University of Colorado, Boulder CU Scholar Applied Mathematics Graduate Theses & Applied Mathematics Dissertations Spring 5-9-2015 Spatially Random Processes in One-Dimensional Maps: The Logistic Map and The Arnold Circle Map An T. Le University of Colorado Boulder, [email protected] Follow this and additional works at: http://scholar.colorado.edu/appm_gradetds Part of the Hydraulic Engineering Commons, Non-linear Dynamics Commons, and the Probability Commons Recommended Citation Le, An T., "Spatially Random Processes in One-Dimensional Maps: The Logistic Map and The Arnold Circle Map" (2015). Applied Mathematics Graduate Theses & Dissertations. Paper 3. This Thesis is brought to you for free and open access by Applied Mathematics at CU Scholar. It has been accepted for inclusion in Applied Mathematics Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact [email protected]. Spatially Random Processes in One-Dimensional Maps: The Logistic Map and The Arnold Circle Map by A. T. Le B.S., University of Colorado, 2015 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Applied Mathematics 2015 This thesis entitled: Spatially Random Processes in One-Dimensional Maps: The Logistic Map and The Arnold Circle Map written by A. T. Le has been approved for the Department of Applied Mathematics Dr. James Meiss Dr. Juan Restrepo Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Le, A. T. (M.S., Applied Mathematics) Spatially Random Processes in One-Dimensional Maps: The Logistic Map and The Arnold Circle Map Thesis directed by Dr. James Meiss One way to model in-situ remediation of contaminated groundwater is to consider spatially random processes in nonlinear systems. Groundwater remediation often requires injecting an aquifer with treatment solution, where degradation reactions break down the toxins. As the treatment solution and contaminated water flow through the aquifer, their movement is limited by the types of sediment found in the aquifer, which act as spatial barriers to mixing. The onset of chaos in this system implies the two solutions are well mixed, and thus the contaminants are rendered inert. The spatially random processes explored in this thesis are meant to mimic the distribution of sediment in the aquifer. These processes were constructed using uniform random variables and normal random variables, and incorporate an exponentially decaying spatial correlation. The three-dimensional model of the fluid flow in the aquifer has been simplified to an in- depth study of two one-dimensional maps: the logistic map and the Arnold circle map. Injection of the treatment solution in the aquifer may be thought of as the initial condition imposed on the map. Numerical simulations of the one-dimensional maps lay the groundwork for future studies of higher-dimensional systems. Simulations indicate evidence of newly stabilized regions of the randomized logistic map, as well as a breakdown of symmetry and stable behavior in the Arnold circle map. The combination of bifurcation diagrams and Lyapunov exponents from the randomized logistic map lead us to hypothesize the spatially random process may stabilize the map in regions previously unstable. In the random circle map, analysis of the Arnold tongues, devil's staircases, and Lyapunov exponents suggest the random processes incur chaotic behavior in typically stable regions. Dedication To my mom, who has always told me, \Just do your best." v Acknowledgements For his extensive patience, support, and guidance, I am truly grateful to Jim Meiss. For being on my committee, I offer thanks to Jim Meiss, Juan Restrepo, and Will Kleiber. For his unwavering confidence in me and continued encouragement, I have sincere gratitude for Swamy Ananthanarayan. vi Contents Chapter 1 Introduction 1 1.1 Groundwater Contamination . .1 1.2 Chaos . .4 1.2.1 Logistic Map . 11 1.2.2 Circle Map . 13 2 Spatially Random Maps 20 2.1 Spatially Random Processes . 21 2.2 Random Dynamics of the Logistic Map . 32 2.3 Random Dynamics of the Circle Map . 33 3 Results 40 3.1 Implementation of Randomness in the Logistic Map . 40 3.2 Implementation of Randomness in the Circle Map . 51 3.2.1 Uniform Distribution . 52 3.2.2 Normal Distribution . 63 4 Conclusion 74 4.1 Spatially Random Processes and 1-Dimensional Maps . 74 4.1.1 Logistic Map . 75 vii 4.1.2 Circle Map . 76 4.2 Future Work . 77 4.2.1 Extension to Higher Dimensions . 77 4.2.2 Basin of Attraction for Chaotic Trajectories . 78 4.2.3 Stabilizing Effects of Spatial Randomness . 79 4.2.4 Dynamic Load Balancing on the Supercomputer . 79 Bibliography 81 Appendix A Load Balancing Tool and Detailed Program Description 83 B MATLAB Code 90 B.1 Logistic Map . 90 B.1.1 Iterating the Logistic Map . 90 B.1.2 Upper and Lower Bounds on the Map . 93 B.1.3 Bifurcation Diagram . 94 B.1.4 Lyapunov Exponent . 97 B.1.5 Period Distribution . 98 B.2 Circle Map . 100 B.2.1 Iterating the Circle Map . 100 B.2.2 Upper and Lower Bounds on the Map . 102 B.2.3 Arnold Tongues Diagram . 103 B.2.4 Lyapunov Exponent . 105 B.2.5 Devil's Staircase and Kernel Density Estimation . 108 B.2.6 Period Distribution . 110 viii Tables Table 1.1 Typical porosity and hydraulic conductivity ranges for clay, sand, and gravel . .2 1.2 Behavior of the deterministic map as r is varied . 12 ix Figures Figure 1.1 Groundwater usage in the United States in 2005 . .2 1.2 Example of a cobweb diagram . 10 1.3 Deterministic logistic map, stable orbit . 11 1.4 Deterministic logistic map, erratic orbit . 12 1.5 Bifurcation diagram for the deterministic logistic map . 13 1.6 Lyapunov exponent in the deterministic logistic map . 14 1.7 Deterministic circle map, stable orbit . 16 1.8 Deterministic circle map, quasiperiodic orbit . 16 1.9 The devil's staircase for the deterministic circle map . 17 1.10 The Arnold tongues for the deterministic circle map . 18 1.11 Lyapunov exponent in the deterministic circle map, varying ! ............. 19 1.12 Lyapunov exponent in the deterministic circle map, varying k ............. 19 2.1 Example of a covariance-spectrum pair . 25 2.2 The correlation function C(x)............................... 26 2.3 Uniform distribution over a square region . 28 2.4 The function R(x)..................................... 29 2.5 A histogram of Tm ..................................... 31 sm 2.6 Random logistic map, stable orbit . 33 x 2.7 Upper and lower bounds on the random logistic map . 34 2.8 Random circle map, stable orbit under the uniform distribution . 34 2.9 Upper and lower bounds on the random circle map, with a uniform distribution, where k = 1;! = 0:3.................................... 35 2.10 Upper and lower bounds on the random circle map, with a uniform distribution, where k = 1:5;! = 0:3................................... 35 2.11 Upper and lower bounds on the random circle map, with a uniform distribution, where k = 1:5;! = 0:7................................... 35 2.12 The function Ω(x)..................................... 36 2.13 Random circle map, stable orbit under the normal distribution . 37 2.14 Upper and lower bounds on the random circle map, with a normal distribution, where k = 1;! = 0:3.................................... 37 2.15 Upper and lower bounds on the random circle map, with a normal distribution, where k = 1:5;! = 0:3................................... 38 2.16 Upper and lower bounds on the random circle map, with a normal distribution, where k = 1:5;! = 0:7................................... 38 3.1 Bifurcation diagrams of the random logistic map, σ = σmax .............. 42 3.2 Bifurcation diagram of the random logistic map, σ = σmax, zoomed in . 43 1 3.3 Bifurcation diagrams of the random logistic map, σ = 2 σmax ............. 44 3.4 Lyapunov exponent in the random logistic map compared to the deterministic map, σ = σmax .......................................... 46 3.5 Lyapunov exponent in the random logistic map compared to the deterministic map, 1 σ = 2 σmax .......................................... 47 3.6 Average fraction of period p orbits for the random logistic map, σ = σmax and r = 3:3 48 3.7 Log-scale plot of the average fraction of period p orbits for the random logistic map, σ = σmax and r = 3:3 ................................... 49 xi 3.8 Average fraction of period p orbits for the random logistic map, σ = σmax and r = 1:2 49 1 3.9 Average fraction of period p orbits for the random logistic map, σ = 2 σmax and r = 3:22........................................... 50 1 3.10 Average fraction of period p orbits for the random logistic map, σ = 2 σmax and r = 1:86........................................... 50 3.11 The Arnold tongues for the random circle map, uniform distribution, α = 10−5 ... 54 1 −5 3.12 The Arnold tongues for the random circle map, uniform distribution, α = 2 10 .. 55 3.13 Lyapunov exponent in the random circle map (uniform distribution) compared to the deterministic map, varying !, α = 10−5 ....................... 56 3.14 Lyapunov exponent in the random circle map (uniform distribution) compared to the deterministic map, varying k, α = 10−5 ....................... 57 3.15 Average fraction of period p orbits for the random circle map (uniform distribution), for α = 10−5, ! = 0:9 and k =1.............................. 58 3.16 Average fraction of period p orbits for the random circle map (uniform distribution), for α = 10−5, ! = 0:6 and k = 1:5............................. 58 3.17 Log-scale plot of average fraction of period p orbits for the random circle map (uni- form distribution), for α = 10−5, ! = 0:6 and k =1..................