Introduction to Analysis Tools


Chapters 2–8 focused on a single objective: finding the equations governing the motion of a system consisting of particles and rigid bodies, a process commonly known as mathematical modeling. Nonetheless, while models constitute the fundamental basis upon which everything else is built, the process of fully understanding and controlling the dynamic behavior of a system cannot be fully realized without analyzing its response, a process commonly known as mathematical analysis.

As you may have noticed in the previous chapters, models of particles and rigid bodies in motion are described by one or more non-linear ordinary differential equations. As such, analyzing the dynamic behavior of a system of particles and/or rigid bodies necessitates learning the tools commonly used to analyze non-linear differential equations. This subject will be the core of this chapter. The analysis of non-linear differential equations can be divided into local and global categories. In local analysis, we inspect the dynamics of the system in the vicinity of some important points where it is possible to implement analytical tools. In a global analysis, we try to draw a global picture of the dynamics using numerical techniques.

The reader should bear in mind that this chapter is only introductory in nature; the analysis of non-linear differential equations is a wide subject, and cannot be covered in a single chapter. For more details, interested readers should consult the book by Nayfeh and Balachandran or that by Strogatz.

9.1Basic Definitions
9.2Equilibrium Solutions of Dynamical Systems
9.3Stability and Classification of Equilibrium Solutions
9.4Phase-plane Representation of the Dynamics
9.5Bifurcation of Equilibrium Solutions
9.6Basins of Attraction

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