[Math] Phase Portraits in 3D differential equations

eigenvalues-eigenvectorslinear algebraordinary differential equations

We know that when we have two equations and two variables, there are certain rules that make the phase portrait a saddle, node, etc… based on whether eigenvalues are positive or negative. For example, if one is positive and one is negative, the phase portrait is saddle. If both are negative, it is a sink node, etc…

I was wondering if there are sets of similar rules with three variables instead of two. This would mean three eigenvalues and I was wondering if there are ways to construct rules with these three eigenvalues. Any input? Thanks.

Best Answer

Yes; the concept of "phase space" is a general one.

Say you have $n$ time-dependent variables $x_1, \ldots, x_n$ related by a system of autonomous linear differential equations:

$$\dot{\mathbf{x}}=A\mathbf{x}$$

The "phase space" is $\mathbb{R}^n$. It is foliated by solution trajectories, just as in the phase plane when $n=2$.

The geometry of this foliation depends on the nature of the eigenvalues of $A$. Roughly speaking:

  • eigenvectors associated with eigenvalues with negative real part span the stable subspace
  • eigenvectors associated with eigenvalues with positive real part span the unstable subspace
  • eigenvectors associated with eigenvalues with zero real part span the center subspace

All this is carefully spelled out in many books on continuous dynamical systems.

See, for example, Section 1.9 ("Stability Theory") of Lawrence Perko, Differential Equations and Dynamical Systems.

You can see some 3D phase space pictures in Section 3.8.2 here.

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