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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. [PDF] 

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).
 
Some brief comments on the method. The choice of the interval extensions can be easily made by applying any sort of basic mathematical property. Care, however, should be taken as the compiler or the routine used to simulate the system can modify (optimize) the original code.
 
Regarding the second step, only a line fitting to the first slope of natural logarithm of the error is necessary. Computationally, a range from typically 10−14 to 10−1 could be defined. 
 
In some sense, the simplicity of the method proposed here is based on two aspects. First, this method is based on the very definition of chaotic systems proposed by Devaney and therefore errors will eventually lead to large scale divergence.
 
Second, a very simple strategy to produce a measure related to these errors is presented using only heuristic arguments. Following this rationale, note that the lower bound error is a measure of the distance between the simulated dynamical systems (or pseudo-orbit) and the real orbit. If a system is chaotic the distance between these two entities must be exponentially divergent, and therefore a slope in a logarithmic plot of the lower bound error is what is needed to capture such a divergence and quantified as a number which is precisely the definition of the positive Lyapunov exponent. 
 
The method does not need any sort of parametrization, embedding dimension and estimation of the linearized flow but uses the original dynamic equation. The estimates of the LLE using the proposed method were shown to be in a good agreement with the values found in the literature for five well-known chaotic systems.
 
Publications that cites this paper may be found in Google Scholar.
 
The Code used in the paper can be verified on Octave Online.

Última atualização: 23/08/2017