Gallery

Homogenous linear systems of ODEs with constant coefficients (two-dimensional)

There are 14 distinct configurations of the eigenvalues in the two-dimensional case.

Real eigenvalues

Distinct

Two real eigenvalues (one positive, one negative):

Two real eigenvalues (both negative):

Two real eigenvalues (both positive):

Two real eigenvalues (one zero, one negative):

Two real eigenvalues (one zero, one positive):

Repeated

Two real eigenvalues (repeated positive with geometric multiplicity of one)

Two real eigenvalues (repeated positive with geometric multiplicity of two)

Two real eigenvalues (repeated negative with geometric multiplicity of one)

Two real eigenvalues (repeated negative with geometric multiplicity of two)

Two real eigenvalues (repeated zero with geometric multiplicity of one)

Two real eigenvalues (repeated zero with geometric multiplicity of two)

Complex eigenvalues

Complex conjugate eigenvalues (positive real part):

Complex conjugate eigenvalues (zero real part / purely imaginary):

Complex conjugate eigenvalues (negative real part):

Norm of the matrix exponential:

Homogenous linear systems of ODEs with constant coefficients (three-dimensional)

There are many more possible configurations of the eigenvalues than in the two-dimensional case; you can find the 41 possible configurations in the three-dimensional case here.

Inhomogenous linear systems of ODEs with constant coefficients (two-dimensional)

Some nonlinear examples

The following set of examples are from Spectra and Pseudospectra by Nick Trefethen and Mark Embree (see section 21).

Pendulum

Chaotic attractors

Three-body problems

20 examples of periodic solutions to the three-body problem. (Source, Wikimedia Commons, CC BY-SA 4.0)

If you are interested in periodic solutions to n-body problems, check out this webpage.