Phase portraits for autonomous linear ODE system

We consider the ODE system

with a nonsingular matrix

. In this case there is one stationary point at

We obtain the following types of stationary points:

REAL EIGENVALUES:

both positive	different	nodal source	unstable
	equal	radial source or twist source (deficient case)	unstable
both negative	different	nodal sink	stable
	equal	radial sink or twist sink (deficient case)	stable
positive & negative		saddle	unstable

Equal eigenvalues: if there are two eigenvectors we get a radial sink/source. If there is only one eigenvector we get a twist sink/source.

NONREAL EIGENVALUES:

real part positive	spiral source	unstable
real part negative	spiral sink	stable
real part zero	center	stable

We get a general solution

In the phase portaits below we always show eight trajectories: We use

and

(for

we just get the constant solution

clearvars

Two different negative eigenvalues: NODAL SINK

A = sym([-4 -3; -1 -2])

A =

The eigenvalues are

[V,D] = eig(A)

V =

D =

phaseportrait(A)

Note: for two different POSITIVE eigenvalues we get a NODAL SOURCE (phase portrait as above, but with arrows pointing in the other direction).

positive and negative eigenvalue: SADDLE

A = sym([1 2;4 -1])

A =

The eigenvalues are

[V,D] = eig(A)

V =

D =

phaseportrait(A)

two equal negative eigenvalues, two eigenvectors: RADIAL SINK

A = sym([-2 0;0 -2])

A =

The eigenvalues are

[V,D] = eig(A)

V =

D =

phaseportrait(A)

Note: for two equal POSITIVE eigenvalues, two eigenvectors we get a RADIAL SOURCE (phase portrait as above, but with arrows pointing in the other direction).

two equal negative eigenvalues, one eigenvector: TWIST SINK

A = sym([-1 -1;1 -3])

A =

The eigenvalues are

. There is only one eigenvector

[V,D] = eig(A)

V =

D =

We have to find a generalized eigenvector

satisfying

: We can use e.g.

[V,D] = jordan(A) % find eigenvectors and generalized eigenvectors

V =

D =

phaseportrait(A)

Note: for two equal POSITIVE eigenvalues, only one eigenvector we get a TWIST SOURCE (phase portrait as above, but with arrows pointing in the other direction).

nonreal eigenvalues, real part negative: SPIRAL SINK

A = sym([-1 5;-2 -3])

A =

The eigenvalues are

[V,D] = eig(A)

V =

D =

phaseportrait(A)

Note: for nonreal eigenvalues with POSITIVE real part we get a SPIRAL SOURCE (phase portrait as above, but with arrows pointing in the other direction).

nonreal eigenvalues, real part zero: CENTER

A = sym([1 -5; 1 -1])

A =

The eigenvalues are

[V,D] = eig(A)

V =

D =

phaseportrait(A)

Function for drawing phase portrait

function phaseportrait(A)
syms y(t) [2 1]
syms C1 C2
sol = dsolve(diff(y)==A*y);             % solve ODE 
ysol = subs(y(t),sol);
f = @(t,y) A*y;                         % define function f(t,y) for vectorfield
vectorfield(f,-5:5,-5:5); hold on
for c1=[-1 0 1]
  for c2=[-1 0 1]
    ys = subs(ysol,{C1,C2},{c1,c2});    % use C1=c1, C2=c2
    fplot(ys(1),ys(2),[-5 5]);          % plot trajectory for t=-5...5
  end
end
hold off; axis equal; axis([-5 5 -5 5])
end