91Ó°ÊÓ

The continuous dynamical systems with complex eigenvalues theorem states.

Consider a linear system,

$\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$

Suppose there exists a complex Eigen basis $\stackrel{→}{v_1},…,\stackrel{→}{v_n}$for A , with associated complex eigenvalues ${\mathrm{λ}}_1,…,{\mathrm{λ}}_n$. Then the general complex solution of the system is:

$\stackrel{\mathbf{→}}{\mathbf{x}}\left(t\right)\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{{\mathbf{λ}}_{\mathbf{1}}\mathbf{t}}\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{+}\mathbf{…}\mathbf{+}{\mathit{c}}_{\mathbf{n}}{\mathbf{e}}^{{\mathbf{λ}}_{\mathbf{n}}\mathbf{t}}\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{n}}}$

Here, $c_i$are arbitrary complex numbers.

Consider the system as follows:

$\frac{d\stackrel{→}{x}}{dt}=\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right]\stackrel{→}{x}$

Determine the eigenvalues as follows:

$A=\left[\begin{array}{cc}3& -2\\ 5& -3\end{array}\right]$

Then, the characteristic equation of A is $det\left|A-\mathrm{λ}I\right|=0$which gives;

$\begin{array}{rcl}det\left|\begin{array}{c}\left[\begin{array}{cc}3-\mathrm{λ}& -2\\ 5& -3-\mathrm{λ}\end{array}\right]\end{array}\right|& =& 0\\ -\left(3-\mathrm{λ}\right)\left(3+\mathrm{λ}\right)+10& =& 0\\ {\mathrm{λ}}^2+1& =& 0\\ \mathrm{λ}& =& Â\pm i\end{array}$

The Eigen spaces that correspond to the Eigen values $\mathrm{λ}=Â\pm i$are:

$\begin{array}{rcl}{E}_i& =& ker\left[\begin{array}{cc}3-i& -2\\ 5& -3-i\end{array}\right]\\ E_{-i}& =& span\left[\begin{array}{c}3+i\\ 5\end{array}\right]\\ E_{-i}& =& span\left[\begin{array}{c}3-i\\ 5\end{array}\right]\end{array}$

Hence, the general solution is $\begin{array}{rcl}\stackrel{→}{x}\left(t\right)& =& c_1{e}^{it}\left[\begin{array}{c}3+i\\ 5\end{array}\right]+c_2{e}^{-it}\left[\begin{array}{c}3-i\\ 5\end{array}\right]\end{array}$.

Solve further to obtain general solution as follows:

$\begin{array}{rcl}\stackrel{→}{x}\left(t\right)& =& e^{it}\left[\begin{array}{c}3+i\\ 5\end{array}\right]+e^{-it}\left[\begin{array}{c}3-i\\ 5\end{array}\right]\\ & =& \left(\cos t+i\sin t\right)\left[\begin{array}{c}3+i\\ 5\end{array}\right]+\left(\cos t-i\sin t\right)\left[\begin{array}{c}3-i\\ 5\end{array}\right]\\ & =& \left[\begin{array}{c}6\cos t-2\sin t\\ 10\cos t\end{array}\right]\end{array}$

Hence, when $\begin{array}{rcl}{c}_1& =& c_2=1\end{array}$the solution of the system is:

$\begin{array}{rcl}\stackrel{→}{x}\left(t\right)& =& \left[\begin{array}{c}6\cos t-2\sin t\\ 10\cos t\end{array}\right]\end{array}$