Limit cycles

Closed trajectories that neighboring solutions spiral into or out of, the Van der Pol oscillator, and criteria that rule periodic orbits in or out of a region.

Notes on Diffy Qs: Differential Equations for Engineers, Limit cycles

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4 exercises

Exercise 101

Show that the following systems have no closed trajectories.

$x'=x+y^2,\quad y'=y+x^2$ ,
$x'=-x\sin^2(y),\quad y'=e^x$ ,
$x'=xy^2,\quad y'=x+x^2$ .

2.5

Exercise 102

Suppose an autonomous system in the plane has a solution $x=\cos(t)(1+e^{-t})$ , $y=\sin(t)(1+e^{-t})$ . What can you say about the system (in particular about limit cycles and periodic solutions)?

2.5

Exercise 103

Show that the limit cycle of the Van der Pol oscillator (for $\mu > 0$ ) must not lie completely in the set where $-1 < x < 1$ . Compare with a figure in the source text.

2.5

Exercise 104

Consider the system $r' = \sin(r)$ , $\theta' = 1$ given in polar coordinates. Find all the closed trajectories.

2.5