%% A nonlinear pendulum
%% The differential equation:
% $$\ddot{\theta}+\omega^2\sin\theta=0,\qquad \theta(0)=0,\qquad \dot{\theta}(0)=1.$$
%% The corresponding system of the first order differential equations: 
% $$\frac{dz_1}{dt}=z_2,\,\, \frac{dz_2}{dt}=-\omega^2\sin(z_1),\,\, z_1(0)=0,\,\, z_2(0)=1$$
%% Input initial conditions:
z0=[0,1];
%% Define the interval on which solution is computed:
tspan =[0,20];
%% Solve the system using |ode45| procedure:
[t,z] = ode45('ode3',tspan,z0);
%% Extract the positions and velocities:
x=z(:,1); v=z(:,2);
%% Plots of the positions and velocities as functions of time:
% *Note:* _The dashed curve indicates velocities_
plot(t,x,t,v,'--')
%% Plot of the phase portrait (velocity as the function of position):
figure(2)
plot(x,v)