Part 2.3: Linearization

(a*b)/c+13*d \[ {\frac {ab}{c}}+13\,d \] 

With(DEtools):
ODE :=[diff(x(t),t)=y(t),diff(y(t),t)=-sin (x(t))-0.1*y(t)];
DEplot( ODE, [x(t),y(t)],t=-10..30,[[x(0)=3,y(0)=0], [x(0)=-3,y(0)=0],[x(0)=0,y(0)=3], 
[x(0)=0,y(0)=-3],[x(0)=0,y(0)=4], [x(0)=0,y(0)=-4],[x(0)=4,y(0)=0], [x(0)=-4,y(0)=0]],
x=-3*Pi..3*Pi,y=-5..5,stepsize=0.05,
dirgrid=[21,21], color=red,linecolor=blue,
axes=BOXED,title=" Damped pendulum: Stable spiral ",arrows=SLIM);
\[ {\frac {ab}{c}}+13\,d \]
Consider the following system:
\begin{equation} \label{EqLinear.1} \frac{{\text d}{\bf x}(t)}{{\text d}t} = {\bf A}\,{\bf x}(t) + {\bf g}\left( {\bf x} \right) , \end{equation}
and suppose that x = 0 is an isolated critical point of Eq.\eqref{EqLinear.1}. This system is called an almost linear system (or local linear) in the neighborhood of x = 0 if
  • g(x) has ontinuous partial derivatives;
  • as x → 0,
    \[ \frac{\| {\bf g}({\bf x}) \|}{\| {\bf x} \|} \rightarrow 0 , \qquad \mbox{where} \quad \| {\bf x}\| = \left( x_1^2 + x_2^2 + \cdots + x_n^2 \right)^{1/2} . \]
If functions f(x, y) and g(x, y) in planar system
\begin{equation} \label{EqLinear.2} \dot{x} = f(x,y) , \qquad \dot{y} = g(x,y) \end{equation}
have two continuous derivatives in the neighborhood of the critical point (x0, y0), then system \eqref{EqLinear.2} is almost linear.