STOCHASTIC MODEL OF EVOLUTION OF HOMOTOPY TYPE OF PHASE Текст научной статьи по специальности «Математика»

ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

MalkhazMumladze

Space of Dynamical System Gori University

STOCHASTIC MODEL OF EVOLUTION OF HOMOTOPY TYPE OF PHASE

DOI: 10.31618/ESSA.2782-1994.2021.1.68.17 Abstract. In this paper, we introduced the concepts of full number invariant for homotopy type and entropy of homotopy type of some topological spaces.

Based on this concept discrete random process describing evolution of the homotopy type of phase space of c closed dynamical system there is built.

In this paper we introduced also the entropy of trajectory of evolution of homotopy type of phase space of closed dynamical system, constructed random value on set this trajectories, proved that the mathematical expectation of this random value coincide to sequence of mathematical expectations in the phase spaces of the constructed random process with respect to the corresponding random values. Key words: Topological space, covering, entropy, random process.

Classification codes 37B02, 37B04, 37H05 .

Introduction

The evolution of real dynamical systems is reflected in the change in their entropy. This fact must be taken into account in the mathematical modeling of such systems, so it is important to define the entropy concepts of the mathematical objects in which the dynamic systems are represented in the models.Often modeling of a dynamical systems(ifs phase spaces) is considered to be a continuous or discrete sequence of topological spaces that describe the continuous or discrete time variability of the system. This sequence we called trajectory of changes of phase space. In the work[11] we introduced the concept of entropy of topological space and built a random process that described the evolution of the phase space to the accuracy of the entropy. In this work on the basis introducing the notion of a complete numerical invariant of the homptopy type of topological spaces and the notion of entropy of the homotopy type of a topological space, we construct a random process describing the evolution of the phase space of a closed dynamical system up to homotopy type.

1. The Number Invariant of Homotopy Type of Topological Space.

Let N the set of natural numbers andP the set of prime numbers. Consider the map^: N ^ P defined so

cp(1) = 2,cp(2)=3, <p(3) = 5, cp(4)=7.....cp(n) = p

wherepis prime number following the prime number^ (n — 1).

Let K is a finite simllicial complex [4] vertices in which are numbered by prime numbers<p(1) = 2,cp(2)=3, cp(3) = 5,cp(4)=7,..., cp(n) = p. Every simplex

a kt

, CL kaj + i,. .. ,CL ka

,.k+1

1,2, ...,k + 1< nwhich vertices are

<p(n±), pfa),... ,v(nk+1), ni<n,i = 1,2, ...,k + 1 < nof K, we correspond to product a^k+i = cp(n1) ■ (p(n2) ■...■ (p(nk+1),ni<n,i = 1,2,...,k + 1 <n. Let {Gaa+1} the family of all simplexes of Kand

{a^ka+i}the family of corresponding numbers. This family uniquely defines complex K Let the sequence

ал+Ш*- ка2

all elements of

а к

а1 + 1

< a ka-,+1 <...<

with essential ordering, i.e. a kap. Each sequence of natural numbers uniquely

Gfv

av

defines a rational number represented by continued fraction[15]. Such, every finite simplicial complex and fixed ordering of its vertices uniquely defines rational number. We will denote this number by L(K,A),A E A where A is the set of all numberings of vertices with prime numbers of simplicial complex K. In set of

numbers [L(K, A)}AeAexist a minimal element. We will Denote this element by

L(K) and call it a simplicial

number of complex K.

Finally we have built a injective mapping from set of finite simplicial complexes to set rational numbers L: № ^ Q.

Since every rational number is uniquely represented as a continued fraction of a finite sequence of natural numbers, knowing the rational number L(K) we can construct simplicial complex K.

Let new radical of number n E Nis the product Rad(n) = (p(n1)cp(n2) ■■■ v(nk+1). Consider a k dimensional Euclidean Simplex kwhich vertices are numbered by <p(n1),(p(n2),...,<p(nk+1) numbers. Every such product uniquely defines Euclidean simplex and each finite sequence of natural numbers n1,n2,n3,...,ni represents a finite Simplicial complex in which simplexes represented by numbers ni± and ni2 have common edge if Rad(nii) andRad(n^) contain common factors. The dimension of the common edge, depends on number of the common factors. Each sequence of natural numbers uniquely defines rational number represented by continued fraction. It follows, that every rational number uniquely defines finite simplicial complex . Let new vertices of complex K is numbered with prime

numbers (p(n1),(p(n2),...,(p(nl), Every simplex of complex K is corresponding to product (p(ni±) ■ V(ni2) ■,...,■ <P(nik+1),k < I, the sequence of such products uniquely defines complex K.

V

a

a

V

Such we have surjective map from set of rational numbers to set of finite simplicial complexes P:Q^ {K}. It is easy, that

corresponding homeomorphisms. Let N(0(X)) is

L(K) E P-1(K), P° L: {K} ^ {K} = id{

covering of X which contains minimal number of

elements n. The number- where k k is the number of

{K}-

Let ^compact topological space which admits a pseudo-convex open covering[11] and the O(X) minimal finite pseudo-convex open covering i.e. O(X) contains minimal number of pseudo- convex open sets. It is known that geometric realization of N(0(X)) nerve[4] is homotopy equivalent to the space X [1]. Rational number L(N(0(X))) uniquely defines simplicial complex^(O(X)), It follows that rational number L(N(0(X))) uniquely define homotopy type of compact topological spaces^ if it admits finite pseudo-convex open covering[11].

Let new^ simplicial complex with countable vertices. We number the vertices of cimplicial complex with prime numbers. If ^simplex in^ then it will represented by productp1p2 ■...■pk+1 If we order these products of primes in a natural way, we get an infinite sequence of natural numbers which defines a continued fraction representing an irrational number[15] and which uniquely defines a given simplicial complex with a countable number of vertices. on the contrary, each irrational number decomposes into a continued fraction corresponding to an infinite sequence of natural numbers and therefore defines a simplicial complex. Note that if the sequence representing a given irrational number contains one, they will be discarded, the resulting sequence is unique with respect to the given irrational number, and therefore this number uniquely determines the syimplicial complex.

Let X topological space which admits countable locally finite pseudo convex covering, and [oa] is minimal such covering i.e. if Oa, Op £ {Oa} then Oa £ Op,Op£Oa. The geometric realization of the simplicial complex N({Oa}) has the same homotopy type as the spaced [1]. From what was said above it follows that the complex N({Oa}) and the fixed ordering of its vertices uniquely corresponds to the real number r. Let an, n = 1,2,3,... suitable fractions of continue fraction of this number. Each of this suitable fractionsa„, n = 1,2,3,... uniquely defines finite simplicial complexK"^ and homotopy type of geometric realization of this complex ^J. Let H([IKanl]entropy of the homotophy type [^J]. If exist, finite or infinite limH([IKa |],then call this

n^tt n

number entropy of homotopy type of space X.

1. Entropy of Homotopy Type of Compact Topological Space and Evolution of Phase Space of Closed Dynamical Systems

In work [11] we called the entropy of the compact topological spaced, f if it admits pseudo- convex open covering, the number— wheren is minimal number of

m

elements of open pseudo- convex coverings among all finite pseudo- convex coverings off and m is the number of orbits in every n element-containing coverings at the actions by covering preserving

orbits in every n element-containing set of vertices of N(0(X)) at the actions by automorphisms of N(0(X)). This numbers are different, therefore number of the covering O(X) preserving homeomorphisms is less than number of the automorphisms of N(0(X)).

Definition 1. We call the entropy of homotopy type of compact topological space the number - where nis minimal number of elements of open pseudo-convex coverings, among all finite pseudo- convex coverings off and k is the number of orbits in every n element-containing set of vertices of N(0(X)) at the actions by automorphisms of N(0(X)) where O(X) is pseudo- convex open covering of X which contains n elements .

Denote entropy of homotopy type of compact topological spaced by Hht(X). In work[11] we denoted entropy of compact topological spaced by H(X). It is clear that

H(X) < Hth(X)

Time is the total characteristic of changes, each a physical system has its own time, when describing the evolution of the phase space of a closed system, the role of time can be taken to change its entropy. Let newX1,X2,X3,...,Xi,... the discrete sequence of topological spaces that describes one realization of the evolution of homotopy tipe of phase space of a some dynamical system in discrete time and letXi, i = 1,2,... compact topological spaces which admits pseudo-convex coverings. If a system is closed, then with the evolution of the homotopi tipe of phase space of such a system, the entropy grows. Therefore

Hth(Xi)<Hth(X2) <...<Hth(Xk) <...

< < k-1 ~ k? ~ kn

Moments of time for each state Xi of the phase space can be identified with the value of the entropyHht(Xi) of homotopy type. Based on such considerations we will built a random process which describes the evolution of the homotopy type of the phase space of a closed system. As we said above, each compact topological space which admits open pseudo convex covering corresponds to a rational number!(N(0(X))) where is N(0(X)) nerve of minimal pseudo-convex covering of this topological space.

Let t1, t2,..., tn,... ascending sequence of points in time i.e. entropy values. Let3tn, n = 1,2,... the set of rational numbers which represents homotopy type of compact topological spaces which admits minimal finite open pseudo-convex covering, and whose entropy of homotopy type is less than tn and that all simplicial complex with vertices from the set of elements of minimal open pseudo convex coverings of

topological spaces which homotopy entropy lees than tn, has the homotopy type of such a compact topological space whose homotopy entropy is less than tn. It is clear that the sets 3 tn, n = 1,2,... are finite. Let Define on the sets 3 t probability measures[2,5,6,12] Htnso: let measure^ of one point subset {ai} is number

ßtn({ a i}) = Tj

Hht(Xgin) %i=iHht(xain)

where I number of elements in the set 3 tn andXa.n topological space which homotopy type represents numbera and measure for subset {«i} i=1,2.....k<i c 3tnis number

Vtn({ai} i=1,2,...k<d = £¿=1 Vtn({ai}) ■

This probability measure defines on the set 3 random value ft with distribution law [2,5,5,12]

ftn =

{«1n},

(a2tn),

Mtn({ain}), Mtn({«2n}),

Mtn({«Ln})

In what follows we mean that 3 is such that the entropy of topological space which homotopy type defined by number E(ftn) has entropy which does not exceed tn , where E(ftn) is the mathematical expectation[2,6,12]

of random value ftn.

The sequence {(3tn, ftn)}, n = 1,2,... is a discrete Markov-type [2,13] random process with transition matrices

E(ftJ = Y!t=iai^tn(ad

Mtn = (Py(afn ^ *tn+1) = ßtn+1(an+1),i =

1,2,..., lt ,j = 1,2,... ,lt CLn £3t ,atin+1 E3t

' ' ' tn'J' ' ' Zn+1' I zn' J zn

Consider sequence

[X^MXJ.....^J-

Where [Xf ],n = 1,2,... class of all topological spaces which have homotopy type defined byE(ftn). This sequence determines the most likely trajectory of the evolution of the phase space of closed dynamical system.

Consider new the family{(3 tn,Btn,^tn)} and

probability measure measure® on the

n n

{®3tn,®Btn} [3,9,10]. Each element of

n n

® 3 ^represents trajectory of realization of the

n

constructed above random process. for this measure has place equality ® ntn(b) = Utn Vtn(btn)where

n

bt element of a -algebraBt , b = bt xbt x.. .x

!n the sequence [XftJ, n = 12... entropy btnx...£x Btn = {bti x bt2 x.. .x btn x... |btn £

H([Xf. ]) ^ m when n ^ m. If lim[Xr. ] = Xthen

ln n^ro ln

Xwill be homotopy equivalent to countable simplicial complex and Hh t([X]) = m.

Bt }and a - algebra® Bt generated by product x Bt .

I

tn®M<1,<2.....af") = 1

ll, 12,...,tntn^" ¿1 ' ~ ¿2 n

I ™ 1Ü2.....tn® Vtn(atí1, «t2.....^n) = I«1,«2.....tn,...® Vtntä1 «V

n

■J 11, i-2>...> t n n^rn 1 2 n t

.) = 1

It follows that the measure ® ^tnis probability Let

n

measure

(at.,at2,...,atn,...) E®3tn

represents one of realization of random process which describes evolution of phase space of closed dynamical system. Consider the random value

f^t^ at^..., atn, ...) = V^t^ at^..., atn...)) =® Vtn(atn),

n

In the set 3 all elements are rational numbers addition operation and the operation of multiplying an therefore in3 we nhave the addition and multiplication element by rational number . It makes it possible a operations, because in the set ® 3t also we have the calculation of the mathematical expectation

n

n

n

n

n

IJ«

Its®]«

E(/) = Z(Btl,Bt2.....atn,...)(at at2.....a^...) p((at ^ at2.....a^...)) ==

(ZaH ^iP^t^ ^..., at„,.. I £«t2 at2P((atl, ^..., at„,..),.., £«tn atnP((atl, ^..., ^..),...)

Theorem . Sequence of mathematical Proof: We prove the theorem by induction with

expectations of random values/tn, n = 1,2,... will be respect to n. Let n = 2 and taken two random value: coincide to mathematical expectation of random value/

= ial1}, {al1}.....[all]

^ ^fta^ ... ^t2([af21}

[a12], [a22].....[ a?2}

ft =

t2 ^(ial2}^ Mt2({a22}), ... Mt2({<2})

Consider set of the sequences with two If /((a^.aj2))i=h2.....¿lj=1,2.....i2=ptl®

members[(a1tl,a/2)}i=1,2.....ii,7-=1,2.....i2. On the thus set ^^((af1,a/2)) = ¡hM1)-^]2) distribution1 of

we have measure^ ® nt2, for which the formula randomvalue

Mt1 ® Mt2((a-1, aj2)) = Mt1(a^1) ■ Mt2(a-2) holds .

j. _ (ati1,aj2)i=1,2.....li,j = 1,2.....I2

Vti®^^1^^ = Mti(ati1) ■Vt2(atj2)'

Then

1 2 1 2 E(f) = ^Y^af1)^^/2) (a^aj2) = Mt2(a]2) afS^af1^ ^(a^a]2))

i = 1 7 = 1 ¿ = 1 7 = 1

And

1 1 2 1 2

E(ft1) = ^ßt1(^ti1)^ti1 = ^ßt1(^ti1)^ti1^ßt2(^'i2) = YY^K^ ■^t2(oj2)ati1 -

¿ = 1 ¿ = 1 7 = 1 ¿ = 1 7 = 1

1 2

1

=1 =1

Similarly we will show tha

tE(ft2) = I,%1[it2(aj2)aj2 = ^^(a^t^^al1) = ^(a^a/2 =

1 2

=YY^®^ ((a[1,ajt2))ajt2.

=1 =1

Let new the theorem is true whenn = k. Consider n = k + 1. The equality (a'1, a2,..., atik+1) =

the case ((a^,a-2,...,a-^),a^+1) holds. From this equality

follows that

lfc+1 lfc+1 i=1

E(ftk+1) = Y^a^na^1 = Y^+M^na^1 £ ® .....aj) =

j=1 j=1 h,h,.,hk

= Y Y®ftk(ait11,aj22.....a;kk)^k+1(a;k+1)a;k+1

'1/2... .'tk 'k+1

k 2 =1

= Y Y®^k(af1,af2.....a^a^a^1 =

1, 2,... , tk =1

£

¿=i zk+1

i1, ^2'...' Lt],• lk+1

.,ar,atik+^)atik+1. "• lk+1' lk+1

tk lk

2

Similarly we will show that,

E f ) = z

i=1

(аЦ,а£ tti < tp < W

: 4 l2 lk lk+1 lp 1 ^

It follows that

IimE(ft ) = lim

k^rn V k^>

=1 zk+1

У ® (at1, а]2.....atk,atk+1)atv

/ , ^ ^t кк 4' i2' ' ik' ik+1J i;

'1' '2'...' hfr' tk+1

=1 гк+1

Z K+1 t

® ^t.(ati1,ati2.....atk,...)a

^ r 11' 12' ' Ifc'

Н'Ь'.'НъЛк'.

The theorem is proved .

Let's call the infinite product Hht([Xatl],[Xat2].....[XBtn],...) = n^H* (a[n)

il ¿2 in

the entropy of the trajectory

[X t1],[X t2].....[X tn],... of evolution of fhomotopy

¡1 ¿2 % type of phase space of closed dynamical system; where

Hht(ain), entropy of homotopy type of spaceXatn

n in

defined by numbera^.

There exist[7,8,14] definitions of entropy trajectory in dynamical system but, it is definition of entropy of trajectory of material point in a phase s pace of dynamical system.

® Vtn Kv a£........) = nnMtn( aZ) = Ппу1пн (tn,

t n £i„Hht(aj„)

This equality shows it is necessary I imHht(atin) = m that the product

n^ro n

® ptn (ai1,ai2,...,ain,...)be non zero. In other cases

always ® Mtn(а¿11, «¿2..., «in...) = 0-

n 1 2 n

Conclusions

1. In paper is introduced the concepts of full number invariant for homotopy type of some topological spaces.

2.In paper is defined entropy of homotopy type of some topological spaces.

3. In paper is constructed discrete random process describing evolution of the homotopy type of phase space of c closed dynamical system.

4. In paper is introduced the concepts of entropy of trajectory of evolution of homotopy type of phase space of closed dynamical system.

4. In paper proved that the mathematical expectation of this random value coincide to sequence of mathematical expectations in the phase spaces of the constructed random process with respect to the corresponding random values.

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