CONSTANT COEFFICIENTS HOMOGENEOUS SYSTEMS

Assume that we are given the first order system ${\bf x}' = A{\bf x}.$

Basic Method

Try ${\bf x}={\bf v}e^{rt}$; then $r{\bf v}e^{rt}=A{\bf v}e^{rt}$, so $(A-rI){\bf v}={\bf0}$
and solutions are determined from $(r,{\bf v})$'s for A.

Graphical Solutions, n = 2 Case:

plots in the x₁-x₂
plane (the phase plane) are called phase portraits.
Direction field is a plot of $A{\bf x}$ vectors for various ${\bf x}$'s.
If both eigenvalues are real, the origin is a
saddle point if the signs are opposite, or a
node if the signs are the same.
If both eigenvalues are complex, the origin is a
spiral point if both real parts are nonzero, or a
center if both real parts are zero.

Hermitian systems:

if A is Hermitian with eigenvalues $r_1,\ \ldots,\ r_n$ and associated eigenvectors ${\bf v}^{(1)}, \ldots, {\bf v}^{(n)}$, then the vectors ${\bf x}^{(1)}={\bf v}^{(1)}e^{r_1t},\ \ldots,\ {\bf x}^{(n)}={\bf v}^{(n)}e^{r_nt}$ form a fundamental set of solutions to ${\bf x}'= A{\bf x}$.

Non-Hermitian systems:

assume a real A; three cases.

• All eigenvalues real and distinct;
  solutions similar to the Hermitian A solutions.
• Some complex eigenvalues;
  need to find real valued solutions.
• Some repeated eigenvalues;
  need to use extra terms of the form ${\bf u}t^je^{rt}$.

${\bf x}'= A{\bf x}$ CONTINUED

Complex Eigenvalues

• A is real, so we want real solutions.
• Complex eigenvalues occur as conjugate pairs $\lambda \pm i\mu$.
• Let $r = \lambda + i\mu$, with eigenvector ${\bf v}={\bf a}+i{\bf b}$;
  then $(A-rI){\bf v}={\bf0}$, so $(A-\bar{r}I)\bar{{\bf v}} = {\bf0}$ and therefore
  the eigenvector for $\bar{r}= \lambda - i\mu$ is $\bar{{\bf v}}={\bf a}-i{\bf b}$.
• Let ${\bf x}^{(1)}={\bf v}e^{rt}=({\bf a}+i{\bf b})e^{\lambda t}(\cos(\mu t)+i\sin(\mu t))$,
  and ${\bf x}^{(2)}= \bar{{\bf v}}e^{\bar{r}t}=({\bf a}-i{\bf b})e^{\lambda t}(\cos(\mu t)-i\sin(\mu t))$.
• Two linearly independent real solutions are
  ${\bf u}=e^{\lambda t}({\bf a}\cos(\mu t)-{\bf b}\sin(\mu t))$ and
  ${\bf w}=e^{\lambda t}({\bf a}\sin(\mu t)+{\bf b}\cos(\mu t))$.