Basins of attraction of nonlinear systems' equilibrium points: stability, branching and blow-up

Nikolai Sidorov, Denis Sidorov, Yong Li

Published 2018-07-15Version 1

The dynamical model based on the differential equation with a nonlinear operator acting in Banach spaces and a nonlinear operator equation with respect to two elements from different Banach spaces is considered. It is assumed that the system has stationary state (rest points or equilibrium). The Cauchy problem with the initial condition with respect to one of the unknown functions is formulated. The second function controls the corresponding nonlinear dynamic process, the initial conditions are not set. The sufficient conditions of the global classical solution's existence and stabilization at infinity to the rest point are formulated. Under suitable sufficient conditions it is shown that a solution can be constructed by the method of successive approximations. If the conditions of the main theorem are not satisfied, then several solutions may exist. Some of them can blow-up in a finite time, while others stabilize to a rest point. Examples are given to illustrate the constructed theory.