Quadratic stochastic dynamical systems of the type $(\sigma |\mu)$ (PhD dissertation discussion)

The thesis is devoted to investigation of the dynamical systems generated by stochastic and non-stochastic cubic matrices.
Basic results of the research are as follows:

• Necessary and sufficient conditions to coefficients of the of quadratic operator to map the simplex ${{S}^{m-1}}$ to itself are found.
• All fixed points of discrete and continuous-time quadratic stochastic and non-stochastic operators defined in the simplex were found and the limit points of the trajectory were studied.
• The dynamics of a one-dimensional quadratic non-stochastic operator is studied in general. Conditions for the parameters of the one-dimensional quadratic nonstochastic operator to form a chaotic dynamical system are given.
• For a class of two-dimensional quadratic non-stochastic operators it is shown that among two-dimensional quadratic non-stochastic operators in particular, there are those that form a chaotic dynamical system and those that do not form a chaotic dynamical system.
• Some multiplication operations for cubic matrices are introduced, and a family of matrices satisfying the Kolmogorov-Chapman equation for these multiplications is determined.
• Several examples of quadratic stochastic processes of type $(\sigma |D)$ have been constructed.