Nonlinear systems' equilibrium points: Branching, blow-up and stability

This article considers the nonlinear dynamic model formulated as the system of differential and operator equations. This system is assumed to enjoy an equilibrium point. 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 sufficient conditions of the global classical solution's existence and stabilisation at infinity to the equilibrium point are formulated. The 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 solutions can blow-up in a finite time, while others stabilise to an equilibrium point. The special case of considered dynamic models are differential-algebraic equations which model various nonlinear phenomena in circuit analysis, power systems, chemical processes and many other processes.