realizationf i( r, t) is

pGf}=C exp ,

where C is such a constant that the path integral

I WMf~= 13 (5.5)

where gu} is the integral path measure. In particular, for these random field at a given instant oftime the mean value of the generating function21

(5.6)

We note that expressions (5.5) and (5.6) are equivalent representations of the Gaussian nature forthe random fieldf.

As follows from Sec. III [see formulae (3.20)] under the random fieldf the motion of thesystem along the manifold R is described by the equation

(5.7)

J. Math. Phys., Vol . 36, No. 10, October 1995



15/18


Averaging the function exp{i($@f)dt} for an arbitrary vector $ over all the possible reali zationsof the random field f we find

where .3@ is the adjoined operator to the projection operator 9 at the given point 4. Comparing(5.6) and (5.8) we obtain that the random field Pf might be treated as a conventional Gaussianrandom field if the projection operator 9 were nondegenerate. In order to make use of theproperties of Gaussian random field we replace the operator @ by the regularized projectionoperator

Pa= P+ SE (5.9)

and in the final formula we shall tend S-t + 0. The operators P and L?~possess the same collec-tion of eigenvectors { qA( 4)) and the corresponding eigenvalues are 1 for A = 0,O for A < 0 and1 + 6 for A = 0, and S for As 0, respectively.

In this way we may describe the random field w by the following distribution function

Tdt(9fl(~1)+I-1~1j@)dr

Expressing Pf from (5.7) as a function of o and ti, and substituting the obtained result in (5.10)we obtain the probability of the system going along a path w(r) on the manifold R:

p{o(t)l= C6 exp[ - j-oTdtZd 0, h)i (5.11)

Here the functional

The vector (d&do) ci, P@) belongs to the plane T+ tangent to the manifold R at the point4, and, thus, is an eigenvector of the operator PJ. So in expression (5.12) we may replace theoperator .P* by l/( 1 + S))E and setting S= + 0 we get

z(~,~}=(~-~~l)lr-*~~-~~l))and the probability of the system going along the path w(t) takes the form

(5..13)

~bW=C6=+0 exp(-; foTdf ZW$)J (5.14)

According to (5.14) the probability density of finding the system at the point 4 at time t providedat initial time to it was at the point +. is represented in terms of the path integral

(5.15)

where all the path {4(r)} on the manifold originate at the point $. and terminate at the point4. Formula (S.15) is the desired expression that characterizes the motion of the system along themanifold fl under the random fieldf.




16/18

5750 V. V. Gafiychuk and 1. A. Lubashevskii: Variational representation of the projection dynami cs

In particular expression (5.15) enables us to obtain the Fokker-Plank equation for the randommotion of the system along the manifold a. By way of example, let us consider the case where themanifold fi is of finite dimension n. In other words, its general ized coordinates consist of thecollection of real variables o = { w 1 * * *, tin}.

The vectors {el=d~ldw,;e2=a~law,;... ;e, = &$l&+} form the basis of the plane T,

tangent to the manifold fi at the point C$and for the vector 4 we get

(5.16)

Due to the vector p#) lying in the plane T+ we also can expand it relative to the basis {ei} andin this way find

9?H1)= E eiPi.i=l

(5.17)

where the coefficients

and the matrix

U=IIUij(eilej)ll.

The substitution of (5.16) and (5.17) into (5.13) yields

(5.18)

(5.19)

Z{~,O}=C (~j-Pj)T,,,l(~i,-Pil), ( (5.20)ii

where rij=(eilI-lej). To find the matrix f*=llrc/l inverse to the matrix IlrlTll we makeuse of the regularization procedure described above in this section. As follows from (5.10) andformula (5.17), (5.18) determining the vector expansion relative to the basis {ei} in the limitS-t+0 the matrix lp-ylcoincides with the correlation matrix of the random variables {pi} beingthe coefficients of the expansion Y= Z&ei.

r;,=x U~U~~,(ejlPf)(ejtIFf)ij

and by virtue of (5.8) we find the desired expression

r;,=s Ug~U~f,(f?~19TPlC?j).ii

(5.21)

In this case the Fokker-Plank equation describing the motion of the system along the manifoldn under random forces takes the form

$h=$$ r:I, * ltr&Pr) T & (PiPA. (5.22)In particular when the system under consideration is Hermitian, i.e., the Frechet derivativeL,{ 4} and, consequently, the projection operator .Y are Hermitian, and the correlation operator is

J. Math. Phys., Vol. 36, No. IO, October 1995



17/18


proportional to the unit operator: r = 2DE where D is a constant, the expressions for the coeffi-cients Pi, ri, may be simplified. Indeed, in this case, from (5.18) and (5.21) we get

and

(5.23)

where Fn is the total evolution operator given at the manifold 0,. Besides, under such conditionsthe functional Z{ $,4} [see expression (5.13)] describing the system motion in terms of randompaths on this manifold may be represented as

We note that the first term on the right-hand side of (5.25) coincides with the conventional Gaussfunctional.

ACKNOWLEDGMENTS

The research described in this publication was made possible in part by support from Ukrai-nian Committee of Science and Technology and by Grant No. Ul 1000 from the InternationalScience Foundation.

T. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcationsofvectorfields (Springer-Verlag, Berlin, 1983).

Cl. Nicolis and I. Prigogine, Exploring complexity. An introduction (Freeman, New York, 1989).D. McLaughlin and A.C. Scott, Multisoliton perturbation theory, in Solitons in Action, edited by K. Lonngren and A.Scott (Academic, New York, 1978).

4C. Folias, G.R. Sell, and ES. Titi, Exponential tracking and approximation of inertial manifolds for dissipative non-linear equations, J. Dynam. Differential Equations 1, 199 (1989).

M. Marion, Approximate Inertial manifolds for reaction - diffusion equations in High Space Dimension, J. Dynam.Differential Equations 3, 245 (1991).

6D.A. Jo nes and E.S. Titi, A remark on quasi-stationary approximate inertial manifolds for the Navier-Stokes equa-tions, SIAM J. Math. Anal. 25, 894 (1994).

VP Bogaevsky and A.Ya. Povzner, Algebraic methods in nonlinear perturbation theory (in Russian) (Nauka, Moscow,1987).

*N.N. Bogolubov and 1u.A. Mitropolsky, Asymptotic methods in theoryof nonlinear oscill ation (in Russian) (Nauka,Moscow, 1974).

9V.N. Biktashev, Diffusion of autowaves. Evolution equation slowly varying autowaves, Physica D 40, 83 (1989).LA. Lubashevskii and V.V. Gafiychuk, *Projection dynamics of highly dissipative systems, Phys. Rev. E. 50, 171(1994).

Y. Morita, H. Ninomiya, and E. Yahagida, Nonlinear perturbation of boundary values for reaction - diffusion systems:inertial manifol ds an d their applications, SIAM J. Math. Anal. 25, 1320 (1994).

B.D. Vujanovic, The practical use of Gauss principle of least constraint, J. Appl. Mech. 98, 491 (1976).13B.D. Vujanovic and SE. Jones, Variational methods in nonconservative phenomena (Academic, London, 1989).14V.V. Gafiychuk and LA. Lubashevskii,15VV Gafiychuk and I.A. Lubashevskii,

Variational principles of dissipative systems, J. Sov. Math. 67, 1 31 (1992).. . Analysis of dissipative structures based on the Gauss variational principle (in

Russian) Ukr. mat. journal No. 9., 1186 (1992) (English transl. in Ukrainian Math. J. May 1993, pp. 1185-1190, PlenumPublish. Corp.).

16V.V. Osipov, Multifunctional variational me thod for description of evolution and dynamics of dissipative structures innonequilibrium systems, Phys. Rev. E 48, 88 (1993).

L.D. Landau and E.M. Lifshiz, Statistical physics, 5 (in Russian) (Nat&a, Moscow, 1976).*G B Whitham The general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid Me ch. 22,

273 ( 1963).

J. Math. P hys., Vol. 36, No. IO, October 1995



18/18


19G.B. Whitham, To timing, variational principles, and waves, J. Fluid Mech. 44, 373 (1970).2oV.V. Gafiychuk, LA. Lubashevskii, and V.V. Osipov, Dynamicsof pattern formation in free bounda ry problem (in

Russian) (Naukova Dumka, Kiev, 1990). C.W. Gardiner, Handbook of srochasric methods, Springer series in synergetics 13 (Springer-Verlag. Berlin, 1983).

jmathphys95

Documents