I tried posting this to my blog but the preview function wasn't rendering the formula's correctly and once I posted it I couldn't edit it.(adsbygoogle = window.adsbygoogle || []).push({});

https://www.physicsforums.com/blog.php?b=1152 [Broken]

Therefor I'll post it here instead.

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 & 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 &

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:

[tex]\dot{x}(t_o)=ax(t_o)^2[/tex]

Keep in mind the choice of using a second order approximation was somewhat arbitrary. We could of equally well, done a third order approximation as follows:

[tex]

\left[ \begin{array}{c}

\dot{x_1} \\

\dot{x_2} \\

\dot{x_3} \end{array} \right]

=

\left[ \begin{array}{ccc}

0 & 1 & 0\\

0 & 0 & 1\\

0 & 2 a & 2 a x(t_o)

\end{array} \right]

\left[ \begin{array}{c}

x_1 \\

x_2 \\

x_3 \end{array} \right]

[/tex]

Where

[tex]

x_2=\frac{dx_1}{dt}, x_3=\frac{d^2x_1}{dt^2}

[/tex]

and

[tex]\dot{x}(t_o)=ax(t_o)^2[/tex]

[tex]\ddot{x}(t_o)=2 a x(t_o) \dot{x}(t_o)=2a^2x(t_o)^3[/tex]

Which can also be solved using the matrix exponential.

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# Approximating Non Linear Systems by Using The Matrix Eponential

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