Linearizing ordinary differential equations

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Hi guys, im trying to linearize a coupled non-linear ode. I used partial derivative, and then jacobian matrix, i have seen paper using state-space model of jacobian matrix. I cant get a proper reference on this state-space model.

Attached is the non-linear ode, the partial derivative of the non-linear ode and the state-space model of jacobian matrix.

Can someone enhance or explain how they got the state-space model as the transformation, is it a fix formulation. Sorry for asking this way cause I cant find any books or reference referring or explaining this. Please help me! thanks!
 
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