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

1. THE MOORE-SPIEGEL
OSCILLATOR
ABHRANIL DAS

2. The System

3. Fixed Points and Stability

4. Numerical Root-finding
% Newton-Raphson method to find roots
disp 'Newton-Raphson Method'
syms x;
i=0;
f=input('f: '); % User inputs function here
y=input('seed x: '); % and seed value here
while (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;
end
x=y % print result
i % and iterations
% Bisection method to find roots
disp 'Bisection Method'
a=input('a: '); % start
b=input('b: '); % and end of starting interval
j=0; % iteration count
syms 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;
end
j=j+1;
end
x=mid
j

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 Exponent
Two 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. Reference
Algebraically Simple Chaotic Flows, J.C. Sprott, S J. Linz,
Intl. J. of Chaos Theory and Applications
A Thermally Excited Non-linear Oscillator, D.W. Moore, E.A.
Spiegel, Astrophysical Journal