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# Linearizing a Nonlinear System: Taylor Expansion

DIFFEQ-TLPYRN

Consider the following first order autonomous nonlinear system

$\cfrac{dx}{dt}=(1+y)\sin x$
$\cfrac{dy}{dt}=1-\cos x-y$

The point $(0,0)$ is a critical point of the system. We can linearize the system around $(0,0)$ by Taylor expansion and the system is reduced to a locally linear system:

$$\frac{d}{dt}\vec{x}=A\vec{x}+\vec{g}(\vec{x})$$

...where:

$$\vec{g}=O(|\vec{x}|^2)$$

...as $|\vec{x}|\to 0$.

What are the eigenvalues of $A$?

A

$r_1=1, r_2=-1$

B

$r_1=1, r_2=0$

C

$r_1=0, r_2=-1$

D

$r_{1,2}=\pm\sqrt{\frac{3}{2}}$