Homogeneous Linear Systems of ODEs with Constant Coefficients (Three-Dimensional)

There are 41 distinct cases in the three-dimensional setting, grouped below by eigenvalue type.

Distinct real eigenvalues (Cases 1–7)

Three distinct positive eigenvalues:

Two positive, one zero:

Two positive, one negative:

One positive, one zero, one negative:

One positive, two negative:

One zero, two negative:

Three distinct negative eigenvalues:

Repeated real eigenvalues — one pair repeated, one distinct (Cases 8–23)

Repeated positive eigenvalue

Geometric multiplicity 2 (semisimple), distinct eigenvalue positive:

Defective, distinct eigenvalue positive:

Geometric multiplicity 2 (semisimple), distinct eigenvalue zero:

Defective, distinct eigenvalue zero:

Geometric multiplicity 2 (semisimple), distinct eigenvalue negative:

Defective, distinct eigenvalue negative:

Repeated negative eigenvalue

Geometric multiplicity 2 (semisimple), distinct eigenvalue positive:

Defective, distinct eigenvalue positive:

Geometric multiplicity 2 (semisimple), distinct eigenvalue zero:

Defective, distinct eigenvalue zero:

Geometric multiplicity 2 (semisimple), distinct eigenvalue negative:

Defective, distinct eigenvalue negative:

Repeated zero eigenvalue

Geometric multiplicity 2 (semisimple), distinct eigenvalue positive:

Defective, distinct eigenvalue positive:

Geometric multiplicity 2 (semisimple), distinct eigenvalue negative:

Defective, distinct eigenvalue negative:

Triple repeated real eigenvalues (Cases 24–32)

Triple positive eigenvalue

Geometric multiplicity 3 (semisimple):

Geometric multiplicity 2:

Geometric multiplicity 1 (fully defective):

Triple zero eigenvalue

Geometric multiplicity 3 (semisimple):

Geometric multiplicity 2:

Geometric multiplicity 1 (fully defective):

Triple negative eigenvalue

Geometric multiplicity 3 (semisimple):

Geometric multiplicity 2:

Geometric multiplicity 1 (fully defective):

One real eigenvalue + one complex conjugate pair (Cases 33–41)

Complex pair with negative real part

Real eigenvalue negative:

Real eigenvalue zero:

Real eigenvalue positive:

Complex pair with zero real part (purely imaginary)

Real eigenvalue negative:

Real eigenvalue zero:

Real eigenvalue positive:

Complex pair with positive real part

Real eigenvalue negative:

Real eigenvalue zero:

Real eigenvalue positive: