Non-Linear Differential Equations and Dynamical Systems

Non-Linear Differential Equations and Dynamical Systems is the second book within Ordinary Differential Equations with Applications to Trajectories and Vibrations, Six-volume Set. As a set, they are the fourth volume in the series Mathematics 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 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 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 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 equation
• Discusses differentials of the first or higher order in two or more variables
• Includes discretization of differential equations as finite difference equations
• Describes parametric resonance of linear time dependent oscillators specified by the Mathieu functions and other methods
• Examines non-linear oscillations and damping of dynamical systems including bifurcations and chaotic motions