Based on statistical data for the years 2014 - 2018 from open sources, a specific model of the Richardson-type arms race is recorded in terms of system of ordinary differential equations (ODE), as traditional. A numerical analysis of its behavior was carried out. For this purpose was used well-known Maple program tools. In real life an answer of any actions can't to be immediate reaction. With the aim to improve adequacy, traditional model is modified by taking into account the time delay in the adversary's response. And new model is rewrites in term of system of functional-differential equations (FDE). There are well-known results from common FDE theory, that fact of involving time lag into study of a dynamic process give the significantly influences on the qualitative characteristics of behavior (stability, type of equilibrium points, bifurcation, etc.). To demonstrate this dependence, phase portraits were constructed for both models, and their comparative analysis was carried out. Differential equation systems, stability, phase portrait, deviated argument.
= = ( ( ), ) ( ( ), )
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Here, = ( ) is a function that, for any given level y of the action performed by party Y, gives the value x of the action given in response by party X, = ( ) – similarly for the other party; ( , ) is the distance (some metrics) between the arguments z and w, according to some criteria, and α and β are positive constants. The idea is that the value of the change in aggressiveness is proportional to the distance between the optimal level, that is, that given by the response function, and the actual level. The response functions chosen by Richardson for his model are linear. Therefore, the differential equations describing conflict escalation in Richardson's model have the form
== (( )) −− (( )) ++ (2)
Here, k and l are protection coefficients, α and β are fatigue and expense coefficients, and p and q are indicators of past complaints (claims) between the two parties.
Solving the system of algebraic equations
+ + + = 0 + = 0 ( 0, 0): 0 = ( 0) і 0 = ( 0)
ˊ ( 0) ˊ ( 0) < 1.
In the linear case (2), which was considered by Richardson, this condition turns into > This can be interpreted as follows - for guaranteed stability, it is necessary that considerations regarding the cost of weapons have more weight than defense issues. 2.1.
In work [
In work [
On the other hand, the nuclear arms race will bring the world to the threat of a nuclear disaster. Unlike conventional weapons, nuclear weapons, as durable goods, can accumulate over time. Based on the above framework, this thesis established the endogeneity of Richardson's one-dimensional nuclear arms race model with a symmetric game.
Richardson-type models were used not only for the study of hostile conflicts. For example, the
purpose of the work [
Also interesting is the fact that the proposed systems of differential equations (1) (actually (2)) are
quite universal. Thus, by applying a different interpretation of them, Lanchester-type combat models
are successfully studied [
We will use the data obtained in work [
Let us denote Ukraine by country X, and Russia, in turn, by Y. Accordingly, the defense and army
expenditures of these countries will be denoted as х = ( ), and = ( ). Based on the data in Tables
1-3, constant coefficients for the model were already calculated in [
We will get the finished model:
= 0.014 ( ) − 0.44 ( ) + 42.37
First, let's find the equilibrium points of our system. The equations are linear, so let's equate the right-hand side of the system to zero and solve the system of equations. Let's get a point:
= 3.3 ( ) − 0.42 ( ) − 121 ( = 116.1717172, = 624.6825397)
Now let's define the type of this point. For this we will find eigen values.
Values are real, different, of the same sign, negative
2.3.
If one takes into account the decision-making time in the development and implementation of types
of weapons, as well as technical delays in their implementation, then the system of ordinary differential
equations with delay is more adequate. Unfortunately, if the solution of a linear stationary system
without delay (2) in an analytical form can be obtained using the usual exponent, then for systems with
even one constant delay, finding the general solution becomes more complicated. Richardson's model
with lagging of general vector-matrix form can be written in the form of a system of
differentialdifference equations with the time-delay argument [
( − ) + ( ), = > 0
Method for analytical finding of the solution for special cases of models of a system with time-delay
(5) was considered in [
Taking as a basis the initial original model of Richardson (2), we modify its elements, without limiting the generality, we will introduce time-delay in the response of one of the parties, for Y: = ( − ) −
( ) + = ( ) −
( − ) + = ( ) −
( − ) + = ( − ) − ( ) + for X:
For the purpose of conducting numerical experiments, it is convenient to use one of the widely used
software packages for solving similar problems (Mathematica, Maple, MatLab Simulink, Sage, etc.)
[
Let's build phase portraits of the arms race systems being studied - first we will do it for model (3), which corresponds to the confrontation between Ukraine and Russia according to the results of 20142018. Fig.1 clearly shows an equilibrium asymptotically stable point - a "node" with coordinates (4) . (5) (6) (7)
Next, we will proceed to the study of the impact of a delay in the response of one of the parties to the conflict. For the sake of clarity, let's focus on the variant of system (6).
We will successively consider certain values of the time delay τ=0.2; 1; 3; 5; 7. For each of the values of the selected time delay, we will construct the corresponding trajectories of the behavior of the system (6).
Graphs of these trajectories, for the purpose of comparison, will be superimposed on the phase portrait of the original system without deviation of the argument (Fig. 2 - Fig. 6).
The given value of the argument deviation of 2 units is too small for the system to undergo any significant changes. We see small deviations, but, in general, the behavior of the system does not differ from the original one.
The next step is to increase the deviation. Analyzing the behavior of the corresponding trajectories of the system depicted in Fig. 3, we see that with a time-delay value equals to 1, the deviation of the integral curves from the initial ones (model (3)) becomes more noticeable, but still there are no qualitative fundamental changes. Curves also go to a equilibrium point "node".
As can be seen from Fig.4, with a delay value of 3 units, we can already see a distortion of the direction of movement of the integral curves, which no longer corresponds to the behavior of the equilibrium point "node", but also does not make it possible to clearly classify the obtained portrait.
As we can see, at a time-delay value of 5 units, the bifurcation of the phase portrait of the dynamic system under study occurred. The asymptotically stable equilibrium point of the "node"-type has finally disappeared. The behavior of the trajectories around this point clearly resembles the "focus"-type point.
By increasing the delay value of the argument to 7 units (see Fig.6), it becomes even more noticeable that the behavior of the integral curves has turned into a "focus". Moreover, a couple of additional points around which the trajectories begin to wind become noticeable.
Therefore, with a significant increase in the value of the delay in the dynamic system, a complete bifurcation of the phase portrait occurs.
These numerical calculations accurately confirmed the well-known theoretical results regarding complex dynamic systems described in terms of functional differential equations.
The problem of the arms race is very urgent today, few countries currently dare to directly escalate
the conflict, at the same time they continue to actively build up their military forces, spending large
percentages of their own budgets. In the framework of this study, Richardson's model of the arms race
is considered, together with the history of its creation and progress. Based on statistical data from [
Thanks to the created software using the MAPLE package, numerical calculations of the models were carried out (without argument deviation and with different gradually increasing time-delay values). They fully demonstrated the confirmation of theoretical mathematical results from the qualitative theory of differential equations.
In the future, trying to expand the scope of research, and with the aim of applying the obtained results in related fields, it is possible to partially use the results presented, for example, in works [2124]. This can make it possible to modernize the conducted research by connecting the latest achievements in the field of artificial intelligence.
Finally, as for the conflicts related to the arms race in general, which can be described by Richardson's models, the main conclusion is one, and it is quite obvious, the delay in response significantly affects the development of the confrontation and is a destabilizing factor.