Find all equilibria for the following system and determine their stability:
$$x'=y^2-4$$ $$y'=x^2-1$$
2026-04-02 11:56:00.1775130960
Equilibria and stability
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We find four critical points as:
$$(x, y) = (-1,-2),(1,-2), (-1,2), (1,2)$$
We evaluate the Jacobian of the system as:
$$J(x, y) = \left( \begin{array}{cc} 0 & 2 y \\ 2 x & 0 \\ \end{array} \right)$$
We now evaluate the eigenvalues at each critical point and find two centers and two saddles.
A phase portrait with nullclines shows: