Phase Portrait
Phase portraits are solution trajectories in phase space; for autonomous systems there is one trajectory through each point of phase space, except at fixed points.
Non-degenerate classification for $n = 2$
For the homogeneous system there is a fixed point at , and for with distinct non-zero eigenvalues the general solution is
The local portrait near the fixed point is determined by and :
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Real eigenvalues of opposite signs: can be chosen real, and the fixed point is a saddle node.
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Real eigenvalues of the same sign, say : both positive gives an unstable node; both negative gives a stable node.
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Complex conjugate eigenvalues with : writing ,
so the trajectories spiral with radial factor . Then gives an unstable spiral, a stable spiral, and a centre with closed elliptical trajectories.
Sketching an overall portrait
Consider a population of prey and predators with the equations
where are positive constants.
Consider a specific case with
The fixed points satisfy
There are three fixed points: , and .
We have
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At , we have
The eigenvalues are and , with eigenvectors and .
Thus, it is a saddle point.
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At , we have
The eigenvalues are and , with eigenvectors and .
Thus, it is a saddle point.
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At , we have
The eigenvalues are found by solving
Since , the fixed point is a stable spiral.
At , we have , so the motion is counter-clockwise.
Now, we can sketch the overall phase portrait.
Related
Stated in
- Definition 9.13 (Phase Portrait)§9.4.2 Non-Degenerate Phase portraits
- Example 9.14 (Predator-Prey Model)§9.5 Non-Linear Dynamical Systems