This fourth volume in the series "Mathematics and Physics Applied to Science and Technology": (i) follows the first three volumes on complex, transcendental and generalized functions that provide a detailed background on the theory of functions; (ii) starts the next set "On Boundary and Initial Value Problems" with the "Ordinary Differential Equations with Applications to Trajectories and Oscillations".The first book of volume IV focused on the simpler "Linear Differential Equations and Oscillators", and the second book (corresponding to the sixth of the series) proceeds to the less simple "Non-linear Differential Equations and Dynamical systems" consisting of the chapters 3 and 4 of the volume IV.The chapter 3 considers non-linear differential equations of first order, including variable coefficients. A first-order differential equation is equivalent to a first-order differential in two variables. The differentials of order higher than the first and with more than two variables are also considered. The applications include the representation of vector fields by potentials.The chapter 4 in the second book starts with linear oscillators with coefficients varying with time, including parametric resonance. It proceeds to non-linear oscillators including non-linear resonance, amplitude jumps and hysteresis. The non-linear restoring and friction forces also apply to electromechanical dynamos. These are examples of dynamical systems with bifurcations that may lead to chaotic motions.Presents general first-order differential equations including non-linear like the Ricatti equationDiscusses differentials of the first or higher order in two or more variablesIncludes discretization of differential equations as finite difference equationsDescribes parametric resonance of linear time dependent oscillators specified by the Mathieufunctions and othermethodsExamines non-linear oscillations and damping of dynamical systems including bifurcations andchaotic motions