Speakers
Description
We study the mathematical model of an abstract society ${\cal S} = \{ a_i \} _{i=1}^m, \ 1< m < \infty$ ($a_i $ - players, opponents or their association) in the form of a complex dynamical system with a conflict interaction between its elements. The states of society $S$ are described by stochastic vectors of players energy ${\bf p}^t=(p_1^t,...,p_i^t,..., p_m^t), \ t=0,1,...,$ which evolve according to the law of conflict interaction. Such models were studied in [1], [2], [3], for example.
Our new step is that we divide the set ${\cal S}$ into two clusters of individuals ${\cal S} = {\cal S}_J\bigcup {\cal S}_I$, where $J=\{j_1,...,j_n\}$ denotes a subset of such indices that individuals $a_j$ receive external support at each step $t>0$ in the form of an additive shift: $ p_j^t \rightarrow p_j^t+b, $ $b>0$ and for $ I=\{i_1,...,i_{m-n}\}$ individuals $a_i $ remain without external support.
Thus, the dynamic system is given by difference equations:
$$
p_i^{t+1}= \frac{p_i^t (1- r_i^t)}{z^t}, \ \
p_j^{t+1}= \frac{(p_j^t+b_j)(1- r_j^t)}{z^t}, \ i \in I, \ \ j \in J, \ t=0,1,..., $$ where $b_{j}=b>0$, $z^t=\sum_k (p_k^t+b \cdot \textbf 1_J(k)) (1-
r_k^t)$, $\textbf 1_J(k) $ — indicator function of a subset $J$. Using of value $ r_i^t=\frac{\sum_{k\neq i} p_k^t}{m-1}=\frac{1-p_i^t}{m-1} $ corresponds to the repulsive interaction of each individual $a_i$ with the rest of society ${\cal
S}^{\perp}_{a_i}$ in the mean field sense. Denominator $z^t$ provides stochastic normalization.
The main results concern systems with three elements (players). In this case, a description of all equilibrium states is given and their stability is investigated with depending on the parameter of external influence. Besides basins of attraction for point attractors are partially described and illustrated.
1.V.D. Koshmanenko, Spectral theory of dynamic conflict systems, Naukova Dumka, Kyiv (2016)(in ukrainian).
2.T. Karataieva, V. Koshmanenko, M.J. Krawczyk, K. Kulakowski, Mean field model of a game for power. Physica A: Statistical Mechanics and its Applications, 525, 535-547 (2019); https://doi.org/10.1016/j.physa.2019.03.110
3.T. V. Karataieva, V. D. Koshmanenko,Equilibrium states of the dynamic conflict system for three players with a parameter of influence of the ambient environment Journal of Mathematical Sciences, 274, No. 6, August, 2023; DOI 10.1007/s10958-023-06649-x.
We are acknowledge a grant from the Simons Foundation (1290607, T. K., V. K.).
