Differential equations

On the properties of finite-dimensional systems of nonlinear equations with multiple solutions

M. V. Bulatov, V. F. Chistyakov

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk

Abstract: Systems of nonlinear equations of the form $F(x)=0$, whose Jacobean is degenerate on the solution, are considered. Definitions of the value of solution's multiplicity are introduced for such problems. A relationship between the values of solution's multiplicity and the values of the index of matrix pencils has been determined; comparing with known definitions of multiplicity has been conducted. Possible applications of the concepts introduced in the theory of differential-algebraic equations, as well as in the process of analysis of non-Morse stationary point of the function and the values of the system's solution multiplicity, which satisfy the stationary point, are discussed.