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1
Vladikavkaz Mathematical Journal 2024, Volume 26, Issue 1, P. 85-99
УДК 517.987
DOI 10.46698/n9080-6847-9986-u
DYNAMICS OF QUADRATIC VOLTERRA-TYPE STOCHASTIC OPERATORS CORRESPONDING TO STRANGE TOURNAMENTS
R. N. Ganikhodzhaev1, K. A. Kurganov1,
M. A. Tadzhieva2, F. H. Haydarov1'3'4
1 National University of Uzbekistan, 4 University St., Tashkent 100174, Uzbekistan; 2 Tashkent State Transport University,
1 Adilkhodjaeva St., Tashkent 100067, Uzbekistan;
3 V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, 9 University St., Tashkent 100174, Uzbekistan; 4 Tashkent International University of Financial Management and Technology,
2 Amir Temur Ave., Tashkent, 100047, Uzbekistan E-mail: rganikhodzhaev@gmail.com, kurganov.k@mail.ru,
mohbonut@mail.ru, haydarov_imc@mail.ru, fa.xaydarov@tift.uz
Abstract. By studying the dynamics of these operators on the simplex, focusing on the presence of an interior fixed point, we investigate the conditions under which the operators exhibit nonergodic behavior. Through rigorous analysis and numerical simulations, we demonstrate that certain parameter regimes lead to nonergodicity, characterized by the convergence of initial distributions to a limited subset of the simplex. Our findings shed light on the intricate dynamics of quadratic stochastic operators with interior fixed points and provide insights into the emergence of nonergodic behavior in complex dynamical systems. Also, the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point defined in a simplex introduces additional complexity to the already intricate dynamics of such systems. In this context, the presence of an interior fixed point within the simplex further complicates the exploration of the state space and convergence properties of the operator. In this paper, we give sufficiency and necessary conditions for the existence of strange tournaments. Also, we prove the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point, defined in a simplex.
Keywords: quadratic stochastic operators of Volterra type, simplex, strange tournaments, Lyapunov functions.
AMS Subject Classification: 37E99.
For citation: Ganikhodzhaev, R. N., Kurganov, K. A., Tadzhieva, M. A. and Haydarov, F. H. Dynamics of Quadratic Volterra-Type Stochastic Operators Corresponding to Strange Tournaments, Vladikavkaz Math. J., 2024, vol. 26, no. 1, pp. 85-99. DOI: 10.46698/n9080-6847-9986-u.
1. Preliminaries
It is known that there are many systems which are described by nonlinear operators. One of the simplest nonlinear case is quadratic one. Quadratic dynamical systems have been proved to be a rich source of analysis for the investigation of dynamical properties and modeling in different domains, such as population dynamics (see [1, 2]) physics, mathematics (see [3]). On the other hand, the theory of Markov processes is a rapidly developing field with numerous
© 2024 Ganikhodzhaev, R. N., Kurganov, K. A., Tadzhieva, M. A. and Haydarov, F. H.
applications to many branches of mathematics and physics [4, 5]. However, there are physical and biological systems that cannot be described by Markov processes. One of such system is given by quadratic stochastic operators (QSO), which are related to population genetics. The problem of studying the behavior of trajectories of quadratic stochastic operators was stated in [6]. The limit behavior and ergodic properties of trajectories of quadratic stochastic operators and their applications to population genetics were studied.
However, such kind of operators and processes do not cover the case of quantum systems. Therefore, in [7, 9] quantum quadratic operators acting on a von Neumann algebra were defined and studied. Certain ergodic properties of such operators were studied in [8]. In these papers, dynamics of quadratic operators were basically defined due to some recurrent rule which marks a possibility to study asymptotic behaviors of such operators.
This paper is devoted to the study of ergodic properties of quadratic stochastic operators of Volterra type defined in the standard simplex. The study of the dynamics of such operators began with the example of Ulam [6]. The non-ergodicity of this operator was proved by Zakharevich [10]. For a general form of quadratic stochastic operators of Volterra type, a proof of non-ergodicity in a two-dimensional simplex can be found, for example, in [11]. In the present paper, we prove the non-ergodicity of quadratic stochastic operators of Volterra type with an internal fixed point. In addition, the notion of a strange tournament is generalized to the case of r-strange tournaments. The application of the non-ergodicity of such operators in genetics is shown.
Let
s m
Sm-i = i x € Rm : x = (xi,... Xm) : X ^ 0, ^ X = 1 ^ i=i
be a standard (m — 1)-dimensional simplex. Put
m
Pij,k > 0, Pij,k = Pji>k, pij,k = 1, i,j, k = 1,..., m. (1.1)
ij=i
For any x € Sm-1 and for all k = 1,..., m we consider a mapping V : Sm-1 ^ Sm-1 which is defined by
m
(Vx)k = ^ Pij,kxixj. (1.2)
i,j=i
Such operator is called a quadratic stochastic operator (Q.S.O.).
For x0 = (x°,x°,... ,xm) € Sm-i trajectory (orbit) is a sequence of iterations x0, Vx0, V 2x0, ..., Vn x0, n = 0,1,2,...
(Q.S.O.) V is called regular if for any x € int Sm-i there exists a unique fixed point x* € Sm-i (i. e. Vx* = x*) such that limn—TO Vnx = x*.
(Q.S.O.) V is called ergodic if for any x € Sm-i there exists a limit
1 n- i
lim - V Vkx.
n—y^o n ^—' k=0
A continuous functional ^ : Sm-i ^ R is called a Lyapunov function if for any initial point x(0) € Sm-i there exists
lim ^(x(n)).
(Q.S.O.) V defined on Sm-1 is called a Q.S.O. Volterra type (Q.S.O.V.T.), if
Pij,k = 0, k/ {i,j}.
(Q.S.O.V.T.) V can be reduced to the form
V : xk = xJ 1 + ^ akixA, k = 1,...,m, (1.3)
^ i=i '
where j«ki| ^ 1, aki = -«ik.
2. Tournaments. Strange Tournaments
Let aki = 0 for k = i. Consider a complete graph with m vertices labeled 1,2,...,m. On the edges of the graph, we define directions as the following way: the edge connecting vertices k and i is directed from k-th vertex to i-th if aki < 0, and has the opposite direction, if aki > 0.
The resulting complete directed (oriented) graph is called a tournament and is denoted by Tm. Two tournaments T'm and T^ are called isomorphic if the graphs are isomorphic graphs.
A tournament is called strong if it is possible to get from any vertex to any other, taking into account the direction of the edges.
The transitivity of a tournament means that any subtournament of the given tournament is not strong.
Stock of a transitive tournament is a vertex from which it is impossible to get to any other vertex, taking into account the direction on the edges. Triangle is a strong tournament with three vertices.
Let us recall the following well-known results:
Denote by S(k) the number of arcs entering to the k-th vertex, while S(k) is called the number of points of the k-th vertex. The set of numbers S(1), S(2), ..., S(m) is called the order of the tournament.
A tournament Tm is called strange if there exists a (strange) vertex i0 such that
1. S(io) = S(k) for all k = io.
2. If S(k) > S(i0) for some k, then the arrow connecting vertices i0 and k is directed from k to i0.
3. If S(k) < S(i0), then the arrow has a direction from i0 to k.
It is easy to check that there are no any strange tournaments in Tm, (m = 2,3,..., 6).
A tournament Tm is called r-strange if there exist vertices i1, i2,..., ir such that
1. S(i) = S(k) for all k = ih l = 1,2,..., r.
2. If S(k) > S(ii) for some k, then the arrow connecting vertices ii and k is directed from k to ii.
3. If S(k) < S(ii), then the arrow has a direction from ii to k.
4. S(ii) = S(i2) = ... = S(ir).
It is clear that for r = 1 the tournament becomes an ordinary strange tournament, and such tournaments appear when r = 2l — 1, l € N.
Lemma 1. 1. Let G = (V, E) be a complete (undirected) graph with jVj =2s + 1, s ^ 1. The graph G can be exchanged to directed graph G such that the number of entering and leaving arcs of any vertex of G are the same.
2. Let G = (V, E) be a complete (undirected) graph with |V| = 2s, s ^ 1. The graph G can be exchanged to directed graph G such that the difference of the number of entering and the number leaving arcs of any vertex of G less than or equal to 1.
< (1) We use method of induction for vertices of the graph G. If |V| = 3 then it will be a cycle in directed graph. Suppose that for |V| = 2s + 1 the Lemma holds, then we shall prove the Lemma for the case | V| = 2s + 3. Let V = {1,2,3,..., 2s + 1,2s + 2,2s + 3} and Vi = {1,2,3,..., 2s + 1}. By induction we obtain directed graph Gl = {Vi, El} which the number of entering and leaving arcs of any vertex of G1 are the same. Now we add the remaining vertices 2s + 2, 2s + 3 as follows:
Jx2s+2 ^ Xi, i € {1, 3, 5,..., 2s + 1}, {£2s+2 ^ Xi, i €{2, 4, 6,..., 2s}.
Similarly,
jx2s+3 ^ Xi, i € {1, 3, 5,..., 2s + 1}, {x2s+3 ^ Xi, i €{2, 4, 6,..., 2s + 2}.
Hence, we obtain directed graph G = (V, E) such that the number of entering and leaving arcs of any vertex of G are the same.
(2) For proving the second part of we use the first part of the Lemma. Let V = {1, 2, 3,... , 2s + 1, 2s + 2} and Vi = {1, 2, 3,... , 2s + 1}. By the first part of Lemma we obtain directed graph Gi = {Vl ,Ei} which the number of entering and leaving arcs of any vertex of Gl are the same. We add the remaining vertex 2s + 2 as follows:
Jx2s+2 ^ Xi, i € {1, 3, 5,..., 2s + 1}, \x2s+2 ^ Xi, i €{2, 4, 6,..., 2s}.
Then S(Xi) = s + 1 for all i € {1,3, 5,..., 2s + 1} and S(Xi) = s for all i € {2,4,6,..., 2s, 2s + 2}. >
Theorem 1. The following statements hold:
1. Let Tm be a strange tournament with vertices {1,2,3,..., m}. If k € {1,2,3,... , m} is a strange vertex, then ^ k ^ 2m~3, m ^ 7.
2. Let ^^ ^ k ^ 2"~3, rri ^ 7. Then there exists a strange tournament Tm with vertices {1, 2, 3, . . . , m} such that k is a strange vertex.
< (1) In the tournament there are k — 1 vertices (without loss of generality {1,2,3,..., k — 1}) such that S(k) < S(i) for all i € {1,2,3,... , k — 1}. Also, we have S(k) > S(j) for all j € {k + 1, k + 2,..., m}. The number of all arcs among the vertices {1,2,3,..., k — 1} is equal to (fc-i). The number of arcs entering to the vertex i € {1,2,3,... ,k — 1} from the vertices: k + 1,k + 2,... ,m less than or equal to m — k. Then total number of arcs entering to the vertices: 1,2,3,..., k — 1 is at most + (k — 1)(m — k). Hence from k is a strange vertex one gets
k—1 2
+ (k — 1)(m — k) >k(k — 1).
By pigeonhole principle, we rewrite the last inequality as follows:
k—1 2
+ (k — 1)(m — k) ^ (k — 1)2
(2.1)
Inequality (2.1) is equivalent to k < 2m3 3. Also, the number of all arcs among the vertices {k + 1, k + 2,..., m} is equal to (m-fc). The number of arcs from strange vertex to the vertices: k + 1, k + 2,..., m is equal to m — k. Then one gets:
m — k\ , , . ,.
2 I + m — k < k(m — k).
Again we use pigeonhole principle and rewrite the last inequality as follows:
m — k
2 j+ m — k ^ (k — 1)(m — k). (2.2)
From (2.2), we have k ^
(2) Let ^ k ^ 2m3~3, m ^ 7. For a fixed fc we construct strange tournament. If m — fc is even number (the case even is similar), then by the first part of Lemma 1 we can show sub-tournament T' with vertices k + 1, k + 2,..., m such that the number of entering and leaving arcs of any vertex of T' are the same. Indeed, by Lemma 1 we can construct directed subgraph with S(i) = S(j) for all i,j € {k + 1, k + 2,..., m}. From the inequality k ^ 2m3~3 we have S(k) > S(k + 1). Analogously, since the second part of Lemma 1 and k ^ we obtain S(k) < S(i), for all i € {1,2,..., k — 1}. >
Corollary 1. There is not any strange tournament among the tournaments Tg. Indeed, there is not any integer k with ^ k ^ In the case ofTj, by Theorem 1 (i. e., 4 ^ k ^ there is a unique (up to permutation of vertices) strange tournament. Since k = 4 and Lemma 1 we can conclude this strange tournament as follows: (4443222) (see Fig. 1).
In genetics, Q.S.O.V.T. V has the following interpretation:
If species 1, 2, 3 dominate in a population under panmixia, and species 5, 6, 7 are on the verge of extinction, then sometimes species 4 can be found, and it ensures the preservation of all species.
Q.S.O.V.T. V corresponding to this weird tournament is:
Vj = xi(1 - 012X2 + 013X3 - 014X4 + 015X5 + ai6X6 + 017X7), x'2 = X2(1 + 012X1 - 023X3 - 024X4 + 025X5 + 026X6 + 027X7), x'3 = X3(1 - 013X1 + 023X2 - 034X4 + 035X5 + 036X6 + 037X7), < X4 = X4(1 + 014X1 + 024 X2 + 034 X3 - 045 X5 - 046X6 - 047X7), x5 = X5 (1 - 015X1 - 025X2 - 035X3 + 045X4 - 056X6 + 057X7), X'6 = X6(1 - 016X1 - 026X2 - 036X3 + 046X4 + 056X5 - 067X7), = x7(1 - 017X1 - 027X2 - 037X3 + 047X4 - 057X5 + 067X6),
(2.3)
where aki € [0; 1] or aki € [—1; 0] at the same time.
Consider only the case aki € [0; 1] otherwise the arcs in the tournament get the opposite direction.
Q.S.O.V.T. (3) except vertices
Mi = (1, 0, 0, 0, 0, 0, 0), M2 = (0,1, 0, 0, 0, 0, 0), ..., M7 = (0, 0, 0, 0, 0, 0,1)
has on the boundary, corresponding to cyclic triples, fixed points:
A1 = 067024 035 + 026047035 + 045027036 + 036024 057 + 037024 056 + 046037025 + 045067023 + 056047025 + 046023057 - 056027034 - 025067034
- 034026057 - 046035027 - 025047036 - 045026037,
A2 = 045067013 + 056047013 + 046013057 + 045016037 + 034016057 + 015067034 + 056017034 + 046035017 + 01504706 - 014056037 - 046037025
- 067014035 - 016047035 - 045017036 - 036014057,
A3 = 047056012 + 047025016 + 046012057 + 045067012 + 025014067 + 026014057 + 045017026 + 056014027 + 046027015 - 047015026 - 015 067024
- 024016 057 - 056017024 - 046 025017 - 045016027,
A4 = 056017025 + 015067023 + 023016057 + 067013025 + 016037025 + 036027015 + 013056027 + 035017026 + 026013057 + 035067012 + 056037012 + 036012057 - 036025017 - 035016027 - 015037026,
C123, C145, C146, C147, C245, C246, C247, C345, C346, C347, C567-
For some of coefficients 0^, there can be an internal fixed point:
where
A5 = a67a34 ai2 + 036 047012 + 023 016047 + 036024057 + 037024056 + 046037025 + 045067 023 + 056^47025 + 046 023 057 — ^56 027034 — ^25^67 034
— 034^26057 — 0-46 035 027 — 025047 036 — 045026037,
A6 = 017035 0-24 + a57 013024 + 014 037025 + 013045 027 + 015034027 + 017 045023
+ 014057 023 + 045037012 + 034 057012 — 037 015024 — 047013 025
— 017034025 — 035 014027 — 047015 023 — 047035012,
A7 = 034056 012 + 046 012035 + 035 026014 + 023015 046 + 014023056 + 015 036 024 + 056013 024 + 034016025 + 025 013046 — 045 036012 — 013045 026
— 034015026 — 016 045023 — 035024 016 — 014036025,
7
A = EA* •
i=1
At first, we shall construct the functionals
1
1,jfc = (;v\^ • 4ifc • x^) with respect to the fixed points of Cj and estimate them by using Young's inequality [12].
145 — (012045 + 014 025 — 015024 )X2 + (013 045 + 015 034 — 014035)^3
A145
(015046 + 014056 — 016045)X6 + (017045 + 014057 — 015047)®7] ,
<Pl46(x) A146
+ (046015 + 056014 — 016045)X5 — (047016 + 067014 — 017046,
<^146(«') ^ - [A146 - (012046 + 026014 ~ aWa24)x2 + (013 046 + 016 034 ~ 014036)^3
^146
11470*0 A147
— (045Oi7 + 057a 14 — 015047)x5 + (oi6047 + O67O14 — O46O17,
<PU7&) < ^ [A 147 - (012047 + 027014 - 024017)3:2 + (013047 + O34O17 - «37^4)^3 ^147
1245(3:0 < [A245 + (012 045 + 014 025 - 0i5024)3:i - (023 045 + 035 024 - 0340,5)^3
^245
— (046O25 + O56O24 — O26O45)X6 + (O27O45 + O57O24 — O47O25)®7] ,
1246(3:0 < [A246 + (012 046 + 014 026 - oi6o24)3:2 - (o23o46 + o36o24 - o34o26),:3
^246
+ (025O46 + O56O24 — O45O26)x5 — (047O26 + O67O24 — O27O46)®7] , 1247(3:0 < [A247 + (oi2o47 + 014 027 - 0170,4)3:1 - (023047 + o37o24 - 0340,7)3:3
^247
— (045O27 + O57O24 — O25O47)X5 + (026O47 + O24O67 — O46O27,
< ^ [a 345 — (O13O45 + O15 O34 — O14O35 )X1 + (023 O45 + O24 O35 — 025034)^2 ^345
— (046O35 + O56O34 — O36O45)X6 + (O37O45 + O57O34 — O47O35)®7] ,
346 — (013O46 + O16 O34 — O14O36 )X1 + (023 O46 + O24 O36 — 026034)^2
^346
+ (046O35 + O56O34 — O35O46)X5 — (047O36 + O67O34 — O37O46)®7] ,
347 — (013047 + 017034 — 014057^1 + (023047 + 024037 — 027034^2
^347
— (045037 + 057034 — 035047)^5 + (^36^47 + 067^34 — 046037)x^ ,
< ^ [A
123 — (014023 + 024013 + 034012)^4 + (015023 + 025013 — 035012)^5
^123
+ (016025 + 026013 + 036012)X6 + (017023 + 027013 + 037012)^7] ,
№.(»0 < ^ [A
567 — (015067 + 016057 + 017056)X1 — (025067 + 026057 + 027056^2
^567
— (035067 + 036057 + 037056)^3 + (045067 + 046057 + 047056^4] • Denoting the inner brackets, we have
<Pl45(x') ^ [A145 - KiX2 + K2x3 - K3x& + K4x7],
^145
¥>146 («O ^ [Ai46 - K5X2 + K6X3 + K3X5 - K7X7] , ^146
^147(^0 ^ ^^^ [A147 - K8x2 + K9x3 - K4X5 + K7Xe],
^147
¥>245 («O ^ ^^ [A245 + K\X\ ~ KWX3 ~ K4X6 + K12XT], ^245
<£>246 (V) ^ [A246 - K5X1 ~ K13X3 + KUX5 - K14X7]
^246
¥>247 (a?') ^ LP2^7<kX^ [A247 - K&x 1 - K15X3 - E12X5 + Kuxe]
^247
¥>345 O*?') ^ [A345 - K2X1 + ^10^2 - KmX6 + K17X7]
345
<^346 («') ^ ^^ [A346 - K&X 1 + ^13^2 + ^16^5 ~ Kl&X7]
346
¥>347 (a:') ^ ^3470*0 [A347 - KgXi + K15X2 - Kux5 + Eissel
347
¥>123 O*?') ^ [A123 - ^19^4 + K20X5 + K2\Xß + ^22X7] :
/1
123
<P567(x') ^ [A567 - ^23^1 - K24X2 ~ K25X3 + E26X4] •
/ 567
If there is no Lyapunov function among the functionals ¥'ijk(x), then there exists an interior fixed point C. We will study only such cases.
Lemma 2. Let x € int S6, x = C, then the limit set of trajectories is infinite and lies on the boundary of the simplex S6, i. e., w(x) C dS6. < Now we consider the following functional
P(x) = LI
i
7 \ Ä
xfM •
,k=1
Because
/
<p(x ) = <p(x) [l - CL12X2 + CL13X3 - auX4 + a\5x5 + oi6*6 + oi7*7] A [l + oi2*i - 0,23X3
Ao
- 024*4 + «25*5 + «26*6 + «27*7] ^ [l - <113*1 " «23*2 " «34*4 + «35*5 + «26*6
A3 A4
+ «27*7] A [l - «14*1 - «24*2 + 034*3 - 045*5 + 0-26*6 + 027*7] A [l - Ol5*l
a5
- 025*2 - 035*3 + 045*4 + 026*6 + 027*7] A [l — Oi6*1 — O26 *2 — 036*3 + 046*4
+ 056*5 - 067*7] A [l - Ol7*l - 027*2 - 037*3 + 047*4 - 057*5 + 067*6]
A7 A
<
l(*)
A
Ai(1 — 0i2*2 + 013*3 — 014*4 + 015*5 + O16 *6 + 017*7) + A2(1 + 0i2*1
— 023*3 — 024 *4 + 025*5 + 026*6 + 027*7) + A3(1 — 013*1 — 023*2 — 034*4 + 035*5 + 026*6 + 027*7) + A4(1 — 014*1 — 024 * 2 + 034*3 — 045*5 + 026*6 + 027*7) + A5(1 — 015*1 — 025*2 — 035*3 + 045*4 + 026*6 + 027*7) + A6(1 — 016*1 — 026*2 — 036*3 + 046*4 + 056*5 — 067*7) + A7(1 — 017*1 — 027*2
— 037*3 + 047*4 — 057*5 + 067*6) = l(*),
^>(Vn*) decreases as n ^ œ, i. e.
lim <^(Vn*) = 0.
Hence w(*) C dS6. >
From the invariance of fixed vertices, edges and faces of S6, the limit set cannot be finite. Thus, (2.3) can be rewritten as following form:
'*1 = *i[1 - (012*2 + 014*4) + (Ol3*3 + Oi5*5 + Oi6*6 + 017*7 *'2 = *2[1 - (023*3 + 024*4) + (012*1 + 025*5 + 026*6 + 027*7 *3 = *3[1 - (013*1 + 034*4) + (023*2 + 035*5 + 026*6 + 027*7 *4 = *4 [1 - (014*1 + 024 *2 + 045*5) + (034*3 + 026*6 + 027*7 *5 = *5[1 - (015*1 + 025*2 + 035*3 + 056*6) + (045*4 + 057*7 *6 = *6[1 - (016*1 + 026*2 + 036*3 + 067*7) + (046*4 + 056*5 k*'7 = *7[1 - (017*1 + 027*2 + 037*3 + 057*5) + (047*4 + 067*6 Let us split the S6 simplex into the following parts:
*1[1 — A24 + A3567], *2 [1 — A34 + A1567], *3 [1 — A14 + A2567], *4 [1 — A123 + A567], *5 [1 — A1236 + A47], *6[1 — A1237 + A45], *7[1 — A1235 + A46].
T1 = {[A24 < A3567] n [A34 < A1567] n [A14 < A2567] n [A567 < A123]
n [A47 < A1236] n [A45 < A1237] n [A46 < A1235U,
T = {[A24 ^ A3567] n [A34 < A1567] n [A14 < A2567] n [A567 < A123]
n [A47 < A1236] n [A45 < A1237] n [A46 < A1235U,
T3 = {[A24 ^ A3567] n [A34 < A1567] n [A14 < A2567] n [A567 < A123]
n [A47 ^ A1236] n [A45 < A1237] n [A46 < A1235]},
T4 = {[A24 ^ A3567] n [A34 ^ Ai567] n [A 14 < A2567] n [A567 < Am]n
n [A47 ^ A1236] n [A45 < A1237] n
T5 = {[A24 ^ A3567] n [A34 ^ A1567] n [A 14 < A2567] n [A567 < A123]
n [A47 ^ A1236] n [A45 ^ A1237] n
T6 = {[A24 ^ A3567] n [A34 ^ A1567] n [A14 ^ A2567] n [A567 < A123]
n [A47 ^ A1236] n [A45 ^ A1237] n
T7 = {[A24 ^ A3567] n [A34 ^ A1567] n [A14 ^ A2567] n [A567 < A123]
n [A47 ^ A1236] n [A45 ^ A1237] n
T8 = {[A24 ^ A3567] n [A34 ^ A1567] n [A14 ^ A2567] n [A567 ^ A123]
n [A47 ^ A1236] n [A45 ^ A1237] n
T9 = {[A24 < A3567] n [A34 ^ A1567] n [A14 ^ A2567] n [A567 ^ A123]
n [A47 ^ A1236] n [A45 ^ A1237] n
T10 = {[A24 < A3567] n [A34 ^ A1567] n [A14 ^ A2567] n [A567 ^ A123]
n [A47 < A1236] n [A45 ^ A1237] n
T11 = {[A24 < A3567] n [A34 < A1567] n [A14 ^ A2567] n [A567 ^ A123]
n [A47 < A1236] n [A45 ^ A1237] n
T12 = {[A24 < A3567] n [A34 < A1567] n [A14 ^ A2567] n [A567 ^ A123]
n [A47 < A1236] n [A45 < A1237] n
T13 = {[A24 < A3567] n [A34 < A1567] n [A14 < A2567] n [A567 < A123]
n [A47 < A1236] n [A45 < A1237] n
T14 = {[A24 < A3567] n [A34 < A1567] n [A14 < A2567] n [A567 ^ A123]
n [A47 < A1236] n [A45 < A1237] n
A46 < Ai235^, A46 < A1235]}, A46 < A1235]}, A46 ^ A1235]}, A46 ^ A1235]}, A46 ^ A1235]}, A46 ^ A1235]}, A46 ^ A1235]}, A46 ^ A1235]}, A46 < A1235]}, A46 < A1235] } .
Lemma 3. For any point x € int S6, x = C, its route is given by the diagram
Ti ^ T2 ^ T3 ^ ... ^ T14 ^ Ti.
< Let x € Tl. Then
xL > x1 , x2 > x2, x3 > x3, x4 > x4, x5 > x5, x6 > x6, x7 > x7
and after some time the trajectory hits T2. The other cases can be verified similarly. It is clear that
A,
Ai
A Xt ~ A Xj'
Close to the vertices Mi (i = 1,7)
A,
A
Ai
A
which will be denoted by * >- *j. Then the partition of the S6 simplex can be carried out as follows:
*1 >- *7 >- *2 >- *6 >- *3 >- *5 >- *4
Gi = {x e S6 G2 = {x e S6 G3 = {x e S6 G4 = {x e S6 G5 = {x e S6
x1 >- x2 >- x7 >- x3 >- x6 >- x4 >- x5 x2 >- x1 >- x3 >- x7 >- x4 >- x6 >- x5 x2 >- x3 >- x1 >- x4 >- x7 >- x5 >- x6 x3 >- x2 >- x4 >- x1 >- x5 >- x7 >- x6 G6 = {x e S6 : x3 >- x4 >- x2 >- x5 >- x1 >- x6 >- x7 G7 = {x e S6 : x4 >- x3 >- x5 >- x2 >- x6 >- x1 >- x7 G8 = {x e S6 : x4 >- x5 >- x3 >- x6 >- x2 >- x7 >- x1 G9 = {x e S6 : x5 >- x4 >- x6 >- x3 >- x7 >- x2 >- x1 G1o = {x e S6 : x5 >- x6 >- x4 >- x7 >- x3 >- x1 >- x2
a
x6 >- x5 >- x7 >- x4 >- x1 >- x3 >- x2 x6 >- x7 >- x5 >- x1 >- x4 >- x2 >- x3 x7 >- x6 >- x1 >- x5 >- x2 >- x4 >- x3 x7 >- x1 >- x6 >- x2 >- x5 >- x3 >- x4
G11 = {x e S6 G12 = {x e S6 G13 = {x e S6 G14 = {x e S6
Put
H1 = G1 u G2, H2 = G3 u G4, H3 = G5 u G6, H4 = G7 u Gß,
H5 = G9 U G10, H = G11 U G12, H7 = G13 U G14.
Choose a neighborhood IJq of G, IJq C intS*6 and Ui = Hi\Uo (i = 1,7) so that they are convex and P|7=1 Ui = 0. These partitions Ti and Gi are almost the same. For example, in the partition Ti, the hyperplane A24 = A3567, which defines the boundaries of T1, passes through the points M1, C and
G45 0, 0, 0,
fl15
a14
a 14 + fl15 a 14 + fl15
0, 0
and in the partition Gi the system of hyperplane inequalities T1 gives a similar part of the simplex whose boundary passes through M1, C and
C45 0, 0 , 0,
A
15
A
14
A14 + A15 A14 + A
0, 0
15
We have changed these partitions in order to easily find the ratio of coordinates near the Mi vertices. Still, the number of iteration steps in both divisions is almost the same. >
Lemma 4. Let x ^ U, Vkx € U for all k = l,n and Vn+l ^ U, where U is one of the regions Ui, * € int S6 and * = C. Then
n > A log2
B
1(x):
where A, B are absolute constants.
< Let, for example, U = Ul. Then x / Ul, x € G14, i. e.,
x7 >- xL >- x6 >- x2 >- x5 >- x3 >- x4. Since x' = Vx € Ul, is precisely x' € Gl, then
xi x7 x2 x6 x3 x5 x4.
Whence x\ > s1^ = av
x
—f ^ min —j = 0:7, xi xeG 14 xi
x
7
x
—7 ^ min —7- =
xi xeGi4 xi
x
6
x
—7 ^ min —f = as,
xi xeGi4 xi
x
3
x
f ^ min —7 = 0:5, —r min —7 = a4,
xi xeGi4 xi xi xeGi4 xi
where we take min in the region G14 C U7
Since the interior of the simplex is invariant, all coefficients of a are positive. For example,
if olj = 0, then there exists x € Gu such that ^f = 0, i. e.,
xi
x7 [1 - (017x1 + 027x2 + 037x3 + 057x5) + (047x4 + 067x6^ = 0. If x7 = 0 then from x7 >- xL >- x6 >- x2 >- x5 >- x3 >- x4 follows
x1 = x2 = . . . = x6 = 0,
which is impossible, but
[1 - (017x1 + 027x2 + 037x3 + 057x5) + (047x4 + 067x6) = 0
never done, because it's never done
(017x1 + 027x2 + 037x3 + 057x5) = 1
in G14. Let Vkx = (xk, xk,..., xk).
Since Vn+1x / Ui, then Vn+1x € G3:
xn+1 . xn+1 , xn+1 , xn+1 , xn+1 , xn+1 , xn+1 min l
x
x
4
x
4
Whence > si^* = ai. Next,
v.n+1
x
n
k=1
k+1
= II i1 + ~ a23^3 - a24^4 + 025^5 + 026^6 + «27^7) < 2™,
2 k=1
0i2xk - 023xk - 024x4 + 025xk + 026xk + 027xk < 1.
Further,
2n >
x
n+1
> -
a
A2
(x2)A2
a
(4)a2 • (x'^Ai . (4)A3 . (^)A4 . (3./ )AB . (a;/ )A6 . (a;/ )Ar
1 A
A2 ' A
[aAi+A2
a
A3 - aA4
.A.
a
5 . ^,A6 . ^Arl Ai
a
a
MVx)]-
i. e. n> ¿log2^y. >
n
n
2
2
x
i
A
2
2
1
x
x
2
2
A
2
Lemma 5. Let U be one of the domains Ui, x € int S6, x = C. Let {ni, mi}°=1 be sequences of natural numbers such that Vnix £ U, Vni+kx € U for all k = l,rrii and vni+mi+1x € U. Then there exists K such that mi > kni.
dp = max \ h - 012x2 + 013X3 - 014X4 + 015X5 + 016X6 + 017X7] Al [1 - 012X1 - 023X3 int S6\Uo I
]A2 1 A3
[1 - 013X1 + 023X3 - 034X4 + 035X5 + 036X6 + 037X7J x [1 + 014X1 + 024X3 - 014X4 + 015X5 + 016X6 + 017X7]Al [1 - 015X1 - 025X2 - 05X3 + 045X4 - 056X6 + 057X7] ab [1 - 016X1 - 026X2 - 036X3 + 046X4 + 056X6 - 067X7]Ae x [1 - 017X1 - 027X2 - 037X3 + 047X4 - 057X5 + 067X6] at j < 1. By Lemma 4
1 1 A' / 1
m > Alog2 W^) > Alog2 ^Ä) =log2 pi =log2 A+n*log2 p =kn>
Theorem 2. Operator V (defined in (2.3)) is non-ergodic, i. e., for any point x € int S6, x / C, the sequence ^ ^fc=o 110
< Assume the opposite, i. e., for any point x € S6 there is a limit
n-1
lim V Vkx = x*.
n—y^o —< k=0
Let x* / U1, {ni, mi} be the same as in Lemma 5. Let d = dist (x*, U1) and
dist (~J2vkx,x*) < d
nk k=0
for sufficiently large n. Since rrii > krii for some x € int S6, x = n.]rm. SfcLi"14
( Ui+mt \ f nt+mt \
y Vkx)+^- U- y V*x),
n k=1 ) ni + mi \mifc=n^l )
where the second term in U\, must be dist (x, x*) < But it contradicts to dist (x, x*) > | for large n. >
The first 3-strange tournament occurs among the T13 tournaments, the order of which is (7777766655555).
Theorem 3. Q.S.O.V.T. corresponding r to strange tournaments is non-ergodic. The proof is similar to Theorem 1. In general, the following holds:
Theorem 4. V has an interior fixed point, then it is non-ergodic.
Acknowledgements. The work supported by the fundamental project (number: F-FA-2021-425) of The Ministry of Innovative Development of the Republic of Uzbekistan.
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Received Jule 1, 2023
Rasul N. Ganikhodzhaev National University of Uzbekistan, 4 University St., Tashkent 100174, Uzbekistan, Professor, Department of Mathematics E-mail: rganikhodzhaev@gmail https://orcid.org/0000-0001-6551-5257
Karim A. Kurganov National University of Uzbekistan, 4 University St., Tashkent 100174, Uzbekistan, Associate Professor, Department of Mathematics E-mail: kurganov. k@mail. ru https://orcid.org/0009-0001-3788-1513
Mohbonu A. Tadzhieva Tashkent State Transport University,
1 Adilkhodjaeva St., Tashkent 100067, Uzbekistan Associate Professor, Department of High Mathematics E-mail: mohbonut@mail.ru https://orcid.org/0000-0001-9232-3365
Farhod H. Haydarov
National University of Uzbekistan,
4 University St., Tashkent 100174, Uzbekistan,
Associate Professor, Department of Mathematics;
V. I. Romanovsky Institute of Mathematics of the Academy
of Sciences of the Republic of Uzbekistan,
9 University St., Tashkent 100174, Uzbekistan,
Postdoctoral Researcher;
Tashkent International University of Financial Management and Technology,
2 Amir Temur Ave., Tashkent, 100047, Uzbekistan Associate Professor, Department of High Mathematics E-mail: haydarov_imc@mail.ru, fa.xaydarov@tift.uz https://orcid.org/0000-0001-9388-122X
Владикавказский математический журнал 2024, Том 26, Выпуск 1, С. 85-99
ДИНАМИКА КВАДРАТИЧНЫХ СТОХАСТИЧЕСКИХ ОПЕРАТОРОВ ТИПА ВОЛЬТЕРРА, СООТВЕТСТВУЮЩИХ СТРАННЫМ ТУРНИРАМ
Ганиходжаев Р. Н.1, Курганов К. А.1, Таджиева М. А.2, Хайдаров Ф. Х.1,3,4
1 Национальный университет Узбекистана им. Мирзо Улугбека, Узбекистан, 100174, Ташкент, ул. Университетская, 4;
2 Ташкентский государственный транспортный университет, Узбекистан, 100174, Ташкент, ул. Адылходжаева 1;
3 Институт математики им. В. И. Романовского АН Республики Узбекистан, Узбекистан, 100174, Ташкент, ул. Университетская, 9;
4 Ташкентский международный университет финансового управления и технологий, Узбекистан, 100047, Ташкент, ул. Амира Темура, 2 E-mail: rganikhodzhaev@gmail.com, kurganov.k@mail.ru, mohbonut@mail.ru, haydarov_imc@mail.ru, fa.xaydarov@tift.uz
Аннотация. Изучая динамику названных операторов на симплексе, уделяя особое внимание наличию внутренней неподвижной точки, мы исследуем условия, при которых операторы проявляют неэр-годическое поведение. Посредством строгого анализа и численного моделирования мы показываем, что определенные режимы параметров приводят к неэргодичности, характеризующейся сходимостью начальных распределений к ограниченному подмножеству симплекса. Наши результаты проливают свет на сложную динамику квадратичных стохастических операторов с внутренними неподвижными точками и дают представление о возникновении неэргодического поведения в сложных динамических системах. Кроме того, неэргодичность квадратичных стохастических операторов типа Вольтерра с внутренней неподвижной точкой, определенной в симплексе, вносит дополнительную сложность в и без того сложную динамику таких систем. В этом контексте наличие внутренней неподвижной точки внутри симплекса еще больше усложняет исследование пространства состояний и свойства сходимости оператора. В данной статье мы приводим достаточные и необходимые условия существования странных турниров. Также доказывается неэргодичность квадратичных стохастических операторов типа Вольтерра с внутренней неподвижной точкой, определенных в симплексе.
Ключевые слова: квадратичные стохастические операторы типа Вольтерра, симплекс, странные турниры, функции Ляпунова.
AMS Subject Classification: 37E99.
Образец цитирования: Ganikhodzhaev, R. N., Kurganov, K. A., Tadzhieva, M. A. and Haydarov, F. H. Dynamics of Quadratic Volterra-Type Stochastic Operators Corresponding to Strange Tournaments // Вла-дикавк. мат. журн.—2024.—Т. 26, № 1.—C. 85-99 (in English). DOI: 10.46698/n9080-6847-9986-u.
