The theory of dynamical systems of conflict in the framework of functional analysis
In this article, we give an introduction to the mathematical setting of problems related to the phenomenon of conflict in terms of constructions in Hilbert spaces. The struggle (conflict, game) between opponents (adversaries, players) will be represented by operator transformations of vectors in Hil...
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Інститут математики НАН України
2023
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| author | Koshmanenko, Volodymyr Кошманенко, Володимир |
| author_facet | Koshmanenko, Volodymyr Кошманенко, Володимир |
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{
"author": "Volodymyr Koshmanenko",
"institution": "Institute of Mathematics of the National Academy of Science of Ukraine"
}
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| description | In this article, we give an introduction to the mathematical setting of problems related to the phenomenon of conflict in terms of constructions in Hilbert spaces. The struggle (conflict, game) between opponents (adversaries, players) will be represented by operator transformations of vectors in Hilbert spaces and probabilistic distributions on the territory of life resources. The phenomenon of conflict as a contradiction between opponents appears in mathematical terms as an intersection the domains of definition for operators and overlapping of corresponding measures. Conflict interaction between opponents in the physical sense is described by the specific transformation of states in a Hilbert space. In turn, this is a mapping that changes the spectral measurements. Thus, a complex dynamical system arises, which we call a dynamical system of conflict. Then the following main problems arise as fundamental questions. What reasonable law of engagement (game or war) should be adopted to resolve the initial intersections? What is a fair limiting distribution of the resource territory? In a more general formulation, solving conflict problems means the detailed describing of all possible outcomes on opponents states of the type: victories, defeats, states of equilibrium, compromises as fixed points together with their basins of attraction. |
| doi_str_mv | 10.3842/trim.v20n1.530 |
| first_indexed | 2026-08-04T01:08:57Z |
| format | Article |
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Збірник праць Ін-ту математики НАН України (2023) т. 20, №1, 843–872
The theory of dynamical systems
of conflict in the framework
of functional analysis
Volodymyr Koshmanenko
Abstract. In this article, we give an introduction to the mathematical set-
ting of problems related to the phenomenon of conflict in terms of construc-
tions in Hilbert spaces. The struggle (conflict, game) between opponents
(adversaries, players) will be represented by operator transformations of vec-
tors in Hilbert spaces and probabilistic distributions on the territory of life
resources. The phenomenon of conflict as a contradiction between opponents
appears in mathematical terms as an intersection the domains of definition
for operators and overlapping of corresponding measures.
Conflict interaction between opponents in the physical sense is described
by the specific transformation of states in a Hilbert space. In turn, this is a
mapping that changes the spectral measurements. Thus, a complex dynam-
ical system arises, which we call a dynamical system of conflict. Then the
following main problems arise as fundamental questions. What reasonable
law of engagement (game or war) should be adopted to resolve the initial
intersections? What is a fair limiting distribution of the resource territory?
In a more general formulation, solving conflict problems means the de-
tailed describing of all possible outcomes on opponents states of the type:
victories, defeats, states of equilibrium, compromises as fixed points together
with their basins of attraction.
Анотація. Стаття присвячена введенню математичної постановки за-
дач, пов’язаних із феномен конфлікту, в термінах конструкцій у гіль-
бертових просторах. Боротьба (конфлікт, гра) між опонентами (супро-
тивниками, гравцями) буде представлена операторними перетвореннями
векторів у гільбертових просторах та ймовірнісними розподілами на про-
сторі життєвих ресурсів. Феномен конфлікту, як суперечність між опо-
нентами, в математичних термінах означає перетин областей визначення
для операторів та перекриття носіїв відповідних мір.
Конфліктна, у фізичному сенсі, взаємодія між опонентами описується
специфічним перетворенням станів у гільбертовому просторі. У свою
чергу, ці відображення змінюють спектральні вимірювання. Таким чи-
ном, виникає складна динамічна система, яку ми називаємо динамічною
The author was partly supported in framework of the project “Mathematical modelling of
complex dynamical systems and processes caused by the state security” (No. 0123U100853)
Keywords: dynamical system of conflict; opponent; Hahn-Jordan decomposition; rigged
Hilbert space; equilibrium
DOI : http://dx.doi.org/10.3842/trim.v20n1.530
843
844 V. Koshmanenko
системою конфлікту. Як основні проблеми постають наступні фунда-
ментальні питання. Який розумний закон конфліктної взаємодії (гри чи
війни) треба прийняти для розв’язання початкових перетинів? Яким має
бути справедливий граничний розподіл території життєвого ресурсу?
У більш загальному формулюванні, розв’язання конфліктних про-
блем означає детальний опис усіх можливих результатів про стани опо-
нентів типу: перемоги, поразки, рівноваги, компроміси, як нерухомі то-
чки, разом з басейнами їх притягання.
Having created the world,
God has appointed for everyone
a place at paradise, but
Devil invented a conflict
1. INTRODUCTiON
1.1. A bit of history. The classical approaches to the conflict phenome-
non and its applications have been discussed in many publications (see, for
example, [5–8], [11], [14–17], [20], [10, 21–23], [27]). Here we shortly recall
only some of well-known relating equations, models and versions:
´ the Malthus-Verhulst population equation describing the dynamics of
internal competition,
dP
dt
= (b´ d)P ´ cP 2,
´ the logistic equation
xn+1 = rxn (1 ´ xn)
which has been used to explain many population phenomena and exis-
tence of different evolution cycles,
´ the Lotka-Volterra equations
Ṅ = aN ´ bNP, Ṗ = ´cP + dNP
with wide spectrum of applications to behavior of hostile essences (for
example, the predator-prey model).
Besides, there are many problems with collision situations reflected in the
game theory (see, for example, [39] “Theory of Games and Economic Be-
havior”, by John von Neumann and Oskar Morgenstern).
1.2. Transition to a new vision. However, it is impossible to understand
the global picture of conflict phenomena on a universal scale, without mov-
ing from classical approaches to posing problems in terms of the modern
theory of dynamic systems, using of Hilbert or Banach spaces, theory of
self-adjoint operators and methods of functional analysis.
Theory of dynamical systems of conflict 845
Today, it is obvious that a wide range of phenomena in the natural
environment, such as, for example, biological populations or the dramatic
evolution of society, can be understood at least partially only with the help
of complex systems theory.
So, in world there are several powerful scientific centers to study of com-
plex systems (of type Santa Fe Institute in Mexico). In this direction the
main mathematical instruments are the non-linear analysis, the theory of
dynamical systems, and the computer modelling. According to the theory
developing in [19, 40] well constructed non-linear dynamical system has
always enough equilibrium states, both repulsive and attractive which sep-
arate the phase space into zones with different regimes of behavior for
trajectories. The equilibrium states here play the same role as eigenvectors
for operator in the linear analysis.
Moreover, we have to use all powerful instruments already developed in
mathematical physics and functional analysis. This transition to using of
contemporary methods is similar to going from classical physics to quantum
mechanics and quantum field theory.
One of the important feature of a new approach to description of results
(instead of construction of deterministic trajectories) is the going to sta-
tistical (probabilistic) interpretation of results. It means that in general
that it is impossible to predict, who will be a winner and how many he
obtains in each single case of the game. Thus, all prediction results of con-
flict interactions will be presented in a form of probabilistic distributions
on infinitely dimensional space. So, we need to use the methods of Hilbert
or Banach spaces.
Further, a notion of the conflict transformation as a some mapping in
the states space may be adequately represented by a specific linear operator
in a Hilbert space. It should correspond to the physical process of conflict
interaction between large (in a real, infinity) amount of alternative sides
(opponents, players, agents).
We start with obvious remark that only two forms of rough interaction
are observed in our living environment, namely, repulsive and attractive.
It follows that the construction of a universal conflict transformation in
the dynamical equations can be based on two operations corresponding to
simple mathematical signs: minus and plus. This is similar to the creation
and annihilation operators in quantum field theory. This is why we have
to look for construction of equations with two basic transformations that
represent the attraction and repulsion in each dynamical conflict model and
that are analogous to the creation and annihilation operators in quantum
field theory. It is important that these equations are necessarily non-linear,
846 V. Koshmanenko
since the value of the terms corresponding to each opponent changes non-
additively at each moment of the conflict game.
Finally, we note that despite the important impact of abstract outcomes
on our understanding of various processes, especially in multi-component
and multi-agent models represented by mean-field games with total payoffs,
from the point of view of concrete applications, the results of an abstract
theory are of little use. In fact, we need to develop a more advanced
universal conflict theory of the type of axiomatic approach in quantum
field theory.
In the next sections, we will consider only the simplest versions of conflict
transformations without taking into account external influences. For more
discussion and results see the constructions in [2], [4], [24]-[25], [28]-[37], [35].
2. THE UNiVERSAL LAW OF CONFLiCT iNTERACTiON
Who should be here,
me or my enemy, that is the question
Here we discuss the mathematical writing of a heuristic version of the
universal law of interaction between alternative opponents. In other words,
we are trying to build a simple mapping that describes the elementary act
of physical collision between abstract adversaries. In the following con-
structions of complex conflict systems, this map will be transformed into a
more convenient and perfect form.
Consider a standard situation. Let A and B be two alternative opponents
living in the common resource space Ω. Alternativeness means that any
interaction between A and B occurs according to the law of mutual repulsion
in the sense probable presence. Therefore, we will use the probabilistic
approach.
Let PA = PA(∆) (PB = PB(∆)) denote probability of presence A (B)
in some disputed region ∆ Ă Ω at the initial moment of discrete time.
It is then natural to expect that any single act of conflict between these
opponents will change these initial probabilities to new ones determined by
simple formulas:
PA
new(∆) = PA(1 ´PB), PB
new(∆) = PB(1 ´PA). (2.1)
The right-hand sides of these equalities contain the products of two val-
ues: the probability of being in the ∆ region for one of the rivals and the
probability of not being in the same region for the second.
We call the law (2.1) a generalized formula of William Shakespeare, con-
sidering PC
new(∆) = PC(1´PC), C = A, B as its usual variant, which was
implemented in the logistic equation.
Theory of dynamical systems of conflict 847
We take this universal law as the basic part of all formulas describing the
conflict struggle. So, in specific models of dynamical systems of conflict,
this law of struggle is consistently repeated until the moment of victory
or defeat, or to a certain kind of balance (compromise), or, finally, to the
achievement of cyclic orbits.
In particular, if continuous time is used, then the above heuristic law of
conflict transformation can be written in the form of a system of nonlinear
differential equations:
d
dt
PA = PA(1 ´PB),
d
dt
PB = PB(1 ´PA).
To clarify this law, we are taking another important step: regionalization
of the resource space of the conflict. Formally, this means the decomposition
Ω to a set of partitioned regions:
Ω =
mď
i=1
Ωi, 2 ď m ď 8. (2.2)
In fact, all real conflict processes take place in some space or territory that
is always there is naturally divided into separate ones parts (we call them
regions) that are separated from each other and in each of which there
are purely local relations between the presence of conflicting parties (oppo-
nents). The structure of such partitions can be significantly different in the
mathematical sense, from Ωi as ideal subsets to manifolds with complex
even fractal supports. In what follows we assume that some decompo-
sition (2.2) are fixed. However, it may change in the course of conflict
resolution similar as it happened when one state intervenes on another.
Next, we assume that some separation (2.2) is fixed. However, this can
change during conflict resolution, such as when one state intervenes in an-
other. Moreover, in models describing the infinite repetition of biological
populations that compete with each other, it is necessary to carry out a
self-similar division of each region Ωi at all moments of time t = 1, 2, . . .:
Ωi = Ωi1 =
mď
i2=1
Ωi1i2 , . . . , Ωi1¨¨¨ik´1
=
mď
ik=1
Ωi1¨¨¨ik , k = t.
Here we will make only one step. Let PA
i ” PA(Ωi, t) and PB
i ”
PB(Ωi, t) denote the independent probabilities of capturing region Ωi by
opponents A and B, respectively, at time t. Then, if we assume the uni-
formity of such distributions into regions Ωi, then the above law takes the
form of a system of 2m differential equations:
d
dt
PA
i = λPA
i (1 ´PB
i ),
d
dt
PB
i = λPB
i (1 ´PA
i ), i P 1,m,
848 V. Koshmanenko
where λ is a normalization factor.
In what follows we use discrete time, so instead of the above differential
equations let’s move on to the differences:
pt+1
i = λpti(1 ´ rti), rt+1
i = λrti(1 ´ pti), t = 0, 1, . . . , (2.3)
where the notation is entered, pti := PA(Ωi, t), rti = PB(Ωi, t). Since we
use a statistical approach, vectors pt = (pt1, . . . , p
t
m), rt = (rt1, . . . , r
t
m) are
stochastic for each t:
mÿ
i=1
pti = 1 =
mÿ
i=1
rti .
It is not difficult to notice that the normalization coefficient λ in (2.3)
depends on time and has the form
λ = 1/zt, zt = 1 ´ θt, θt := (pt, rt) =
mÿ
i=1
ptir
t
i .
Difference equations (2.3) establishes the simplest probabilistic law for
conflict transformation which we will denote as ‹. Models of dynamical
conflict systems
tp0, r0u ‹, tÝÝÑ tpt, rtu, t = 1, 2, . . .
generated by the difference system equations of the type (2.3), has already
been studied in a number of publications (see [28–34, 36, 37]). One of our
main ones results confirm the convergence of all trajectories of the dy-
namic systems described above to equilibrium states. We call this result
the conflict theorem. It can be formulated as follows (see [32] and references
therein).
Theorem 2.1. Each trajectory tpt, rtu of CDS generated by the system of
equations (2.3) starting with of an arbitrary point tp0, r0u given by a pair of
stochastic vectors p0, r0 which are different, (p0, r0) ‰ 1, converges to the
limit state (fixed point),
tpt, rtu ÝÑ tp8, r8u, t ÝÑ 8,
which consists with two orthogonal vectors, p8 K r8. That is,
´ if at the initial moment of time the inequality p0i ą r0i was fulfilled for
some coordinates, then
p8
i ą 0, r8
i = 0,
´ if p0k ă r0k, then
p8
k = 0, r8
k ą 0,
´ and if p0j = r0j , then
p8
j = r8
j = 0.
Theory of dynamical systems of conflict 849
Values of non-zero boundary coordinates p8
i ą 0, r8
k ą 0 are proportional
to the initial differences di = p0i ´ r0i , dk = r0k ´ p0k, i.e.
p8
i = di/D, r8
k = dk/D,
where the proportionality coefficient D is independent of indices of non-zero
coordinates.
Thus, the struggle of opponents for possession of regions in space (2.2)
is only able to redistribute the initial values probabilities of their presence
in different regions in accordance with the law alternative conflict of the
form (2.3). That is, the dominance of one of the players in some Ωi leads to
the disappearance of another in the same region. Presence is redistributed
in such a way that opponents are located in different regions that do not
overlap. Therefore, all trajectories of the dynamic system (2.3) converge
to the equilibrium states given by orthogonal vectors in the sense of Rm if
m ă 8 or in the Hilbert space l2 if m = 8. Any compromise states in pure
alternative dynamics are impossible.
3. CONNECTiON WiTH PERTURBATiON THEORY
Perturbation theory in mathematical physics provides a powerful tool for
its use in conflict theory. First, let’s briefly recall the background.
3.1. Perturbed operators and their scattering. Let a Hilbert space
H be the state space of some physical system and H be its free Hamil-
tonian. For example, H = L2(R3), and H = ´∆ – the Laplace operator.
Then, according to quantum mechanics, the dynamics is described by the
Schrödinger equation. Its solutions are written by vector functions of the
form Ψ(t) = exp´itH Ψ(0), which describe the time evolution of the trajec-
tories, starting from states Ψ(0) P H.
Let us now consider two perturbations V1 and V2 of H, which can be con-
sidered as the influence of alternative opponents on free evolution. Suppose
both sums, Hk = H + Vk, k = 1, 2, are well-defined self-adjoint operators
of H. Then, again according to the abstract Schrödinger equation
i
d
dt
Ψ(t) = HkΨ(t),
there appear two different time evolutions:
Ψ1(t) = e´itH1Ψ(0), Ψ2(t) = e´itH2Ψ(0).
The physical collision between such different behaviors is described in
perturbation theory by the so-called scattering operator
S =W+(W´)˚,
850 V. Koshmanenko
where
W˘ = lim
tÑ˘8 e
itH2e´itH1
is defined as the wave operators [26]. The existence problem of the wave
and scattering operators are non-trivial and has long history. Especially in
the case of singular perturbations (see, for example, [33]).
Here we want to look at Hk as operator strategies that correspond to
the alternative behavior of a certain entity (population, political ideolo-
gies, opinions, etc.). Then Ψk(t) represent its independent time evolutions
with the initial state Ψ(0). But instead of the collision in the above form
of the scattering operator, we propose to find a new description of the con-
frontation between opposite sides in the form of a discrete-time sequence
of acts of redistribution for the initial positions in the life space according
to the universal nonlinear rule.
3.2. Infinite-dimensional space of living resources. Thus, we need to
define the conflict composition ‹ in terms of vector-states of Hilbert space,
Ψ1(t) ‹ Ψ2(t) =?, t = 0, 1, 2, . . .
The transformation ‹ will correspond to interaction between alternative
sides for presence in the living (resources) space H.
One of the way is to engage an observation the position operator Q. Its
spectrum, Ω := σ(Q), we will treat as the living (or resources) space for
alternative opponents. Then the independent time evolutions in such space
may be described in terms of probability measures
µtk(∆) := (EQ(∆)Ψt
k,Ψ
t
k), ∆ P B,
where B is the Borel algebra of subsets on Ω, and EQ(∆) denotes the
operator spectral measure (the identity resolution of operator Q).
It is worth recalling the physical interpretation. According to the princi-
ples of quantum physics (see, for example, J. von Neumann “Mathematis-
che Fundamentals of Quantum Mechanics”) the values µt1(∆), µt2(∆) can be
considered as the independent probabilities of finding opposite sides repre-
sented by states Ψi in the subset ∆ Ď Ω. In other words, its are the math-
ematical expectations to observe (to measure) the presence of opponents
with strategies H1, H2 in states Ψt
1,Ψ
t
2 restricted to subspace EQ(∆)H.
The conflict interaction causes a certain deformation of these indepen-
dent expectations.
Now we will explain the idea for construction of the mathematical trans-
formation ‹ corresponding to a conflict interaction in a Hilbert space.
We suppose that each conflict dynamics has its own time, continuous or
discrete, in general different from the independent (free) time evolutions of
physical systems. Here we develop an abstract approach to definition of the
Theory of dynamical systems of conflict 851
conflict transformation. Let µti, νti denote a couple of probability measures
of above type on the measurable space (Ω = σ(Q),B). Putting
Pt
µ1
(∆) := µt1(∆), Pt
µ2
(∆) := µt2(∆), ∆ P B,
we define conflict transformation in terms of these measures:
"
µt1
µt2
*
‹, tÝÝÝÑ
#
µt+1
1
µt+1
2
+
(µ01 = µ1, µ
0
2 = µ2),
where we use discrete time t = 0, 1, . . .. At each step, the measures are
modified using a generated William Shakespeare type formula (2.1):
µt+1
1 (∆) = Pt+1
µ1
(∆) » Pt
µ1
(∆) ¨ (1 ´Pt
µ2
(∆)),
µt+1
2 (∆) = Pt+1
µ2
(∆) » Pt
µ2
(∆) ¨ (1 ´Pt
µ1
(∆)).
The inverse problem, i.e., the reconstruction of vectors Ψt
1, Ψt
2 on the mea-
sures µt1, µt2 will be considered in further sections.
3.3. Singular perturbation as a cause of conflict.
Conflict is caused
by the singular structure of matter
Here we show that alternative behavior strategies corresponding to the
above operators Hk arise naturally under a singular perturbation of the
free Hamiltonian.
Briefly, the splitting of unity (free Hamiltonian) into contradiction sides
(Hk operators) can be described within the framework of singular pertur-
bation theory.
In our approach, the phenomenon of conflict between alternative parties
looks like a kind of explosion of free evolution, ”exit from paradise”. Math-
ematically, this means splitting the free Hamiltonian H on two or more
branches of conflicting evolution. We associate this path with the singular
perturbation H. But first we mention the usual approach again.
Let us consider a free energy operator H with domain D(H) in a Hilbert
space H. Let
D = tΨ P D(H), }Ψ} = 1u
denote some initial set of states for the physical system associated with
operatorH. We note that in general in real situation, each concrete physical
system involves in the “life” not all vectors from D(H). So, D is only a part
of D(H), but it is assumed that D is a dense subset in H. Each vector Ψ P S
has a pure deterministic free time evolution described by the Schrödinger
equation, Ψ(t) = exp´itH Ψ.
852 V. Koshmanenko
If the physical influence on the system is described as a “rough” pertur-
bation written in the form H1 = H+V1, where V1 is a fairly good operator
that has an interpretation of the external field, then the problem is to de-
scribe the perturbed picture of the new evolution, including the spectral
analysis of H1. Moving on to another perturbation V2, we get some new
evolution picture, Ψ(t) = exp´itH2 Ψ evolution. There is no conflicting
phenomenon on this way.
Further we need in the following
Definition 3.3.1 ([33]). A self-adjoint operator rH is called (purely) sin-
gularly perturbed with respect to H if the linear set
DΓ =
␣
f P Dom(H) X Dom( rH) | Hf = rHf
(
is dense in the Hilbert space H.
Here, Γ is associated with some extremely small set in physical space
that is responsible for the singular perturbation.
For more detailed facts connected with definition of singular perturbation
see [33]
Formally every singular perturbed operator may appear in the following
way. At first one consider a restriction of H into some dense linear subset
DΓ = D Ď D(H) with consequent extension to any a new self-adjoint
operators Hi such that:
HiæDΓ = HæDΓ, i ě 2.
We will always assume that Di = DΓ X D(Hi) are dense in H.
Now, the evolution of the physical system associated with different oper-
ators Hi will have a certain kind of uncertainty because these operators are
quite close (they are identical on a dense set) but still different (due to a sin-
gular perturbation on a very small set Γ). Hence, opposite and conflicting
paths may arise for the evolutions started from the vectors Ψ P XiDi. The
theory of the dynamic system of conflict is designed to give a description
of this kind of evolution and to solve the problem of “fair” redistribution of
the conflict territory Ω = σ(Q) which the spectrum of Q.
To this aim, we need to select and use an explicit law of conflict interac-
tions that will govern the time dependence of the trajectories of the conflict
dynamical system.
3.4. The law of conflict interaction in terms of states. Here we will
describe one of the possible variants of conflict dynamics in terms pairs of
non-orthogonal states Ψ P D1, Φ P D2 which correspond to two conditional
opponents. It will be performed by a composition marked ‹ and operating
in the Hilbert space H.
Theory of dynamical systems of conflict 853
Let Q denote the self-adjoint operator in H corresponding to the obser-
vation, which is called a position. Its spectrum represents the resource area
for all other observations.
Put in correspondence to a couple Ψ, Φ their spectral representations
with respect the operator Q:
Ψ Ñ µ(∆) := (EQ(∆)Ψ,Ψ),
Φ Ñ ν(∆) := (EQ(∆)Φ,Φ), ∆ P B,
where EQ(∆) stands for the operator spectral measure of Q and B denotes
the Borel σ-algebra. In what follows we denote this map by symbol K.
These probability measures µ, ν we interpret as two starting indepen-
dent distributions for a couple of some opponents (individuals) along the
territory Ω = σ(Q) where σ(Q) denotes the spectrum of Q. If the states of
the opponents are orthogonal, Ψ K Φ, then the measure carriers µ, ν have
a zero intersection and there is no conflict between the opponents. But if
supp(µ) X supp(ν) ‰ ∅,
then the opponents start to struggle (conflict interaction) with the aim of
displacing each other from the territory of joint coexistence.
The simplest version for the evolution law ‹ of opponent sides,
"
µ0
ν0
*
‹, tÝÝÝÑ
"
µt
νt
*
, (µ0 = µ, ν0 = ν), t ě 0,
may be described by the nonlinear equations of a view:
d
dt
µt » Θtµt ´ ηt,
d
dt
νt » Θtνt ´ ηt, (3.1)
where Θt = Θ(µt, νt) := (HΦt,Ψt) stands for the multiplicative Hamilton-
ian of the system and ηt = ηt(µ, ν) denotes the so-called conflict occupation
measure. Here we used an isometric correspondence
K : H Q Ψt ÐÑ µtΨ ÐÑ ψ(t) P L2(Ω, dEQ(x)), x P Ω (3.2)
(ψ(t) is the density of µtΨ with respect to the spectral measure of Q) which
is constructed on the basis of the spectral theorem for the operator Q
(see [9] for details). In fact formulas (3.1) are analogies of the products
Pt
µ(∆) ¨ (1 ´Pt
ν(∆)) from the above Shakespeare’s formula.
Now we need to give some explanation about the concept of the conflict
occupation measure in the abstract situation. Let µ, ν be a pair of positive
measures on (Ω,B). Of course, we can assume that as above
µ(∆) = (EQ(∆)Ψ,Ψ)H, ν(∆) = (EQ(∆)Φ,Φ)H
with the assumption that supp(µ) X supp(ν) ‰ ∅.
854 V. Koshmanenko
Suppose that the conflict territory Ω is separated into a finite amount of
regions: Ω =
Ť
Ωi. Then we define the conflict occupation (intervention)
measure for the starting couple tµ, νu as η := ην + ηµ, where
ην(∆) := Varµ(ν) = sup
∆=Y∆i
ÿ
i
χω(∆i)ν(∆i), ∆,∆i P B,
χω(∆i) =
#
1, if ω(∆i) ě 0,
0, otherwise
and
ηµ(∆) := Varν(µ) = sup
∆iĎ∆
Ş
Ωi
ÿ
i
χ´ω(∆i)µ(∆i),
χ´ω(∆i) =
#
1, if ´ω(∆i) ě 0,
0, otherwise
where ω := µ´ ν is the signed measure associated with µ, ν. In particular,
if ∆ is a set of absolute domination for µ:
µ(∆1) ě ν(∆1), @∆1 Ď ∆,
then ην(∆) = ν(∆). And similarly for ηµ, if ν absolute dominates on some
∆, i.e.,
ν(∆1) ě µ(∆1), @∆1 Ď ∆,
then ηµ(∆) = µ(∆). Thus, the value ην(∆) estimates the “intervention
strength” of opponent for µ on a set where it has absolute dominance,
and vice versa, ηµ(∆) has a similar but opposite meaning, it evaluates the
strength of occupation of another opponent which is represented by µ on a
set ∆, where now ν has absolute dominance.
It should be noted that in general the conflict occupation measure is
not probabilistic, i.e., η(Ω) ă 1. In addition, we can say that conflict
confrontation is very weak whenever η(Ω) „ 0 and extremely strong if η(Ω)
is close to 1.
We propose some illustrative example: ην(∆) may estimate how many
English-speaking persons lives in Ukraine, while ηµ(∆) gives values of
Ukrainian-speaking persons there are in some fixed region ∆ in the USA.
These values refer to the initial time t = 0 and will change according to the
conflict dynamics in a form (3.1).
Theorem 3.5 (Theorem of conflict in terms of states in a Hilbert space).
Let states of a couple opponents at a time moment t = 0 are given by two
unite non-orthogonal vectors Ψ,Φ P H. Using the mapping (3.2) put in
Theory of dynamical systems of conflict 855
correspondence to Ψ,Φ measures µΨ, νΦ. Assume that the set
supp(µΨ)
č
supp(νΦ) ‰ ∅.
Then the trajectory of the conflict dynamical system
"
Ψ0
Φ0
*
‹, tÝÝÝÑ
"
Ψt
Φt
*
, (Ψ0 = Ψ, Φ0 = Φ), t ě 0,
with the conflict composition ‹ generated by equations of type (3.1) in terms
of measures µΨ(¨) = (EQ(¨)Ψ,Ψ), νΦ(¨) = (EQ(¨)Φ,Φ), converges to a fixed
point tΨ8,Φ8u. Thus, there exist the limits in the strong sense in H
Ψ8 = lim
tÑ8Ψt, Φ8 = lim
tÑ8Φt.
That is
Ψ8 K Φ8
and
supp(µΨ8)
č
supp(νΦ8) = 0.
Proof. Here we give only some sketch of our arguments.
At first, we come from equations (3.1) to its difference variants, i.e., to
the conflict dynamics at the discrete time
"
Ψ0
Φ0
*
‹, tÝÝÝÑ
"
Ψt
Φt
*
, t = 0, 1, . . .
where each couple Ψt,Φt is defined by the iteration procedure in accordance
with dynamics of the associated measures:
µt+1
Ψ (∆) =
µtΨ(∆)(1 + Θt) ´ ηt(∆)
1 + Θt +W t
,
νt+1
Φ (∆) =
νtΦ(∆)(1 + Θt) ´ ηt(∆)
1 + Θt +W t
,
(3.3)
where ∆ P B, Θt = (HΨt,Φt), and W t = ηt(Ω).
Further, for proving of the limiting measures
µ8
Ψ = ω+/zµ, ν8
Φ = ω´/zν ,
we use a suitable version of arguments (see, for instance [32]), where ω+/zµ,
ω´/zν denote the normalized components of the Hahn-Jordan decomposi-
tion for signed measure ω = µΨ´νΦ = ω++ω´. And finally, we come back
from L2(Ω, dEQ(x)) to the Hilbert space using the inverse transformation
K´1:
Ψ8 = K´1µ8
Ψ , Φ8 = K´1ν8
Φ . □
856 V. Koshmanenko
A more general theorem is also true if we modify the conflict interaction
law (3.3). Namely, instead of Θt = (HΨt,Φt), we take Θt
1 = (H1Ψ
t,Φt) and
Θt
2 = (H2Ψ
t,Φt), where H1, H2 is a pair of singularly perturbed operators.
They arise as a pair of self-adjoint extensions of the symmetric restriction
H to the domain DΓ, where Γ Ă Ω is the zero set with respect to the
spectral measure of the operator Q:
ż
Γ
}ϕ(x)}2dEQ(x) = 0, @ϕ P L2(Ω, dEQ(x)).
3.6. The method of rigged spaces. Quite often, the conflict struggle
arises between rather close, almost identical living conceptions. This kind
of proximity can be accurately described in terms of singular perturbations
of the general free Hamiltonian. Then, as in the real situation, each ad-
versary will have its own Hamiltonian, which determines the strategy of its
evolution in time.
Let us describe this approach in more details.
Let H be a strongly positive self-adjoint operator in a Hilbert space H
and
H´ Ą H Ą H+ = DomH
denotes the associated rigged (equipped) space (for details see [9]). Here Ą
stands for the dense inclusion. This rigged space is only some part of the
so-called H-scale of Hilbert spaces,
H´k Ą H = H0 Ą H+k = DomHk/2, k ą 0.
The concept of a singular perturbation first appears in physical consid-
erations (see [3]) related to the problem of the expression
´∆+ λδ, λ P R1,
where ´∆ is the Laplace operator, and δ stands for the Dirac delta func-
tion treated as a singular one-point potential. The corresponding linear
functional
lδ(φ) = xφ, δy :=
ż
R3
δx0(x)φ(x)dx = φ(x0), φ P C(R3)
is singular in L2(R3, dx) since its null set ker(lδ) creates a dense domain
in L2. In an abstract approach, the singularity property was extended to
quadratic forms in the rigged spaces.
Definition 3.6.1 ([33]). A positive quadratic form γ(¨, ¨) on H+ is called
singular in H if for each Ψ P H there exists a sequence φn P H+ such that
φn Ñ Ψ and γ[φn] = γ(φn, φn) Ñ 0 as n Ñ 8.
Theory of dynamical systems of conflict 857
It is clear that a quadratic form γ is singular if its null set
ker(γ) = tφ P H+| γ[φ] = 0u =: D0
creates a dense domain in H, i.e. Dcl
0 = H (cl = closure). Thus, γ contains
practically unobservable physical information.
Despite this “almost zero information”, singular quadratic forms can exert
a strong physical influence on the H Hamiltonian. In particular, singular
quadratic forms can carry singular perturbations H, which significantly
change the behavior of evolutions in time.
Let γ be fixed and its null set ker(γ) = D0 be dense in H. Then the
restrictions
H := H|D0
defines some symmetric operator in H. We assume that its deficiency in-
dices are equal and nonzero:
n+(H) = n´(H) ‰ 0.
The family of its self-adjoint extensions
t rH = rH˚ | rH|D0 = HD0u
represents all possible candidates for a singular perturbed operator. The
resolvent of each operator rH allows an explicit construction according to
the so-called Krein’s formula. We will give only the most famous sample of
Krein’s formula:
rH´1 = H´1
F + rBPN0 ,
where PN0 denotes the orthogonal projector onto the defect subspace N0
of H, and rB represents the extension parameter. It is some self-adjoint
operator in N0. Above HF denotes the Friedrichs extension of H which
often coincides with H. Depending on the operator rB, the corresponding
singular perturbed extension rH can have many new spectral properties in
comparison with the original H.
We note that each operator rB is closely connected with singular quadratic
form γ. Since it is usually assumed that γ is continuous on H+ there exists
the associated bounded operator Bγ : H+ Ñ H´. And all rB in fact is
also obtained using some singular quadratic form γB in the rigged Hilbert
space.
It is important that all operators rH are the same on the dense in H
domain D0. So, one able to consider the conflict dynamical system with
many, m ě 2, opponents. Every of them will have its own strategy of
behavior in a form of the operator Hi. Thus, for any family of self-adjoint
extensions tHiumi=1 defined by the above formula,
H´1
i = H´1
F + rBiPN0 , (3.4)
858 V. Koshmanenko
we have
HiΨ = HΨ, Ψ P D0.
This fact causes a certain uncertainty of the evolution in time of arbi-
trary vectors Ψ P H, if they are interpreted as the states of some biological
system. Since the singular perturbation is located on a very small, in the
physical sense, set, we denote it Γ (for example, at a single point, as a delta
potential, or on a fractal that has zero Lebesgue measure), it is difficult
to take into account, it is not visible. From the point of view of biological
essence, this kind of violation does not exist. But nevertheless, different
elements of the biological population under the influence of a singular influ-
ence choose different strategies of behavior corresponding to Hamiltonian
among the set t rHu. Thus, any singular perturbation can be interpreted as
the cause of the splitting of the behavior of the biological population into
different branches of evolution over time with subsequent confrontational
struggle between them.
So, if we fix some family of self-adjoint extensions tHiumi=1, m ě 2 and
associate to each Hi some kind of “society” (as a linear set of vectors Di
from H), then its evolution admits description similar to the previous ones
for a couple of players. Thus, we have the Theorem of conflict for a complex
system with many opponents fighting each one against all others.
Theorem 3.7. For Ψi P H, i P 1,m, put µi(¨) = (EQ(¨)Ψi,Ψi) and de-
fine the “mean field” measures νi(¨) = 1
m´1
ř
k‰i µk. Consider the conflict
dynamical system with trajectories
tµ01, . . . , µ0mu ‹, tÝÝÝÑ tµt1, . . . , µtmu, t ě 0
produced by formulas
d
dt
µti = Θt
iµ
t
i ´ ηti , µ0i = µi, (3.5)
where Θt
i := (H´1
i Ψt
i, 1/(m ´ 1)
ř
k‰iΨk) with Hi defined by the Krein
formula (3.4) and ηi = ηµi + ην̃i is the occupation measure for a couple
µi, νi. Assume that all differences
Dii1 = rBi ´ rBi1 , i, i1 P 1,m
are compact operators. Then, under some pure technical additional as-
sumptions on the family µi, for each ∆ P B there exist limits
µ8
i (∆) = lim
tÑ8µ
t
i(∆), µ8
i K µ8
i1 , i ‰ i1.
That is,
µ8
i = ω+
i /zωi ,
Theory of dynamical systems of conflict 859
where ω+
i /zωi denotes the normalized positive component of the Hahn-
Jordan decomposition for family of signed measures ωi = µi ´ νi.
In particular, for the vectors Ψt
i := K´1µti we have convergence in the
strong sense:
lim
tÑ8Ψt
i = Ψ8
i ,
where all vectors Ψ8
i := K´1µ8
i are orthogonal in H.
One may consider the family of the limiting vectors tΨ8
i umi=1 as the
equilibrium state for the starting conflict system.
The proof of this theorem faces new difficulties due to the fact that
in (3.5) the functional Θt
i is not the same for all components, but depends
on Hi. Besides, it is necessary to coordinate the procedure of the Hahn-
Jordan decomposition for family of signed measures ωi. In application this
decomposition means the separation of the common living territory between
alternative players.
We recall that according to the well-known Hahn-Jordan theorem [13],
there exist two kinds of decomposition connected with a signed measure
of a view ωi = µi ´ νi. Namely, for the set, Ωωi = supp(ωi) and for
ωi, ωi = ω+
i ´ ω´
i , where measures ω+
i and ω´
i are orthogonal. The first
decomposition has a form
Ωωi = Ωω+
i
ď
Ωω´
i
,
where Ωω˘
i
= supp(ω˘
i ). That is
Ωω+
i
č
Ωω´
i
= ∅,
and where the positive and negative components ω+
i , ω
´
i are uniquely defi-
nition as follows,
ω+
i (∆) = V ar+(ωi,∆) := sup
∆1Ď∆
ωi(∆
1),
ω´
i (∆) = V ar´(ωi,∆) := ´ inf
∆1Ď∆
ωi(∆
1), ∆1,∆ P B.
This theorem states that
µti(∆) Ñ µ8
i (∆) ě 0, if ∆ Ď Ωω+
i
,
µti(∆) Ñ 0, if ∆ Ď Ωω´
i
.
The set
ΩΨ1,...,Ψm =
ď
i
supp(ωm
i=1)
860 V. Koshmanenko
naturally to treat as the common living territory for m players associated
with vectors Ψ1, . . . ,Ψm. Due to conflict transformation,
Ωω+
i
= supp(ω+
i ) = supp(µ8
i ),
and therefore the set ΩΨ1,...,Ψm was separated into new family of regions
without intersections:
ΩΨ1,...,Ψm =
mď
i=1
Ωω+
i
=
ď
i
supp(µ8
i ).
Thus, the conflict is resolved since there appears the equilibrium state
Ψ8
1 , . . . ,Ψ
8
m described in terms of the associate limiting measures:
$
’&
’%
µt1
...
µtm
,
/.
/-
‹, tÝÝÝÑ
$
’&
’%
µ8
1 = ω+
1 /zµ1
...
µ8
m = ω+
m/zµm
,
/.
/-
, µ8
i K µ8
i1 , i ‰ i1
where
supp(µ8
i ) = Ωω+
i
.
The last equality shows that every measure µi8 is concentrated in region
of the initial absolute domination of µi over all µi1 , i1 ‰ i.
On this way we may consider a model of an abstract society represented
by two conflict clusters of vectors S = S1
Ť
S2. Each subsystem S1, S2
contains a finite number, m1,m2, of players represented by unite vectors
Ψi P S1 Ă D1 Ă D(H), Φk P S2 Ă D2 Ă D(H).
Then there appear two clusters of the associated probability measures:
µi(¨) = (EQ(¨)Ψi,Ψi),Ψi P D1, νk(¨) = (EQ(¨)Φk,Φk),Φk P D2
and their “mean field” alternative variants
µ̃i = 1/m2
ÿ
k
νk, ν̃k = 1/m1
ÿ
i
µi.
Theorem 3.8 (Conflict between two clusters of opponents). All trajecto-
ries of the conflict dynamical system
tµ01, . . . , µ0m1
, ν01 , . . . , ν
0
m2
u ‹, tÝÝÝÑ tµt1, . . . , µtm1
, νt1, . . . , ν
t
m2
u, t ě 0,
generated by formulas of type
d
dt
µti = Θtµti ´ ηtν ,
d
dt
νtk = Θtνtk ´ ηtµ,
converges to a fixed point (an equilibrium state):
tµ8
1 , . . . , µ
8
m1
, ν8
1 , . . . , ν
8
m2
u, µ8
i K ν8
k , @i, k.
Theory of dynamical systems of conflict 861
3.9. Towards physical examples. LetH denote the Hamiltonian of some
physical system of some elements (particles) with +/´ charge or +1/ ´ 1
sign of spines.
Then subspaces Di Ă D(H), i = 1, 2 differ due to chosen priority with
respect to positive-negative charges or right-left sign of spins.
At starting time moment there are many vectors Ψi,Φk with the mixed
distribution along an above physical property. The conflict interaction bet-
ween possessing to one of them produces the separation all mixed states
into two orthogonal subspaces:
H = H1 ‘ H2.
The close pictures appear in the Ising model describing the behavior of
nuts in a lattice of magnetic dipoles with +1 or ´1 spins, as well as in po-
tential theory which deals with the existence problem of minimizing signed
measures supported on a condenser A = A+
Ť
A´.
Else one example gives a model of quantum harmonic oscillator. Let
H = L2(R1, dx), H = 1/2m(Q2 + P 2), Q » x, P » id/dx.
The operator H has the discrete spectrum:
Hei = λiei, λi = (i+ 1/2)hω, ω is the oscillation phase,
where
ei(x) » exp(´x2/2)Hi(x), Hi(x) are Hermite polynomials.
In the simplest case we consider the system tΨ,Φu with two unit vectors,
Ψ,Φ P D(H):
Ψ =
8ÿ
i=0
aiei, Φ =
8ÿ
i=0
biei, ai, bi P C,
}Ψ} =
ÿ
i
pi = 1 =
ÿ
i
ri = }Φ}, pi = |ai|2, ri = |bi|2
The conflict interaction between Ψ,Φ deforms their free dynamics given by
the Schrödinger equation. The problem of “right” redistribution of starting
priorities along the spectrum we write in a symbolic form asΨt‹Φt? In other
words we regard Ψ,Φ as opponents whose projection weights (amplitudes)
pi, ri on ei show their priority relations with respect to eigenvalues λi.
Let us separate all basic oscillators ei into two subsets DΨ,DΦ using the
priority domination produced by Ψ and Φ:
ei P DΨ, if i P NΨ = ti | pi ě riu,
ek P DΦ, if k P NΦ = tk | rk ą pku,
862 V. Koshmanenko
where recall, pi (rk) is a probability to find Ψ (Φ) in a pure state ei, (ek).
Now we are able to define the occupation discrete measure η(Ψ,Φ) = η:
η = (ηi)
8
i=0, ηi =
#
pi, if pi ď ri
ri, if ri ă pi
We assert the existence of an equilibrium limiting state for such conflict
dynamics at discrete time. Namely, by the Theorem of conflict there exist
the limits
lim
tÑ8 p
t
i = p8
i , lim
tÑ8 r
t
k = r8
k ,
such that p8
i = 0, i P NΦ, r8
k = 0, k P NΨ. The system tΨ8,Φ8u with the
limit vectors
Ψ8 =
ÿ
iPNΨ
a8
i ei, Φ8 =
ÿ
kPNΦ
b8
k ek
creates a fixed point, Ψ8 K Φ8. It is easy to see that this equilibrium state
is extremely unstable.
4. CONNECTiONS WiTH POTENTiAL THEORY
In this section, we are going to substantiate our idea about the connection
between the well-known Gaussian minimization problem in potential theory
and the existence of a limit state of equilibrium in conflict theory. In other
words, we want to show that the minimizing measure in the potential theory
as the equilibrium charge on the capacitor essentially coincides with state
of compromise redistribution in dynamical system of conflict (DSC). To
formulate our goal in more detail, we need additional preparations.
But at first we recall shortly some facts from the classical theory of
capacities of compact sets [12,38] following the papers [41–45]. This theory
was initiated by Wiener and developed by many scientists, we remind only
Frostman, Riesz, de la Vallee-Poussin. The modern notion of inner and
outer capacities was originated by Cartan. He observed that the cone of all
positive measures on R3 with finite Newtonian energy is complete in the
energy norm. The using of the strong and the so-called vague topologies
enabled Fuglede to extend a theory of capacities for measures on a locally
compact space X for positive definite kernels κ on X. Ohtsuka developed
this approach for vector-valued Radon measures µi, i P I on X, where
dim I ď 8.
The last fact is important for application in theory of dynamical systems
of conflict when we want to study the models with an arbitrary amount of
opponents associated with vectors Ψi, i P I in a Hilbert space H of type
L2(Ω, dEQ(x)) with Ω = X Ď Rn where Q is so-called the position operator.
Theory of dynamical systems of conflict 863
In [45] was developed and investigated the theory of inner capacities and
inner capacitary measures and proved a series of theorem on convergence
of above measures and their potentials for monotone families of sets. We
used some results from [45] to prove Theorem 4.1 (see below).
Let M,M+ denote the sets of all signed and positive measures on X,
and M+
1 (X) denotes the set of all probability measures on X.
Let G(x, y), x, y P X be a positive definite kernel on a locally compact
Hausdorff space X. For a given kernel G the function
Uµ
G(x) ” G(x, µ) :=
ż
G(x, y)dµ(y), µ P M
is called the potential of a measure µ and the value
EG[µ] ” G(µ, µ) :=
żż
G(x, y)dµ(x)dµ(y)
is its energy.
Consider a compact set K Ă X. Given G and a compact set K Ă X,
the value
c(K) ” capG(K) :=
[
inf tEG[µ] : µ P M+
1 (X), µ(K) = 1u]´1
is called [1, 12,38] the interior capacity of a set K with respect to G.
The capacity of the set can be defined in another way [1] in terms of
smooth functions,
c(K) := inf
␣}φ}2H+
: φ P C8
0 , φ(x) ě 1 on K
(
,
where H+ means a positive Hilbert space built on a given kernel G (in fact,
it is some Sobolev space).
One of main results of the potential theory asserts that under rather
wide assumptions on a set K there exists the so-called equilibrium measure
γ P M+ supported on K such that
c(K) = EG[γ] = }γ}2H+
and
Uγ
G(x) = 1, @x P K,
where } ¨ }H+ denotes the norm in the Hilbert space associated with G.
Further we assume that X Ď Rn and that a kernel G is perfect in the B.
Fuglede’s sense [18]. In particular, G(x, y) is a symmetric and lower semi-
continuous. So, using the notation from [45], we may put G(x, y) = κ(x, y),
where κ(x, y) denotes one of the perfect kernels of type:
´ Newton
κ(x, y) = |x´ y|2´n, n ě 3,
´ Riesz
κ(x, y) = |x´ y|α´n, 0 ă α ă n,
864 V. Koshmanenko
´ or general Green ones
κ(x, y) = gΓ(x, y), Γ Ă Rn.
For given strictly positive definite kernel κ on X we define the Hilbert
space Hκ ” H+ by the standard procedure of compactification and fac-
torization of the linear space from signed measures M(X) with the inner
product
(ω1, ω2)Hκ := κ(ω1, ω2)
and the norm }ω} =
a
κ[ω].
According to [45] there are various ways for solution for “the problem
of minimizing energy integrals over various unite charge distributions” on
a set K with a presence of an extreme field. This problem is frequently
referred to as the Gauss variational ones.
In particular, for arbitrary K Ă Rn and the kernel κ denote by M+(K)
the cone of all positive measures µ concentrated on K and which have finite
energy Eκ[µ]. It is known that this cone is strongly complete in the norm of
Hκ. Then for any K Ă Rn with finite inner capacity c(K), there exists the
equilibrium measure γK which is uniquely determined by the two relations
κ[γK ] = }γK}2Hκ
= c(K),
κ(x, γK) = 1 on K.
It is remarkable that similar kind of the result is true for a set K which
is replaced by the condenser of a view Ω = Ω´
Ť
Ω+ and instead of positive
measures µ need to take signed measures ω. In the simplest reading, the
solution of the minimizing problem meant that for wide kind of sets of type
a condenser with finite κ-capacity there exists the signed measure
γ = γ+ ´ γ´ P M(Ω), (4.1)
γ+ P M+(Ω+), γ´ P M+(Ω´), such that
Eκ[γ] = κ(γ, γ) = cκ(Ω), κ(x, γ) = 1 γ ´ almost everywhere.
Here cκ(Ω) denotes a capacity of the condenser Ω with respect a given
kernel κ(x, y).
Our hypothesis is that the minimizing measure γ may be constructed as
the limiting distribution in the DSC. In a symbol we are going to write (see
below Theorem 4.1)
µ8
+ = γ+, µ8́ = γ´,
with γ+ and γ´ defined as follows:
lim
tÑ8tµtA ‹ µtBu = tµ8
+ , µ
8́u,
Theory of dynamical systems of conflict 865
where ‹ denotes the conflict interaction in the space of probability measures
on Ω. In other worlds it means that the measures µtA, µtB corresponds to
two alternative opponents A,B after the conflict interaction at the time
moment t creates a sequence of signed measures ωt which converges to the
minimizing measure γ under a fixed condenser Ω. That is, the conflict
interaction mapping ‹ have to be constructed with using the kernel κ, and
we extend our considerations to construction of the DSC in the Hilbert
space Hκ with the inner product
(ω1, ω2)Hκ :=
ż
κ(x, y)d(ω1 b ω2)(x, y)
and the norm
}ω}Hκ =
a
κ(ω, ω).
So we formulate our intentions in the form of a theorem, although we do
not have a complete proof of it today.
Theorem 4.1. Let H,Q be a self-adjoint operators in Hilbert space H.
Assume the spectrum Q is absolutely continuous, σ(Q) = σac = X Ď Rn,
n ě 1, H is strongly positive, and its inverse has an integral representation
in L2(σ(Q), dEQ(x)) given by a kernel:
(H´1Ψ,Φ) = (µΨ, µΦ)L2(σ(Q),dEQ(x)) =
ż
σ(Q)
κ(x, y)dµΨ(x)dµΦ(y),
where
µΨ(¨) = (EQ(¨)Ψ,Ψ), µΦ(¨) = (EQ(¨)Φ,Φ)
and the associated with H´1 integral kernel κ(x, y) is perfect in sense Fu-
glede (see [18]).
Let HA,HB denote a couple self-adjoint extension of the symmetric ope-
rator
H := HD0 , D0 Ă D(H)
which has a nontrivial deficiency indices n+(H) = n´(H) ‰ 0. These
operators, HA,HB, are corresponded to the Hamiltonians of two opponent
sides, A and B. They admit interpretations as the strategies of the dynam-
ical behavior.
Let the DSC associated with HA,HB is fixed by a number of components
mA +mB = m ă 8 and a decomposition of σ(Q) ” Ω into some kind of
condenser:
Ω = Ω´
ď
Ω+, Ω´ =
ď
jPN´
Ωj , Ω+ =
ď
iPN+
Ωi,
N´ = t1, . . . ,mBu, N+ = t1, . . . ,mAu,
(4.2)
866 V. Koshmanenko
such that
Ωcl
i
č
∆cl
j = H, i ‰ j.
The states of this DSC are consisted from a set of m vectors
tΨt
i P D(HA)uiPN+ , tΦt
j P D(HB)ujPN´
which change #
Ψt
i
Φt
j
+
‹ÝÑ
#
Ψt+1
i
Φt+1
j
+
,
in the discrete time t = 0, 1, . . . in accordance with the law of conflict
interaction ‹ given in the terms of associated measures:
µt+1
Ψi
(∆) =
µtΨi
(∆)(1 + Θt) ´ ηti(∆))
1 + Θt +W t
A
,
µt+1
Φj
(∆) =
µtΦj
(∆)(1 + Θt) ´ ηtj(∆))
1 + Θt +W t
B
,
(4.3)
where ∆ P B,
Θt =
ÿ
iPIA,jPIB
(HΨt
i,Φ
t
j), W t
A =
ÿ
i
ηti(X), W t
B =
ÿ
j
ηtj(X), (4.4)
and where the occupation measures ηti(¨) and ηtj(¨) are defined under as-
sumption that every Ωi is a set of absolute domination for µΨi in the sense
that:
µΨi(∆
1) ě νi(∆
1), @∆1 Ď Ωi, (4.5)
and similarly every Ωj, j P IB is a set of absolute domination for µΦj :
µΦj (∆
1) ě νj(∆
1), @∆1 Ď Ωj , (4.6)
where measures νi, νj are defined as the “mean fields” created by the opponent
sides:
νi(∆) = 1/mB
ÿ
jPIB
µΦj (∆) + 1/(mA ´ 1)
ÿ
k‰i
µΨk
(∆),
νj(∆) = 1/mA
ÿ
iPIA
µΨi(∆) + 1/(mB ´ 1)
ÿ
k‰j
µΦk
(∆).
That is, ηti(¨), ηtj(¨) are defined as it was described in Subsection 3.4 starting
with signed measures ωi := µΨi ´ νi and ωj := µΦj ´ νj.
Then the minimizing problem for above fixed condenser Ω = σ(Q) of
view (4.2) with above assumptions (4.5) and (4.6), admits unique solution
γΩ in the following space of probability measures
M1
+(Ω) ˆ M1
+(Ω) :=
Theory of dynamical systems of conflict 867
:=
␣
µΨi , µΦj ,Ψi P D(HA),Φj P D(HB), }Ψi} = }Φj} = 1
(
.
In particular, it means that there exists the trajectory of DSC that converges
to the equilibrium signed measure γΩ = γΩ´ + γΩ+ where
γΩ+ = lim
tÑ8
ÿ
iPIA
µtΨi
, γΩ+ = lim
tÑ8
ÿ
jPIB
µtΦj
.
Here we represent only some arguments for proving of the above theorem.
At first, we remark that in according Theorems 3.5 and 3.7, there exist
limiting measures
µ8
Ψi
= lim
tÑ8µ
t
Ψi
, µ8
Φj
= lim
tÑ8µ
t
Φj
, i P IA, j P IB,
which are supported on the corresponded subsets Ωi or Ωj , respectively.
The last fact follows from assumption that for every starting measure µΨi
or µΦj the set Ωi or respectively Ωj is a set of absolute domination:
µΨi(∆
1) ě νi(∆
1), @∆1 Ă Ω,
µΦj (∆
1) ě νj(∆
1), @∆1 Ă Ω.
By this, the values of occupation measures ηti(∆1), ηtj(∆1), which, we re-
call, estimate the “intervention strength” of opponents and the “strength of
competition” with µtΨk
, k ‰ i and µtΦk
, k ‰ j, respectively, on sets Ωi, Ωj
converges to zero with t Ñ 8 for any ∆1 Ď Ωi and ∆1 Ď Ωj . And vice
versa, by the same reason of absolute dominance, values of all measures
µtΨi
(∆1) and µtΦj
(∆1) goes to zero when ∆1 did not belong to Ωi and Ωj ,
respectively.
Note that ηµ(∆) has a similar but opposite meaning, it evaluates the
strength of occupation of another opponent which is represented by µ on a
set ∆, where now ν has absolute dominance.
Here it is worth recalling the property of absolute dominance in the
abstract case. Consider a couple of probability measures µ, ν P M+(X),
µ ‰ ν. Assume that for some ∆ one of the measures µ or ν has a local
priority with respect other. It means that for any Borel ∆1 P ∆ following
inequality is fulfilled:
µ(∆1) ě ν(∆1) or µ(∆1) ď ν(∆1).
Using this tool we introduce on of the main characteristic of an opponent,
its dominant territory as the maximal subset where one of above inequalities
are fulfilled. In the case of two measures, this problem has a solution in
terms of the classical Hahn decomposition for the charge ω = µ´ ν.
In the theory for conflict dynamical systems with many alternative op-
ponents associated with an arbitrary family of probability measures tµiu
on the measurable space (Ω,B) there appears a similar task as a part of the
868 V. Koshmanenko
equilibrium states problem. Importantly that supports of limiting measures
corresponding to the equilibrium state just coincide with subsets from de-
composition of Ω into the maximal regions of dominance for each measure
µi over all others in the sense, µi(F ) ě µk(F ), @k ‰ i.
We assert that under fixed integral kernel κ constructed by a suitable
operator H, the minimizing measures γi for each subset Ωi admits construc-
tion among a family of the limiting measures µ8
i which appear in DSC with
arbitrary sets of starting opponent components m = mA +mB ă 8.
It should be noted that above the decomposition (4.2) was originally
fixed, while in theory the complex systems composed with a family ofm ě 2
advisories, say A,B,C, . . . which are associated with probabilities measures
µi, i ě 3 on a space (Ω,B) the analogous decomposition
Ω =
mď
i=1
Ω+
i
appears as result of long fighting. Here Ω+
i is not unique and dependents
on the starting relations between measures µi. Therefore, it coincides with
the union of all subsets of absolute dominance for one of the opponents.
Equivalently, this means that every µi exceeds all other measures in the
sense that
µi(F ) ě µk(F ), @k ‰ i, @F Ď Ω+
i
č
B.
Although the same set also appears as support of the limit measure µ8
i .
In fact, it is a nontrivial problem to generalize the above kind of decom-
position for a case of several opponents. From the one hand side it may be
performed using the classic Hahn-Jordan decomposition for each couple of
measures µi, νi which define a signed measure ωi = µi ´ νi, where
νi := 1/(m´ 1)
ÿ
k‰i
µk.
The difficulties connected with a fact that µi(F ) ě νi(F ) does not imply
that µi(F ) ě µk(F ) for k ‰ i and therefore the set of absolute domination
for µi in general is less than a positive subset of ωi. So we need to extend the
classic Hahn-Jordan decomposition into positive and negative components,
ω = ω´ + ω+, on the case of multipolar expansions for a family of signed
measures: ωi,k = µi ´ µk, i, k P [1,m],m ě 3. On the other hand in the
theory of DSC with many opponents, this problem is closely related to
the problem of finding equilibrium states and the limiting redistribution of
resource space as a result of the conflict interaction.
4.2. A case of conflict interaction between a finite number of ab-
stract societies. Let Ai, i P 1,m denote m ą 1 abstract societies living
in the resource territory Ω Ă X, which is a compact set in Rn. Match each
Theory of dynamical systems of conflict 869
Ai with a set of some individuals. Mathematically, this means a family of
vectors Ψαi , αi P Ii ă 8 in the Hilbert space H or probability measures
µαi P M+
1 (Ω) associated with the vectors Ψαi in the spectral representation
of the position operator Q with o(Q) = Ω. We assume that each Ai has
priority in some region Ωi Ă Ω of the resource area. In other words, Ωi
denotes the domain of absolute dominance for the family of measures µαi :
Ωi = YF
␣
µ P M+
1 (X) | µαi(E) ě µαk
(E), @E Ď F, @k ‰ i
(
.
Let tAi,Ω =
Ť
iΩi, ‹u denote the dynamical system of conflict between
abstract societies Ai. Here ‹ and corresponds to the law of conflict inter-
action and controls the behavior of trajectories in terms of vectors Ψt
αi
or
measures µtαi
, t = 0, 1, . . . The mathematical definition of ‹ depends on the
specific real model. In general, in what form this mapping is implemented,
the question is open. It can be constructed similarly to formulas (4.3) with
additional terms corresponding to the division of the entire system into clus-
ters. The question about the form of the energy functional Θt, see (4.4),
is especially important since it is defined by some Hamiltonian H whose
inverse H´1 and its quadratic form (H´1¨, ¨)H is used for construction of
the integral kernel κH(¨, ¨) ” κ(¨, ¨). The question of the form of the energy
functional Θt, see (4.4), is particularly important, since it is determined
by some Hamiltonian H, the inverse of which H´1 and its quadratic form
(H´1¨, ¨)H is used to construct the integral kernel κH(¨, ¨) ” κ(¨, ¨).
Thus, under all technical assumptions about the structure of the resource
space Ω =
Ť
iΩi, as a generalized condenser, the kernel κH , and the prop-
erties of the potential function κ(x, µ), where the measures of µ we take
from M+
1 (Ω) (for details see [41–45]), we can formulate our hypothesis in
the form of a theorem as follows.
Theorem 4.3. The minimizing measure γΩ =
ř
i γi for which
κ(γΩ, γΩ) = }γΩ}2 and κ(γi, γi) = cκ(Ωi),
where cκ(¨) is the capacity of the set Ωi can be defined among the family
of limit measures µ8
i that arise in the above-described dynamical system of
conflict with arbitrary sets of starting combinations µi P Mi(Ω).
It is important that the minimizing measure γΩ as an equilibrium state
of the conflict system allows approximate construction using iterative se-
quences according to formulas of the type (4.3).
Finally, we note that an essential technical trick in the proof of the above
theorem is based on the notion of inner balayage for measures in the sense
of the following definition [45].
870 V. Koshmanenko
For the set Ωi Ă Ω, the inner balayage µΩi µ P M+(Ω) is defined as the
measure with minimum energy κ(ν, ν) in the class
Γ+
Ωi,µ
:=
␣
ν P M+(Ω) | κ(¨, ν) ě κ(¨, µ) on Ωi
(
,
i.e.
}µΩi}2 = min
νPΓ+
Ωi,µ
}ν}2.
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Volodymyr Koshmanenko
INSTiTUTE OF MATHEMATiCS OF NAS OF UKRAiNE, KYiV, 01004, UKRAiNE
Email: koshman63@gmail.com
ORCID: 0000-0002-7108-4964
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| spelling | oai:trim.imath.kiev.ua:article-5302023-08-17T09:06:54Z The theory of dynamical systems of conflict in the framework of functional analysis Теорія динамічних систем конфлікту в рамках функціонального аналізу Koshmanenko, Volodymyr Кошманенко, Володимир In this article, we give an introduction to the mathematical setting of problems related to the phenomenon of conflict in terms of constructions in Hilbert spaces. The struggle (conflict, game) between opponents (adversaries, players) will be represented by operator transformations of vectors in Hilbert spaces and probabilistic distributions on the territory of life resources. The phenomenon of conflict as a contradiction between opponents appears in mathematical terms as an intersection the domains of definition for operators and overlapping of corresponding measures. Conflict interaction between opponents in the physical sense is described by the specific transformation of states in a Hilbert space. In turn, this is a mapping that changes the spectral measurements. Thus, a complex dynamical system arises, which we call a dynamical system of conflict. Then the following main problems arise as fundamental questions. What reasonable law of engagement (game or war) should be adopted to resolve the initial intersections? What is a fair limiting distribution of the resource territory? In a more general formulation, solving conflict problems means the detailed describing of all possible outcomes on opponents states of the type: victories, defeats, states of equilibrium, compromises as fixed points together with their basins of attraction. Стаття присвячена введенню математичної постановки задач, пов’язаних із феномен конфлікту, в термінах конструкцій у гільбертових просторах. Боротьба (конфлікт, гра) між опонентами (супротивниками, гравцями) буде представлена операторними перетвореннями векторів у гільбертових просторах та ймовірнісними розподілами на просторі життєвих ресурсів. Феномен конфлікту, як суперечність між опонентами, в математичних термінах означає перетин областей визначення для операторів та перекриття носіїв відповідних мір. Конфліктна, у фізичному сенсі, взаємодія між опонентами описується специфічним перетворенням станів у гільбертовому просторі. У свою чергу, ці відображення змінюють спектральні вимірювання. Таким чином, виникає складна динамічна система, яку ми називаємо динамічною системою конфлікту. Як основні проблеми постають наступні фундаментальні питання. Який розумний закон конфліктної взаємодії (гри чи війни) треба прийняти для розв'язання початкових перетинів? Яким має бути справедливий граничний розподіл території життєвого ресурсу? У більш загальному формулюванні, розв'язання конфліктних проблем означає детальний опис усіх можливих результатів про стани опонентів типу: перемоги, поразки, рівноваги, компроміси, як нерухомі точки, разом з басейнами їх притягання. Інститут математики НАН України 2023-08-17 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/530 10.3842/trim.v20n1.530 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 20 No. 1 (2023): Modern problems of mathematics and its applications, III; 843-872 Сборник Трудов Института математики НАН Украины; Том 20 № 1 (2023): Сучасні проблеми математики та її застосувань III; 843-872 Збірник Праць Інституту математики НАН України; Том 20 № 1 (2023): Сучасні проблеми математики та її застосувань III; 843-872 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/530/515 Авторське право (c) 2023 Volodymyr Koshmanenko http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Koshmanenko, Volodymyr Кошманенко, Володимир The theory of dynamical systems of conflict in the framework of functional analysis |
| title | The theory of dynamical systems of conflict in the framework of functional analysis |
| title_alt | Теорія динамічних систем конфлікту в рамках функціонального аналізу |
| title_full | The theory of dynamical systems of conflict in the framework of functional analysis |
| title_fullStr | The theory of dynamical systems of conflict in the framework of functional analysis |
| title_full_unstemmed | The theory of dynamical systems of conflict in the framework of functional analysis |
| title_short | The theory of dynamical systems of conflict in the framework of functional analysis |
| title_sort | theory of dynamical systems of conflict in the framework of functional analysis |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/530 |
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