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# The Moore-Spiegel Oscillator

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An analysis of the behaviour of the Moore-Spiegel oscillator with computer simulations.

An analysis of the behaviour of the Moore-Spiegel oscillator with computer simulations.

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### Transcript

• 1. THE MOORE-SPIEGELOSCILLATORABHRANIL DAS
• 2. The System
• 3. Fixed Points and Stability
• 4. Numerical Root-finding% Newton-Raphson method to find rootsdisp Newton-Raphson Methodsyms x;i=0;f=input(f: ); % User inputs function herey=input(seed x: ); % and seed value herewhile (abs(subs(f,x,y)/subs(diff(f),x,y))>1e-15) % termination criterion y=y-subs(f,x,y)/subs(diff(f),x,y); i=i+1;endx=y % print resulti % and iterations% Bisection method to find rootsdisp Bisection Methoda=input(a: ); % startb=input(b: ); % and end of starting intervalj=0; % iteration countsyms x;while (b-a>0.000001) % termination criterion mid=(a+b)/2; if subs(f,x,b)*subs(f,x,mid)<0 a=mid; else b=mid; endj=j+1;endx=midj
• 5. Roots for T=6 and R=20 Root Seed (Newton- Interval Raphson) (Bisection) 3 5 [0,5] 0.4495 0 [0,1] -4.4495 -5 [-5,0]
• 6. Phase-Space Plots with RK4/5 (General Code) t=10; N=10000; h=float(t)/N; l=range(3) T=6; R=20 x=list(input(Starting x,y,z: )) file=open(msplot.txt, w) def f(x): return [x[1], x[2], -x[2]-(T-R+R*x[0]**2)*x[1]-T*x[0]] for iter in range(N): print>> file, x[0],x[1],x[2] k1=[h*f(x)[i] for i in l] k2=[h*f([(x[j]+k1[j]/2) for j in l])[i] for i in l] k3=[h*f([(x[j]+k2[j]/2) for j in l])[i] for i in l] k4=[h*f([(x[j]+k3[j]) for j in l])[i] for i in l] x=[x[i]+(k1[i]+2*k2[i]+2*k3[i]+k4[i]) for i in l] file.close() import Gnuplot g=Gnuplot.Gnuplot() g(splot msplot.txt w l) g(pause -1) global T; global R; T=0; R=20; [tarray,Y] = ode45(@mseq,[0 1000],[-1 1 0]); function dy = mseq(t,y) global T; global R; dy = zeros(3,1); dy(1) = y(2); dy(2) = y(3); dy(3) = -y(3)-(T-R+R*y(1)^2)*y(2)-T*y(1); end
• 7. Phase-Space Plots: Periodic
• 8. Phase-Space Plots: Chaos
• 9. Projection: x-y plane
• 10. Projection: x-z plane
• 11. Projection: y-z plane
• 12. Lyapunov ExponentTwo particles were released from close points in the flow, (-1, 1, 0)and (-1, 1.0001, 0). Characteristic time is ~0.7s:
• 13. Lyapunov Exponent
• 14. Lyapunov Exponent
• 15. Poincaré Sections of projections P=[]; for i=1:length(Y)-1 if (Y(i,2))<0 && (Y(i+1,2))>0 P(end+1)=Y(i,1); end end P=P; plot(P,.);
• 16. Poincaré Sections: Zoomed in
• 17. Bifurcation Diagrams global T; global R; T=0; R=20; B=[]; while T<20 [tarray,Y] = ode45(@mseq,[0 1000],[-1 1 0]); P=[]; for i=1:length(Y)-1 if (Y(i,2))<0 && (Y(i+1,2))>0 P(end+1)=Y(i,1); end end P=P; P=P(end-10:end); for i=1:length(P) B(end+1,:)=[T P(i)]; end T=T+.1 end
• 18. Bifurcation Diagrams R=20
• 19. Bifurcation Diagrams T=6
• 20. ReferenceAlgebraically Simple Chaotic Flows, J.C. Sprott, S J. Linz,Intl. J. of Chaos Theory and ApplicationsA Thermally Excited Non-linear Oscillator, D.W. Moore, E.A.Spiegel, Astrophysical Journal