Mendes E. M. A. M. and Nepomuceno E. G. (2016), "A very simple method to calculate the (positive) Largest Lyapunov Exponent using intervak extensions", International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. Vol. 26(13), pp. 1650226. []
In this paper, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. In this page, we give a more informal description about its main ideas.
The very simple, but robust, fast and easy to implement method proposed in this work can be summarized in some steps, considering the logistic map:
1) Choose two interval extensions of the system under investigation. Certain properties of real arithmetic are not valid in floating-point arithmetic, such as associativity of addition. Thus, it is possible to find two mathematical equivalent sequence of arithmetic operations that can lead to two different results. |
xn+1 = µxn(1 − xn) xn+1 = µxn − µxnxn |
2) With exactly the same initial conditions, step size and discretization scheme, simulate the two interval extensions. In this case: x(1) = 2/3; r = 4; N = 200; x1 = x; x2 = x;
for k=1:N
x1(k+1) = r*x1(k)*(1-x1(k));
x2(k+1) = r*x2(k)-r*x2(k)*x2(k);
end
plot(x1,'-or')
hold on
plot(x2,'-xk')
box off
xlabel('t')
ylabel('X1, X2')
axis([0 60 0 1])
|
Result of simulation for x1 (−o−) and x2 (−x−) for Logistic map. |
3) Use the least squares method to fit a line to the slope of the ln curve of the absolute value of the natural algorithm error (the lower bound error).
T=1:k+1;
lbe=abs(x1-x2)/2;
figure(1)
plot(T,log(lbe),'-k.')
hold on
t1=T';
aux=[find(log10(lbe)>-1)];
aux=[find(log10(lbe)>-15)];
deldata=aux(1);
lbe(1:aux(1))=[];
t1(1:aux(1))=[];
aux=[find(log10(lbe)>-1)];
lbe(aux(1):end)=[];
t1(aux(1):end)=[];
aux1=polyfit(t1',log(lbe),1);
lyap=aux1(1)
plot(t1,aux1(1)*t1+aux1(2),'r','LineWidth',2)
axis([0 210 -40 0])
box off
xlabel('t','FontSize',18)
ylabel('log(|X1-X2|)','FontSize',18)
text(64,-20,sprintf('%1.3f%1.3f',aux1(1),aux1(2)))
|
The lower bound error for the logistic map. The red line is the least squares fit. In the figure, the equation of the line is also shown, where the first value is the estimate of the LLE. The x-axis is time and y-axis is the natural logarithm of the absolute value of lower bound error. Source: Fig 1(a) (Mendes and Nepomuceno, 2016). |
Última atualização: 23/08/2017