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# Approximating Non Linear Systems by Using The Matrix Eponential

Posted Jul17-09 at 10:39 PM by John Creighto

For simplicity let's consider a very simple ODE.

$$\dot{x_1}=a x_1^2$$

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

$$\left[ \begin{array}{c} \dot{x_1} \\ \dot{x_2} \end{array} \right] = \left[ \begin{array}{ccc} 0 & 1) \\ 0 & \frac{d f(x_1)}{d x_1} \end{array} \right] \left[ \begin{array}{c} X_1) \\ X_2 \end{array} \right]$$

Where

$$X_2=\frac{dX_1}{dt}$$

Or in the simple case mentioned above we have:

$$\left[ \begin{array}{c} \dot{x_1} \\ \dot{x_2} \end{array} \right] = \left[ \begin{array}{ccc} 0 & 1) \\ 0 & \frac{2x_1(t_o) \end{array} \right] \left[ \begin{array}{c} X_1) \\ X_2 \end{array} \right]$$

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

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

Where:

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