Systems of ODEs and Phase Portraits

Linear Systems

For $x' = Ax$ , eigenpairs of $A$ give the solutions.

Distinct real eigenvalues $r_i$ with eigenvectors $v_i$ : $x = \sum_i c_i e^{r_i t} v_i$ .
Complex pair $r = \lambda \pm i\mu$ with eigenvector $v = a + ib$ : the real solutions are $e^{\lambda t}\big(a\cos(\mu t) - b\sin(\mu t)\big)$ and $e^{\lambda t}\big(a\sin(\mu t) + b\cos(\mu t)\big)$ .
Repeated eigenvalue $r$ with one eigenvector $v$ : find a generalized eigenvector $(A - rI)w = v$ . Then $x = c_1 e^{rt} v + c_2 \big(t e^{rt} v + e^{rt} w\big)$ .

Phase Portraits of 2-by-2 Systems

Get the eigenvalues and eigenvectors and draw the eigenvector lines. As $t \to \infty$ , trajectories line up with the eigenvector of the larger eigenvalue.

$\lambda_1, \lambda_2 < 0$ distinct: nodal sink, asymptotically stable.
$\lambda_1, \lambda_2 > 0$ distinct: nodal source, unstable.
$\lambda_1 \lambda_2 < 0$ : saddle point, unstable.
complex $\lambda \pm i\mu$ : spiral point, asymptotically stable if $\lambda < 0$ , unstable if $\lambda > 0$ . The direction of rotation follows from evaluating $x'$ at a point, e.g. on an axis.
purely imaginary $\pm i\mu$ : center, closed elliptical orbits around the origin, stable but not asymptotically stable.
repeated $\lambda$ : a proper (star) node if $A = \lambda I$ , an improper node otherwise. Asymptotically stable if $\lambda < 0$ , unstable if $\lambda > 0$ .

Trajectories

For $x' = F(x,y)$ , $y' = G(x,y)$ , the trajectories satisfy

\frac{dy}{dx} = \frac{G(x,y)}{F(x,y)}

and solving this, when possible, gives a conserved quantity $H(x,y) = c$ whose level sets are the trajectories.

Periodic Solutions and Limit Cycles

In polar coordinates $x = r\cos\theta$ , $y = r\sin\theta$ , $r^2 = x^2 + y^2$ ,

r\frac{dr}{dt} = x\frac{dx}{dt} + y\frac{dy}{dt}, \qquad r^2 \frac{d\theta}{dt} = x\frac{dy}{dt} - y\frac{dx}{dt}

A limit cycle is an isolated closed trajectory. It is asymptotically stable if trajectories on both sides approach it, unstable if trajectories on both sides move away, and semistable if they approach on one side and move away on the other.

Bendixson's criterion If $F_x + G_y$ has the same sign everywhere in a simply connected region $D$ , there is no closed trajectory lying entirely in $D$ .