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In this paper the authors have the dynamical system

$$\begin{align} T_f \dot{y}_f & = -y_f + (1-\alpha(v))\varphi(z,d) \\ T_r \dot{y}_r & = -y_r + \alpha(v) \varphi(z,d) \\ \dot{z} & = -\varphi(z,d) + y_r + u \end{align}$$

and they state in eqns (8-10) that the eigenvalues of the linearization at the equilibrium points $(\overline{y}_f, \overline{y}_r, \overline{z})$ are

$$\begin{align} \lambda_1 & = -T_f^{-1} \\ \lambda_2 + \lambda_3 & = -\varphi_z(\overline{z},d) - T_r^{-1} \\ \lambda_2 \lambda_3 & = T_r^{-1} \phi_z(\overline{z},d)(1-\alpha(\overline{v})) \end{align}$$

Can someone explain to me how these are derived?

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    You'll get a better response if you type out the dynamical equations and the eigenvalues here, rather than just linking to a paper and saying "How did they derive that?" Alternatively, you could email the authors of the paper (who will be experts on that paper, unlike people here) but it seems that your main problem is that you don't understand how to linearize a dynamical system around a fixed point and calculate its stability there. You could try reading eg [these](http://math.colgate.edu/~wweckesser/math312Spring05/handouts/Linearization.pdf) lecture notes.2011-06-01
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    @Chris Taylor I tried to paste an image, but my rate<10. I can't wait the answer from authors. Thanks for link.2011-06-02
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    I edited to include the equations. Please go and read the notes at that link (they give a general explanation of what the authors are doing in that paper) and then feel free to ask here if you still have questions.2011-06-02
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    @Chris Taylor, thanks a lot!2011-06-06

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