% The first example of specific equations undergoing the catastrophe % (Blue Sky Catastrophe) was given by N. Gavrilov and A. Shilnikov % [Gavrilov and Shilnikov, 2000; L. Shilnikov et al, 2001] % % dx/dt = x(2+mu-beta((x^2)+(y^2)))+(z^2)+(y^2)+2*y % dy/dt = -(z^3)-(1+y)((z^2)+(y^2)+2y)-4x+ mu y % dz/dt = (1+y)(z^2)+(x^2)-eta. % % Early development of the blue sky catastrophe in this system begins % with a homoclinic connection to an equilibrium state with the % characteristic exponents (0,-+i.omega); this is indeed a cod-2 bifurcation named % after Gavrilov-Guckenheimer or aka the homoclinic Fold-Hopf. % The blue sky catastrophe has turned out to be a typical phenomenon % in slow-fast systems [L. Shilnikov et al, 2001; A. Shilnikov et al,2005]. global mu beta eta mu=0.456; beta=10; eta=0.0357; x0=[0 0 0]; [t, X] = ode45(@BSCgeneric, [0 1000], x0); % For t, an interesting parameter setting % can be used [0 10000] rather than [0 100] % m1=min(X(:,1));m2=min(X(:,2));m3=min(X(:,3)); M1=max(X(:,1));M2=max(X(:,2));M3=max(X(:,3)); plot3(X(:,3),X(:,2),X(:,1)); xlabel('x'); ylabel('y'); zlabel('z'); axis([m3 M3 m2 M2 m1 M1]) view([0, 0, 1]); grid box; % "Complex and Chaotic Nonlinear Dynamics. % Advances in Economics and Finance, % Mathematics and Statistics" % T.Vialar, Springer 2009 % Copyright(c).