Dynamical System Modeling and Stability Investigation�DSMSI-2023

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Dedicated to the 77th anniversary of the outstanding Ukrainian scientist

professor Denys Khusainov

December 19-21, 2023, Kyiv, Ukraine

Simulation the Impact

of Time-Delay in Richardson

Arms Race Models

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Denys Khusainov,Andriy Shatyrko,

Taras Shevchenko National University of Kyiv�

Abstract

• Based on statistical data for the years 2014 - 2018 from open sources, a specific model of the Richardson-type arms race is recorded. A numerical analysis of its behavior was carried out. The model is modified by taking into account the time delay in the adversary's response. Phase portraits were constructed for both models, and their comparative analysis was carried out.

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Introduction

• Lewis Richardson was born in 1881 in Newcastle and studied mathematical psychology. Richardson believed that each state steadily increases its equipment, as if obliged, forced to do so, which may be related to primitive instincts, or to the lack of a spiritual and moral basis for establishing borders. Based on this hypothesis, he built a mathematical model of the arms race [1,2]. There are many different definitions of arms races, but for the purposes of this paper they can be seen as long-lasting rivalries between pairs of hostile states that encourage the competitive acquisition of military power. One option is a two-person game, specifically a prisoner's dilemma, where the choice is to arm or not to arm, and the dominant strategy for both is not Pareto optimal [3,4]. The other is Richardson's model, as an action-reaction process represented by a pair of differential equations [5-7]. There is a sufficient number of scientific and popular science works devoted to both, the support and development of Richardson's ideas in this direction, and their substantial criticism, which sometimes reaches the point of complete denial [6,8,9]. The answer to this question is beyond the scope of this work.

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Formulation of the problem

The basic idea of Richardson's model is that in a situation where there are two parties who are potential enemies, each of them responds to the actions of the other, increasing or decreasing its level of aggressiveness, or, as he puts it, "readiness for war." Let's call X and Y the two sides, respectively. The dynamics of aggressiveness between two parties X and Y can be described by the following system of differential equations:

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(1)

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Formulation of the problem

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

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(2)

• Solving the system of algebraic equations

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We will obtain the equilibrium point

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It will be stable if

In the linear case (2), which was considered by Richardson, this condition turns into

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Russia-Ukraine military confrontation

We will use the data obtained in work [16] for the model of the arms race between Russia and Ukraine (the original data source is the websites of the State Statistics Service of Ukraine and the Federal Statistics Service of the Russian Federation for Ukraine and Russia, respectively 

• Table 1 Expenditures of Ukraine and Russia in 2014-2018 (billion dollars)

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• Table 2 Gross domestic product of Ukraine 2012-2017

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Russia-Ukraine military confrontation

• Table 3 Gross domestic product of the Russian Federation 2012-2017

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Based on the data in Tables 1-3, constant coefficients for the model were already calculated in [16], so we will take them in the form of Table 4 and substitute them in equations system (2).

• Table 4 Coefficients of the arms race model

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Russia-Ukraine military confrontation

We will get the finished model:

(3)

The equilibrium points of our system

(4)

Eigen values are real, different, of the same sign, and negative

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Therefore, the equilibrium point is a "stable node"

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Consideration of the response delay factor

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Numerical simulations

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 2014-2018. Fig.1 clearly shows an equilibrium asymptotically stable point - a "node" with coordinates (4):

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Fig.1

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Numerical simulations

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).

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Fig. 2 Fig. 3 Fig. 4

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Numerical simulations

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Fig. 5 Fig. 6

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 known theoretical results regarding complex dynamic systems described in terms of functional differential equations [22]

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Conclusion

• Based on statistical data from [16], which were calculated based on the GDP of countries and their defense spending, a mathematical model of the arms race between Ukraine and Russia starting in 2014 (Conflict in Donbas) was built.

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• Numerical calculations (MAPLE) 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.

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• 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.

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

References

[2]. Richardson L.F., Arms and insecurity: A mathematical study of the causes and origins of war, Ed. by N. Rashevsky and E. Trucco, Boxwood Press, Pittsburgh, 1960.

[6]. Smith R.P., The influence of the Richardson arms race model, In: Lewis Fry Richardson: His intellectual legacy and influence in the social sciences, Ed. by Gleditsch N.P., Springer, Cham, 2019, pp. 25–34/ DOI: 10.1007/978-3-030-31589-4.

[7]. A.V. Shatyrko, D.Ya. Khusainov, B. Puzha, V. Novotna. The Dynamics of One Arms Race Mathematical Model with a Delay // Journal of Automation and Information Sciences – 2020, 52(12), pp.26-38. DOI: 10.1615/JAutomatInfScien.v52.i12.30

[16]. Geseleva N.V., Novik A.S., Trends in defense spending in the Russia–Ukraine confrontation (according to the model of the Richardson arms race). Matematychni metody, modeli ta informatsiyni tekhnologii v ekonomitsi, 2019, No. 27, pp.375–381. (in ukrainian).

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Dynamical System Modeling and Stability Investigation, DSMSI-2023

Thank you for your attention

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