It partly depends on what exactly you want. A quick and dirty approach would be to solve the ODE over a fixed interval using ode45, and plot the results:
t = 0:0.001:5;
y0 = [1; 2; 5];
f = @(t, y) [-y(2) - y(3); y(1) + y(2); 1 + y(1)*y(3)];
[~, y] = ode45(f, t, y0);
figure;
comet3(y(:, 1), y(:, 2), y(:, 3));
If you want to plot the results as you incrementally solve the equation, you can use the techniques described in the FAQ or the notes on animation, but you'd have to be careful to ensure that the ODE is well suited to an explicit integration scheme, and that your step size is appropriate:
f = @(t, y) [-y(2) - y(3); y(1) + y(2); 1 + y(1)*y(3)];
y0 = [1; 2; 5];
t0 = 0;
hf = figure;
ha = axes;
hp = plot3(ha, NaN, NaN, NaN, '.', 'MarkerSize', 50);
ht = title('t = ');
set(ha, 'XLimMode', 'manual', 'YLimMode', 'manual', 'ZLimMode', 'manual');
set(ha, 'XLim', [-20, 20], 'YLim', [-90, 10], 'ZLim', [-50, 200]);
dt = 0.001;
tmax = 5;
t = t0;
y = y0;
while t < tmax
t = t + dt;
dy = f(t, y)*dt;
y = y + dy;
set(hp, 'XData', y(1), 'YData', y(2), 'ZData', y(3));
set(ht, 'String', ['t = ', num2str(t, '%5.3f')]);
drawnow;
end
Using one of the ODE solvers is definitely safer, if you care at all about the accuracy of the results...
