Dynamic Behavior of a Simple Canonical System

Consider the following system of ODEs:
The eigenvalues of this simple canonical system are . The extremum, , is shown as a green dot.
If , the extremum is an unstable focus.
If , the extremum is a stable focus.
If , the dynamic behavior is that of a limit cycle and the critical point is a center.
If , the trajectories spiral clockwise around the origin.
If , the trajectories spiral counterclockwise around the origin.
The red curve is the parametric plot of the solution of the system of ODEs with an initial condition (shown as a cyan dot).

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DETAILS

S. Lynch, Dynamical Systems with Applications using Mathematica, Boston: Birkhäuser, 2007.
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