This book focuses on the development of Melnikov-type methods applied to high dimensional dynamical systems governed by ordinary differential equations. Although the classical Melnikov's technique has found various applications in predicting homoclinic intersections, it is devoted only to the analysis of three-dimensional systems (in the case of mechanics, they represent one-degree-of-freedom nonautonomous systems). This book extends the classical Melnikov's approach to the study of high dimensional dynamical systems, and uses simple models of dry friction to analytically predict the occurrence of both stick-slip and slip-slip chaotic orbits, research which is very rarely reported in the existing literature even on one-degree-of-freedom nonautonomous dynamics. This pioneering attempt to predict the occurrence of deterministic chaos of nonlinear dynamical systems will attract many researchers including applied mathematicians, physicists, as well as practicing engineers. Analytical formulas are explicitly formulated step-by-step, even attracting potential readers without a rigorous mathematical background. Sample Chapter(s). Chapter 1: A Role of the Melnikov-Type Methods in Applied Sciences (137 KB). Contents: A Role of the Melnikov-Type Methods in Applied Sciences; Classical Melnikov Approach; Homoclinic Chaos Criterion in a Rotated Froude Pendulum with Dry Friction; Smooth and Nonsmooth Dynamics of a Quasi-Autonomous Oscillator with Coulomb and Viscous Frictions; Application of the Melnikov-Gruendler Method to Mechanical Systems; A Self-Excited Spherical Pendulum; A Double Self-excited Duffing-type Oscillator; A Triple Self-Excited Duffing-type Oscillator. Readership: Graduate students and researchers in dynamical systems.