Quadratic stochastic operators, Lyapunov functions and tournaments. (Russian) Zbl 0766.47037

The discrete dynamic system of the form \[ x_ k'=x_ k\bigl(1+\sum^ m_{i=1}a_{ki}x_ i\bigr),\quad k=\overline{1,m},\quad a_{ki}=- a_{ik},\quad| a_{ki}|\leq 1, \tag{1} \] acting on the simplex \(S^{m-1}\) is studied. The existence of the Lyapunov function of the form \(\varphi(x)=x_ 1^{p_ 1}\cdots x_ m^{p_ m}\) is proved. An algorithm is offered for finding isolated fixed points of the mapping \(V:S^{m-1}\to S^{m-1}\) where \(Vx=x'=(x_ 1',\dots,x_ m')\). The connection between fixed points of \(V\) and Lyapunov functions for (1) is investigated. It is proved that \(V:S^{m-1}\to S^{m-1}\) is a homeomorphism. Convergence of “negative” trajectories \(\{V^{-n}x^ 0\}_{n\in N}\) and, as a rule, non-regular behaviour of positive trajectories \(\{V^ nx^ 0\}\) are typical for the dynamic system (1). For estimating the set of limit points of trajectories, elements of tournament theory are used. The absence of periodical orbits for dynamic systems of the form (1) is proved. A biological interpretation of the obtained results is given.
Reviewer: R.N.Ganikhodzhaev


47J05 Equations involving nonlinear operators (general)