Stability and classification of isolated critical points

Almost linear systems, eigenvalue tests for nodes, saddles, spirals and centers, and the borderline cases where a linearization refuses to decide stability.

Notes on Diffy Qs: Differential Equations for Engineers, Stability and classification of isolated critical points

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4 exercises

Exercise 101

For the systems below, find and classify the critical points.

$x'=-x+x^2$ , $y'=y$
$x'=y-y^2-x$ , $y'=-x$
$x'=xy$ , $y'=x+y-1$

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Exercise 102

Find the implicit equations of the trajectories of the following conservative systems. Next find their critical points (if any) and classify them.

$x''+ x^2 = 4$
$x''+ e^x = 0$
$x''+ (x+1)e^x = 0$

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Exercise 103

The conservative system $x''+x^3 = 0$ is not almost linear. Classify its critical point(s) nonetheless.

2.5

Exercise 104

Derive an analogous classification of critical points for equations in one dimension, such as $x'= f(x)$ based on the derivative. A point $x_0$ is critical when $f(x_0) = 0$ and almost linear if in addition $f'(x_0) \neq 0$ . Figure out if the critical point is stable or unstable depending on the sign of $f'(x_0)$ . Explain. Hint: see the “Autonomous equations” section.

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