%%% Reminder: Normal Forms of Supercritical Bifurcations.%%%%%%%%%%% %% %% %% (1): dx/dt= mu - x^2 : Node-Saddle %% %% (2): dx/dt= mu x - x^2 : Transcritical (Stability Exchange) %% %% (3): dx/dt= mu x - x ^3 : Pitchfork %% %% (4): dz/dt= (mu + i gamma)z - z|z|^2 (z=complex) : Hopf %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Example of a Nonlinear Dynamical System (in which a Hopf % bifurcation can occur for appropriate values of parameters). % The dynamical system is defined as follows: % (1): dx/dt=[µ-(x^2+y^2)]x-gamma y, % (2): dy/dt=gamma x+[µ-(x^2+y^2)]y. % In a general way, it is said that: % . The stationary solution is z=0 (i.e. x=y=0). % . There exists another solution such that |z|^2 is independent from time, % i.e. |z|^2=x^2+y^2=µ, this condition defines the equation of a circle (in % the plane) whose radius is (µ)^1/2. % % The routine attempts to depict (in a discrete way) the orbit for -0.5<µ<3 % while gamma varies from -1 to 1. % % ATTENTION: Considered as an exercise the following question: % IS THIS DISCRETE SIMULATION ("Dynamic2.m") VALID OR NOT? clear all figure(1) disp(' ') disp('PLEASE, ENLARGE FIGURE, THEN, PRESS ANY KEY TO CONTINUE,') disp(' ') disp('TO STOP SIMULATION, TYPE "CTRL-C" IN THE COMMAND WINDOW,') disp(' ') pause s = 1000; k = 700; x0 = rand(1,s); y0 = rand(1,s); A = linspace(-0.5,3,s); x = zeros(k,s); x(1,:) = x0; y = zeros(k,s); y(1,:) = y0; for v=0:0.01:2.02; gamma=v-1; for s = 1:k; x(s+1,:) = (A-((x(s,:).^2)+(y(s,:).^2))).*x(s,:)-gamma.*y(s,:); y(s+1,:) = gamma.*x(s,:)+(A-((x(s,:).^2)+(y(s,:).^2))).*y(s,:); end pause(0.00000001) Ha=ones(1000,1); VX=Ha*A; SX=VX(1:701,:); NX=SX(467:701,:); plot3(NX,x(floor(2.*end/3):end,:),y(floor(2.*end/3):end,:), '.','markersize',1); view(-15,29) axis([-0.5 3 -1.8 1.8 -1.8 1.8]); title(['Orbit of the system: dx/dt=[\mu-(x^2+y^2)]x-\gammay, dy/dt=\gammax+[\mu-(x^2+y^2)]y with \gamma= ', num2str(gamma)]); xlabel('\mu-abscissa: -0.5 \leq \mu \leq 3'); ylabel('X'); zlabel('Y'); grid; %box; end % "Complex and Chaotic Nonlinear Dynamics. % Advances in Economics and Finance, % Mathematics and Statistics" % T.Vialar, Springer 2009. % Copyright(c).