Phase plane

In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc.

The phase plane method refers to graphically determining the existence of limit cycles in the solutions of the differential equation.

With enough of these arrows in place the system behaviour over the regions of plane in analysis can be visualized and limit cycles can be easily identified.

The flows in the vector field indicate the time-evolution of the system the differential equation describes.

In such cases one can model the rise and fall of reactant and product concentration (or mass, or amount of substance) with the correct differential equations and a good understanding of chemical kinetics.

[1] More commonly they are solved with the coefficients of the right hand side written in matrix form using eigenvalues λ, given by the determinant: and eigenvectors: The eigenvalues represent the powers of the exponential components and the eigenvectors are coefficients.

If the solutions are written in algebraic form, they express the fundamental multiplicative factor of the exponential term.

The eigenvectors and nodes determine the profile of the phase paths, providing a pictorial interpretation of the solution to the dynamical system, as shown next.

Then the phase plane is plotted by using full lines instead of direction field dashes.

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.

Complex eigenvalues and eigenvectors generate solutions in the form of sines and cosines as well as exponentials.

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.