Phase plane

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Systems of differential equations are collectively of the general form

dx/dt = Cx

where C may be any combination of constants in order to create linear combinations with x on the right side; here x is in bold to indicate it is actually a vector, not a scalar.

To solve such systems, they may be solved algebraically (as seen here), or more commonly the coefficients of the right hand side are written in matrix form, and the system solved using eigenvalues and eigenvectors. The eigenvalues represent the terms of the exponential components and the eigenvectors are the constants which, if the solutions are written in algebraic form, express the fundamental multiplicative factor of the exponential term. Due to the nonuniqueness of eigenvectors it is important to note that every solution arrived at this way has undetermined constants c₁, c₂, and so on, up to the number of eigenvectors.

For the special case of a two-by-two matrix representing a system of differential equations, the solutions are:

$x = \begin{bmatrix} k_{1} \\ k_{2} \end{bmatrix} c_{1}e^{\lambda_1 t} + \begin{bmatrix} k_{3} \\ k_{4} \end{bmatrix} c_{2}e^{\lambda_2 t}$

Here, $λ$ ₁ and $λ$ ₂ are the eigenvalues, and the two matrices containing (k₁, k₂), (k₃ and k₄) are the basic eigenvectors. The constants c₁ and c₂ account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system.

The phase plane is then first set-up by drawing straight lines representing the two eigenvectors (which represent stable situations where the system either converges towards those lines or diverges away from them). Then the phase plane is plotted by using full lines instead of direction field dashes, and the signs of the eigenvalues will tell how the system's phase plane behaves:

If the signs are opposite, the intersection of the eigenvectors is a saddle point.
If the signs are both positive, the eigenvectors represent stable situations which the system diverges away from, and the intersection is an unstable node.
If the signs are both negative, the eigenvectors represent stable situations which the system converges towards, and the intersection is a stable node.

The above can be visualized by recalling the behavior of exponential terms in differential equation solutions.

This page covers only the case for real, separate eigenvalues. Real, repeated eigenvalues require solving the coefficient matrix with an unknown vector and the first eigenvector to generate the second solution of a two-by-two system. However, if the matrix is symmetric, it is possible to use the orthogonal eigenvector to generate the second solution.

Complex eigenvalues and eigenvectors generate solutions in the form of sines and cosines as well as exponentials, and one of the simplicities in this situation is that only one of the eigenvalues and one of the eigenvectors is needed to generate the full solution set for the system. This will be offered without proof, but the details can be seen in any standard textbook on the subject.

More generally, phase planes are useful in visualizing the behavior of physical systems; in particular, of oscillatory systems such as predator-prey models (see Lotka-Volterra equations), which can "spiral in" towards zero, "spiral out" towards infinity, or reach neutrally stable situations called centres where the path traced out is circular, elliptical, or ovoid, or some variant thereof. This is useful in determining if the population dynamics are stable or not.

Other examples of oscillatory systems are certain chemical reactions with multiple steps, some of which involve equilibria rather than reactions that go to completion. In such cases one can easily model the rise and fall of reactant and product concentration (or mass, or number of moles) with the correct differential equations and a good understanding of chemical kinetics.