Formation of stochastic behavior and explosive solutions in the infinitely remote phase space of dynamic systems

The article examines the conditions under which phase variables undergo a blow-up regime, tending toward the Poincaré circle in finite time. It also explores systems where, alongside explosive solutions, stochastic behavior of trajectories is observed in some cases. The role of separatrices and separatrix cycles is analyzed both before perturbation and under non-autonomous small periodic perturbations of the right-hand sides of the original dynamic systems. These perturbations give rise to homoclinic structures in the phase space, leading to stochastic trajectory behavior. Various cases of soliton formation during trajectory bifurcations are also considered.