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

This paper presents a nonlinear dynamical model which consists the system of differential and operator equations. Here differential equation contains a nonlinear operator acting in Banach space, a nonlinear operator equation with respect to two elements from different Banach spaces. This system is assumed to enjoy the stationary state (rest points or equilibrium). The Cauchy problem with the initial condition with respect to one of the desired 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. It is demonstrated that a solution can be constructed by the method of successive approximations under the suitable sufficient conditions. If the conditions of the main theorem are not satisfied, then several solutions may exist. Some of solutions can blow-up in a finite time, while others stabilize to a rest point. The special case of considered dynamical models are nonlinear differential-algebraic equation (DAE) have successfully modeled various phenomena in circuit analysis, power systems, chemical process simulations and many other nonlinear processes. Three examples illustrate the constructed theory and the main theorem. Generalization on the non-autonomous dynamical systems concludes the article.