^{1}

### Abstract

The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Analytic results for global Lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In the case of a tent map with N control parameters, the fraction of parameter space for which the global Lyapunov exponent is positive is calculated. The case of bi-parameter maps of period N are investigated.

The logistic map, and to a lesser extent the tent map, are classic pedagogic examples of how chaotic behaviour can occur in a simple one dimensional iterated map. In each case, the behaviour of the map under repeated iteration is characterised by a single control parameter, with variation of this parameter determining the long term behaviour over many iterations. Depending on the value chosen for the control parameter orbits can be either periodic or chaotic. In the investigation of a particular one dimensional map, the control parameter is typically fixed and the resulting behaviour of the map is analysed. However, relatively little attention has been given to one dimensional maps where the control parameter itself varies as the map is iterated. In this paper, we investigate a number of such multi-parameter scenarios for both the logistic and tent maps and investigate the behaviour of the resulting Lyapunov exponents. We are able to present analytic results for the tent map and a number of numerical results for the logistic map.

I. INTRODUCTION

II. MULTI-PARAMETER MAPS

III. BI-PARAMETER MAPS

IV. CONCLUDING REMARKS

## Figures

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).

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).

(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 10^{4} 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 R^{2} = 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 10^{4} 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%.

(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 10^{4} 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 R^{2} = 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 10^{4} 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%.

(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 10^{4} 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 10^{4} steps with a starting value of x_{0} = 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.

(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 10^{4} 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 10^{4} steps with a starting value of x_{0} = 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.

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 10^{4} 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.

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 10^{4} 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.

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.

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.

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.

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.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content