% Example of Supercritical Hopf bifurcation % From a stable node to a stable limit cycle shown in the Phase-plane. % Phase-plane plots of the components [X(1),X(2)] from µ=-1 to µ=2. %%% Reminder: Normal Forms of Supercritical Bifurcations.%%%%%%%%%%% %% %% %% (1): dx/dt= µ - x^2 : Node-Saddle %% %% (2): dx/dt= µ x - x^2 : Transcritical (Stability Exchange) %% %% (3): dx/dt= µ x - x ^3 : Pitchfork %% %% (4): dz/dt= (µ + i gamma)z - z|z|^2 with z=complex : Hopf %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all disp('To stop simulation, Press "ctrl-c" in the command window') global mu for mu=-1:0.1:2; tfinal=100; % First set of initial values x0a=[-1.5 -1.5]; x0b=[0 -1.5]; x0c=[1.5 -1.5]; x0d=[1.5 0]; x0e=[1.5 1.5]; x0f=[0 1.5]; x0g=[-1.5 1.5]; x0h=[-1.5 0]; % Initial values close to the origin (0,0). x0ia=[-0.001 -0.001]; x0ib=[0 -0.001]; x0ic=[0.001 -0.001]; x0id=[0.001 0]; x0ie=[0.001 0.001]; x0if=[0 0.001]; x0ig=[-0.001 0.001]; x0ih=[-0.001 0]; [t xa]=ode23(@hopf,[0 tfinal],x0a); [t xb]=ode23(@hopf,[0 tfinal],x0b); [t xc]=ode23(@hopf,[0 tfinal],x0c); [t xd]=ode23(@hopf,[0 tfinal],x0d); [t xe]=ode23(@hopf,[0 tfinal],x0e); [t xf]=ode23(@hopf,[0 tfinal],x0f); [t xg]=ode23(@hopf,[0 tfinal],x0g); [t xh]=ode23(@hopf,[0 tfinal],x0h); [t xia]=ode23(@hopf,[0 tfinal],x0ia); [t xib]=ode23(@hopf,[0 tfinal],x0ib); [t xic]=ode23(@hopf,[0 tfinal],x0ic); [t xid]=ode23(@hopf,[0 tfinal],x0id); [t xie]=ode23(@hopf,[0 tfinal],x0ie); [t xif]=ode23(@hopf,[0 tfinal],x0if); [t xig]=ode23(@hopf,[0 tfinal],x0ig); [t xih]=ode23(@hopf,[0 tfinal],x0ih); pause(0.001) plot(xa(:,1),xa(:,2),'b',xb(:,1),xb(:,2),'b',xc(:,1),xc(:,2),'b',xd(:,1),xd(:,2),'b',... xe(:,1),xe(:,2),'b',xf(:,1),xf(:,2),'b',xg(:,1),xg(:,2),'b',xh(:,1),xh(:,2),'b',... xia(:,1),xia(:,2),'r',xib(:,1),xib(:,2),'r',xic(:,1),xic(:,2),'r',xid(:,1),xid(:,2),'r',... xie(:,1),xie(:,2),'r',xif(:,1),xif(:,2),'r',xig(:,1),xig(:,2),'r',xih(:,1),xih(:,2),'r'); title(['Hopf bifurcation with -1 \leq \mu \leq 2; \mu: ', num2str(mu)]); grid; xlabel('X1'); ylabel('X2'); end % "Complex and Chaotic Nonlinear Dynamics. % Advances in Economics and Finance, % Mathematics and Statistics" % T.Vialar, Springer 2009 % Copyright(c).