Page 634 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 634
DYNAMICAL SYSTEMS AND ORDINARY DIFFERENTIAL EQUATIONS AND
APPLICATIONS
classes of PWS. In addition, we discuss how the developed theory may be applied to study
dynamics of strongly nonlinear systems e.g. the van der Pol equation.
Limit states of multi-component discrete dynamical systems
Oksana Satur, oksana@satur.in.ua
Institute of Mathematics of the National Academy of Sciences of Ukraine, Ukraine
Object. We study models of multicomponent discrete dynamic conflict systems with attractive
interaction, which are characterized by a positive value that is called the attractor index. Con-
sider the set of discrete probability measures µi ∈ M1+(Ω) on finite space Ω = {ω1, . . . , ωn},
i = 1, m. Each of these measures µi can be identified with a stochastic vector pi = (pij)jn=1,
where
pij = µi(ωj), i = 1, m, j = 1, n.
Consider the mapping
{pt1, pt2, . . . , ptm} −→,t {pt1+1, pt2+1, . . . , ptm+1}, (1)
which generates multi-component discrete dynamical systems with trajectories (1), where the
coordinates of each vector pit = (pitj)nj=1 are changed according to equations
pitj+1 = 1 ptij(θt + 1) + τjt , t = 0, 1, . . . . (2)
zt
Here θt = θ(p1t , pt2, . . . , ptm) is a finite positive function, T t = (τjt)nj=1 is a vector with non-
negative coordinates (attractor index), and zt = θt + 1 + W t is normalizing denominator,
Wt = n τjt.
j=1
Main results.
Theorem 1. Let all coordinates of vector wt = (wjt )jn=1, wjt := τjt be bounded and monotonic
Wt
(increase or decrease independently one to other). Then for all i = 1, m there exist
p∞i = lim pti
t→∞
and all limit vectors pi∞ coincide with the vector w∞, i.e.
pi∞j = τj∞ ∀j.
W∞
Let us consider the different variants of attractor index T t:
τjt := τjt,min = miin{pitj}, (3)
(4)
τjt := τjt,max = miax{pitj},
(5)
1 n (6)
m
τjt := τ t = pitj ,
j
i=1
τjt1 = τjt2 > 0, j1, j2 = 1, n.
632
APPLICATIONS
classes of PWS. In addition, we discuss how the developed theory may be applied to study
dynamics of strongly nonlinear systems e.g. the van der Pol equation.
Limit states of multi-component discrete dynamical systems
Oksana Satur, oksana@satur.in.ua
Institute of Mathematics of the National Academy of Sciences of Ukraine, Ukraine
Object. We study models of multicomponent discrete dynamic conflict systems with attractive
interaction, which are characterized by a positive value that is called the attractor index. Con-
sider the set of discrete probability measures µi ∈ M1+(Ω) on finite space Ω = {ω1, . . . , ωn},
i = 1, m. Each of these measures µi can be identified with a stochastic vector pi = (pij)jn=1,
where
pij = µi(ωj), i = 1, m, j = 1, n.
Consider the mapping
{pt1, pt2, . . . , ptm} −→,t {pt1+1, pt2+1, . . . , ptm+1}, (1)
which generates multi-component discrete dynamical systems with trajectories (1), where the
coordinates of each vector pit = (pitj)nj=1 are changed according to equations
pitj+1 = 1 ptij(θt + 1) + τjt , t = 0, 1, . . . . (2)
zt
Here θt = θ(p1t , pt2, . . . , ptm) is a finite positive function, T t = (τjt)nj=1 is a vector with non-
negative coordinates (attractor index), and zt = θt + 1 + W t is normalizing denominator,
Wt = n τjt.
j=1
Main results.
Theorem 1. Let all coordinates of vector wt = (wjt )jn=1, wjt := τjt be bounded and monotonic
Wt
(increase or decrease independently one to other). Then for all i = 1, m there exist
p∞i = lim pti
t→∞
and all limit vectors pi∞ coincide with the vector w∞, i.e.
pi∞j = τj∞ ∀j.
W∞
Let us consider the different variants of attractor index T t:
τjt := τjt,min = miin{pitj}, (3)
(4)
τjt := τjt,max = miax{pitj},
(5)
1 n (6)
m
τjt := τ t = pitj ,
j
i=1
τjt1 = τjt2 > 0, j1, j2 = 1, n.
632
