Non-Linear Differential Equations and Dynamical Systems is the second book withinOrdinary Differential Equations with Applications to Trajectories and Vibrations, Six-volume Set.As a set, they are the fourth volume in the seriesMathematics and Physics Applied to Science and Technology. This second book consists of two chapters (chapters 3 and 4 of the set).The first chapter considers non-linear differential equations of first order, including variablecoefficients. 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 second chapter in the 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 toelectromechanical dynamos. These are examples of dynamical systems with bifurcations that may leadto 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 Mathieu functions and other methodsExamines non-linear oscillations and damping of dynamical systems including bifurcations and chaotic motions