Linearization and Stability Analysis of Nonlinear Problems

The focus of this paper is on the use of linearization techniques and linear differential equation theory to analyze nonlinear differential equations. Often, mathematical models of real-world phenomena are formulated in terms of systems of nonlinear differential equations, which can be difficult to solve explicitly. To overcome this barrier, we take a qualitative approach to the analysis of solutions to nonlinear systems by making phase portraits and using stability analysis. We demonstrate these techniques in the analysis of two systems of nonlinear differential equations. Both of these models are originally motivated by population models in biology when solutions are required to be non-negative, but the ODEs can be understood outside of this traditional scope of population models. In fact, allowing solutions for these equations to be negative provides some very interesting mathematical problems, and demonstrates the utility of the analysis techniques to be described in this article. We provide stability analysis, phase portraits, and numerical solutions for these models that characterize behaviors of solutions based only on the parameters used in the formulation of the systems. The first part of this paper gives a survey of standard linearization techniques in ODE theory. The second part of the paper presents applications of these techniques to particular systems of nonlinear ODEs, which includes some original results by extending the analysis to solutions lying anywhere in the plane, rather than only those in the first quadrant