% Evaluate the van der Pol ODE for µ=0.2 and µ=1, then plot the outcomes. % % The van der Pol equation (y_1)"-µ*(1-(x_1)^2)*y'+(y_1)=0 is equivalent % to a system of coupled first-order differential equations % (y_1)'=(y_2) % (y_2)'=µ*[1-(y_1)^2]*(y_2)-(y_1). % % The ODEs commands are as follows: x0=[0.1 0.1]; [t,Q] = ode45('vdpeq1',[0 100],x0); subplot(221); plot(t,Q(:,1),'-',t,Q(:,2),'r-.'); xlabel('time t'); ylabel('solution y_1 and y_2'); title('van der Pol with \mu=0.2'); axis([0 100 -2.5 2.5]); subplot(222); plot(Q(:,1),Q(:,2)); xlabel('y_1'); ylabel('y_2'); title('van der Pol with \mu=0.2'); axis([-2.5 2.5 -2.5 2.5]); x0=[1 0];%x0=[0.9 0.9]; [t,W] = ode45('vdpeq2',[0 30],x0); subplot(223); plot(t,W(:,1),'-',t,W(:,2),'r-.'); xlabel('time t'); ylabel('solution y_1 and y_2'); title('van der Pol with \mu=1'); axis([0 30 -3 3]); subplot(224); plot(W(:,1),W(:,2)); xlabel('y_1'); ylabel('y_2'); title('van der Pol with \mu=1') axis([-2.1 2.1 -3 3]); % Alternately, µ can be taken equal to µ=10, % so the ODE command could be as follows: % [t,W] = ode15s('vdpeq3',[0 100],x0); % with x0=[2 0]; % "Complex and Chaotic Nonlinear Dynamics. % Advances in Economics and Finance, % Mathematics and Statistics" % T.Vialar, Springer 2009 % Copyright(c).