Linearization

Let us consider a nonlinear system of the form  

$$\dot x=f(x),$$ (59)

where x takes values in ${\rm I\!R}^n$ and f is a smooth mapping from ${\rm I\!R}^n$to ${\rm I\!R}^n$. Suppose there is an equilibrium solution x=x_c, i.e. we have f(x_c)=0. To study the behavior of solutions of (2.1), it is natural to set x=x_c+y, and if y is small, we expect to get a first approximation to the behavior of solutions if we linearize, i.e. we neglect terms of order $\Vert y\Vert^2$ in (2.1). In this fashion, we find the linearized system

$$\dot y_i=\sum_{j=1}^n {\partial f_i\over\partial x_j}(x_c)y_j.$$ (60)

Example 10

The system

   $$\begin{eqnarray} \dot x_1&=&x_1-2+\sin x_2,\nonumber\ \dot x_2&=&x_1^2-4\end{eqnarray}$$

has the solution x₁=2, x₂=0. The linearized system is

   $$\begin{eqnarray} \dot y_1&=&y_1+y_2,\nonumber\ \dot y_2&=&4y_1.\end{eqnarray}$$

In this chapter, we explore the question how qualitative features of the linearized system, e.g. stability, persist in the nonlinear system.