Dynamic Behavior of a Simple Canonical System

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Consider the following system of ODEs:


d​xd​t=α x+β y&IndentingNewLine;d​yd​t=-β x+α y.

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).


Contributed by: Housam Binous (March 2011)
Open content licensed under CC BY-NC-SA



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

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