Case of complex (nonreal) eigenvalues ,

Example: consider the ODE system

A = sym([4 5;-4 0])

A = 

This gives the eigenvalues . Note that .

r = eig(A) % find eigenvalues of A

r = 

Find the eigenvectors: For each eigenvalue r find with

[V,D] = eig(A) % find matrix V with eigenvectors, diagonal matrix D with eigenvalues

V = 

D = 

For we get the eigenvector . This gives the complex solution

For we get the eigenvector . This gives the complex solution

RECIPE: Find a fundamental set of real solutions

• Pick a complex solution
• Let
• Let

Here we have and . This gives the fundamental set of solutions

syms y(t) [2 1] % define vector valued function y(t) = [y1(t);y2(t)]

syms C1 C2

sol = dsolve( diff(y)==A*y ); % find general solution

ysol = simplify( subs(y(t),sol) ) % in y(t) substitute the values from sol

ysol = 

Y1 = subs(ysol,{C1,C2},{1,0})

Y1 = 

Y2 = subs(ysol,{C1,C2},{0,1})

Y2 = 

Plot vector field and trajectories in the phase plane

Plot the curves , for

f = @(t,y) A*y; % define function f(t,y) for vectorfield

vectorfield(f,-20:2:20,-20:2:20); hold on

fplot(Y1(1),Y1(2),[-2,2],'b'); % plot trajectory for Y1 in blue

fplot(Y2(1),Y2(2),[-2,2],'k'); % plot trajectory for Y2 in black

fplot(-Y1(1),-Y1(2),[-2,2],'b:'); % plot trajectory for -Y1 in blue, dotted

fplot(-Y2(1),-Y2(2),[-2,2],'k:'); hold off; % plot trajectory for -Y2 in black, dotted

axis equal; axis([-20 20 -20 20]); xlabel('y_1'); ylabel('y_2')

title('four trajectories in the phase plane')