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Lyapunov exponents for multi-parameter tent and logistic maps
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10.1063/1.3645185
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1 School of Computing and Mathematics, University of Ulster, Newtownabbey, Northern Ireland
Chaos 21, 043104 (2011)
/content/aip/journal/chaos/21/4/10.1063/1.3645185
http://aip.metastore.ingenta.com/content/aip/journal/chaos/21/4/10.1063/1.3645185
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Figures

FIG. 1.

The fraction of parameter space for the multi-parameter tent map (9) for which the global Lyapunov exponent is positive as given by (15). The dots represent (15) and the dashed and dotted lines give the upper and lower bounds as given by (21).

FIG. 2.

(a) Graph of (24), which is the fraction of (A, B) parameter space which is chaotic for the bi-parameter tent map where the control parameter varies as (22). Note that which is the fraction of parameter space which is chaotic for the single parameter tent map (2).(b) The fraction of (A,B) parameter space which is chaotic for the bi-parameter logistic map where the control parameter varies as (22). Note that which is the fraction of parameter space which is chaotic for the single parameter logistic map (1). The map is iterated over 104 steps and averaged over 50 uniformly distributed starting values. The incremental change in each control parameter is performed over a mesh of 10 001 × 10 001 points.(c) Data from (b) for p ≥ 70 plotted as with corresponding linear fit. Linear regression gives an equation of the form with R2 = 0.9995.(d) The fraction of (A,B) parameter space which leads to chaotic behaviour as starting value is varied. As with (b) the map is iterated over 104 steps and the incremental change in each control parameter is performed over a mesh of 10 001 × 10 001 points. Note that though for small values of p the variation appears significant it is, for the data presented here, a variation of at most 3.3% when p = 5 (the third data cluster from the left). For values of p > 70 the variation is less than 0.1%.

FIG. 3.

(a) Global Lyapunov exponent for the bi-parameter logistic map where the control parameter varies via (25) with p = 6. The map is iterated over 104 steps and averaged over 50 uniformly distributed starting values. The incremental change in each control parameter is performed over a mesh of 10 001 × 10 001 points. White areas signify regions where the global Lyapunov exponent is less than −3, black areas where it is greater than 0. (b) Global Lyapunov Exponent for the bi-parameter logistic map where the control parameter varies via (25) with p = 6. The map is iterated over 104 steps with a starting value of x0 = 0.1. The incremental change in each control parameter is performed over a mesh of 10 001 × 10 001 points. White areas signify regions where the global Lyapunov exponent is less than −3, black areas where it is greater than 0. (c) As for (b), but with starting value x 0 = 0.2. (d) As for (b), but with starting value x 0 = 0.3. (e) As for (b), but with starting value x 0 = 0.4. (f) Log of the standard deviation of the global Lyapunov exponents generated from the 50 uniformly distributed starting values used to generate (a). White areas signify regions where the log of the standard deviation is less than −3, dark grey areas where it is greater than −1.

FIG. 4.

Example of the dependence of the global Lyapunov exponent on starting value x 0 for the bi-parameter logistic map where the control parameter varies via (25) with A = 3.0 and p = 6. The global Lyapunov exponent is calculated over 104 iterations. White areas signify regions where the global Lyapunov exponent is less than −1, black areas where it is greater than 0. Examples of corresponding attractors are shown in Figure 5.

FIG. 5.

Feigenbaum diagrams for the bi-parameter logistic map where the control parameter varies via (25) with A = 3.0 and p = 6 with (a) x 0 = 0.2, (b) x 0  = 0.4. These starting values correspond to a sample of those whose global Lyapunov exponent is presented in Figure 4.

FIG. 6.

Feigenbaum diagrams for the bi-parameter logistic map where the control parameter varies via (25) with A = 2.8 and p = 40 with (a) x 0  = 0.2, (b) x 0  = 0.4.

/content/aip/journal/chaos/21/4/10.1063/1.3645185
2011-10-11
2014-04-24

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