Approximating Non Linear Systems by Using The Matrix Eponential

For simplicity let's consider a very simple ODE.

[tex]\dot{x_1}=a x_1^2[/tex]

We can approximate this first order system with a second order ODE as follows:

[tex]\left[ \begin{array}{c}
\dot{x_1} \\
\dot{x_2} \end{array} \right]
=
\left[ \begin{array}{ccc}
0 & 1) \\
0 &
\frac{d f(x_1)}{d x_1}
\end{array} \right]

\left[ \begin{array}{c}
X_1) \\
X_2 \end{array} \right][/tex]

Where

[tex]X_2=\frac{dX_1}{dt}[/tex]

Or in the simple case mentioned above we have:

[tex]\left[ \begin{array}{c}
\dot{x_1} \\
\dot{x_2} \end{array} \right]
=
\left[ \begin{array}{ccc}
0 & 1) \\
0 &
\frac{2x_1(t_o) \end{array} \right]

\left[ \begin{array}{c}
X_1) \\
X_2 \end{array} \right][/tex]

Using the matrix exponential the solution to the linear approximation of this stem as follows:

[tex]\left[ \begin{array}{c}
\x(t) \\
\dot{x(t)} \end{array} \right]
=
exp(\left( \left[ \begin{array}{ccc}
0 & 1) \\
0 &
\frac{2x_1(t_o) \end{array} \right] (t-t_o) \right)

\left[ \begin{array}{c}
X(t_o)) \\
\dot{X(t_o)} \end{array} \right][/tex]

Where:

dot{X(t_o)}=aX_1(t_o)^2