Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction
Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and di...
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| Published in | Systems (Basel) Vol. 4; no. 4; p. 37 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Basel
MDPI AG
01.12.2016
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| Abstract | Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems’ behavior. |
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| AbstractList | Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems' behavior. |
| Author | Boeing, Geoff |
| Author_xml | – sequence: 1 givenname: Geoff orcidid: 0000-0003-1851-6411 surname: Boeing fullname: Boeing, Geoff |
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| Cites_doi | 10.1016/0098-3004(86)90047-6 10.1007/978-3-642-24544-2 10.1007/BF01608556 10.1007/BF01646553 10.1109/ICCSA.2008.7 10.1068/b38124 10.7551/mitpress/9399.001.0001 10.1126/science.156.3775.636 10.1007/978-3-540-48625-1 10.1103/PhysRevE.86.036209 10.1155/S1026022699000151 10.1080/13658810210137013 10.1016/0167-2789(83)90125-2 10.1161/01.CIR.101.1.47 10.1103/PhysRevLett.45.712 10.1038/35023206 10.1007/BF01020332 10.1063/1.3156832 10.1007/978-1-4612-2150-0 10.1136/jech.2006.054254 10.1038/261459a0 10.1177/1473095206061022 10.1126/science.238.4827.632 10.1016/j.cnsns.2011.06.014 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 10.1007/3-540-32023-7 10.1088/1751-8113/46/25/254009 10.14445/22315381/IJETT-V13P256 10.1063/1.4927643 10.4324/9780203773895 10.1080/00029890.1975.11994008 10.1016/j.envsoft.2005.12.018 10.4324/9780203864302 10.1126/science.186.4164.645 10.1016/j.amc.2011.05.079 10.1068/b2617 10.1063/1.4922973 10.1119/1.13295 10.1057/palgrave.ivs.9500082 10.4172/2155-9880.1000346 10.1080/01944369108975471 10.1016/j.chaos.2006.05.018 10.1364/JOSAA.7.001055 10.1016/0167-2789(83)90112-4 10.1142/S0218127415300116 10.1002/9780471740360 10.1007/978-81-322-2556-0 10.1016/0167-2789(85)90011-9 10.1007/s11071-013-1065-7 10.1103/PhysRevLett.50.346 10.3998/mpub.14623 10.1007/s00004-008-0088-8 10.1146/annurev.es.24.110193.000245 10.1007/BFb0091903 10.1016/j.autneu.2013.05.004 10.1063/1.4917289 10.1016/j.arcontrol.2009.01.002 10.1016/j.chaos.2006.07.051 |
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| References | Chen (ref_16) 2008; 35 Shilnikov (ref_46) 2002; 3 Hoshi (ref_5) 2013; 177 ref_58 ref_57 ref_12 ref_11 ref_10 Mandelbrot (ref_56) 1967; 156 Benguigui (ref_14) 2000; 27 Li (ref_42) 2011; 218 ref_19 Cartwright (ref_24) 1991; 57 Wolf (ref_67) 1985; 16 Alpigini (ref_28) 2004; 3 Singh (ref_4) 2012; 17 Oxley (ref_21) 2007; 22 May (ref_32) 1976; 261 ref_60 Feigenbaum (ref_52) 1983; 7 ref_25 ref_69 ref_68 ref_66 ref_20 Farmer (ref_54) 1983; 7 Rickles (ref_2) 2007; 61 (ref_53) 1976; 50 ref_29 Stewart (ref_38) 2000; 406 Hunt (ref_70) 2015; 25 ref_27 ref_26 Sprott (ref_65) 2015; 25 Chen (ref_17) 2008; 36 Grassberger (ref_55) 1983; 50 Huikuri (ref_59) 2000; 101 Feigenbaum (ref_51) 1978; 19 Sander (ref_64) 2015; 25 ref_34 Babbs (ref_6) 2014; 5 (ref_18) 2009; 33 Ostwald (ref_23) 2013; 40 ref_30 Kekre (ref_62) 2014; 13 Glass (ref_7) 2009; 19 ref_39 ref_37 Ruelle (ref_45) 1971; 20 Shen (ref_15) 2002; 16 Packard (ref_33) 1980; 45 Hastings (ref_1) 1993; 24 Theiler (ref_61) 1990; 7 Batty (ref_13) 1999; 3 May (ref_41) 1974; 186 Wu (ref_49) 2014; 75 Kantz (ref_71) 2013; 46 Chettiparamb (ref_36) 2006; 5 Hamouche (ref_22) 2009; 11 Li (ref_50) 1975; 82 ref_44 ref_43 ref_40 Grebogi (ref_47) 1987; 238 Suetani (ref_3) 2012; 86 ref_48 Clarke (ref_63) 1986; 12 ref_9 ref_8 Bradley (ref_35) 2015; 25 Lorenz (ref_31) 1963; 20 |
| References_xml | – volume: 12 start-page: 713 year: 1986 ident: ref_63 article-title: Computation of the Fractal Dimension of Topographic Surfaces Using the Triangular Prism Surface Area Method publication-title: Comput. Geosci. doi: 10.1016/0098-3004(86)90047-6 – ident: ref_9 – ident: ref_26 doi: 10.1007/978-3-642-24544-2 – volume: 50 start-page: 69 year: 1976 ident: ref_53 article-title: A Two-Dimensional Mapping with a Strange Attractor publication-title: Commun. Math. Phys. doi: 10.1007/BF01608556 – volume: 20 start-page: 167 year: 1971 ident: ref_45 article-title: On the Nature of Turbulence publication-title: Commun. Math. Phys. doi: 10.1007/BF01646553 – ident: ref_37 doi: 10.1109/ICCSA.2008.7 – volume: 40 start-page: 644 year: 2013 ident: ref_23 article-title: The Fractal Analysis of Architecture: Calibrating the Box-Counting Method Using Scaling Coefficient and Grid Disposition Variables publication-title: Environ. Plan B doi: 10.1068/b38124 – ident: ref_27 doi: 10.7551/mitpress/9399.001.0001 – volume: 156 start-page: 636 year: 1967 ident: ref_56 article-title: How Long Is the Coast of Britain? publication-title: Science doi: 10.1126/science.156.3775.636 – ident: ref_34 doi: 10.1007/978-3-540-48625-1 – ident: ref_39 – volume: 86 start-page: 036209 year: 2012 ident: ref_3 article-title: Manifold Learning Approach for Chaos in the Dripping Faucet publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.86.036209 – volume: 3 start-page: 109 year: 1999 ident: ref_13 article-title: Self-Organized Criticality and Urban Development publication-title: Discret. Dyn. Nat. Soc. doi: 10.1155/S1026022699000151 – volume: 16 start-page: 419 year: 2002 ident: ref_15 article-title: Fractal Dimension and Fractal Growth of Urbanized Areas publication-title: Int. J. Geogr. Inf. Sci. doi: 10.1080/13658810210137013 – volume: 7 start-page: 153 year: 1983 ident: ref_54 article-title: The Dimension of Chaotic Attractors publication-title: Phys. Nonlinear Phenom. doi: 10.1016/0167-2789(83)90125-2 – volume: 101 start-page: 47 year: 2000 ident: ref_59 article-title: Diamond Study Group. Fractal Correlation Properties of RR Interval Dynamics and Mortality in Patients with Depressed Left Ventricular Function after an Acute Myocardial Infarction publication-title: Circulation doi: 10.1161/01.CIR.101.1.47 – volume: 45 start-page: 712 year: 1980 ident: ref_33 article-title: Geometry from a Time Series publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.45.712 – volume: 406 start-page: 948 year: 2000 ident: ref_38 article-title: The Lorenz Attractor Exists publication-title: Nature doi: 10.1038/35023206 – volume: 19 start-page: 25 year: 1978 ident: ref_51 article-title: Quantitative Universality for a Class of Nonlinear Transformations publication-title: J. Stat. Phys. doi: 10.1007/BF01020332 – volume: 3 start-page: 349 year: 2002 ident: ref_46 article-title: Bifurcations and Strange Attractors publication-title: Proc. Int. Congr. Math. – volume: 19 start-page: 028501 year: 2009 ident: ref_7 article-title: Introduction to Controversial Topics in Nonlinear Science: Is the Normal Heart Rate Chaotic? publication-title: Chaos doi: 10.1063/1.3156832 – ident: ref_8 – ident: ref_58 doi: 10.1007/978-1-4612-2150-0 – volume: 61 start-page: 933 year: 2007 ident: ref_2 article-title: A Simple Guide to Chaos and Complexity publication-title: J. Epidemiol. Commun. Health doi: 10.1136/jech.2006.054254 – ident: ref_48 – ident: ref_69 – volume: 261 start-page: 459 year: 1976 ident: ref_32 article-title: Simple Mathematical Models with Very Complicated Dynamics publication-title: Nature doi: 10.1038/261459a0 – volume: 5 start-page: 71 year: 2006 ident: ref_36 article-title: Metaphors in Complexity Theory and Planning publication-title: Plan Theory doi: 10.1177/1473095206061022 – volume: 238 start-page: 632 year: 1987 ident: ref_47 article-title: Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics publication-title: Science doi: 10.1126/science.238.4827.632 – volume: 17 start-page: 521 year: 2012 ident: ref_4 article-title: A New Iterative Approach to Fractal Models publication-title: Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2011.06.014 – volume: 20 start-page: 130 year: 1963 ident: ref_31 article-title: Deterministic Nonperiodic Flow publication-title: J. Atmos. Sci. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 – ident: ref_43 doi: 10.1007/3-540-32023-7 – volume: 46 start-page: 254009 year: 2013 ident: ref_71 article-title: The Problem of Spurious Lyapunov Exponents in Time Series Analysis and Its Solution by Covariant Lyapunov Vectors publication-title: J. Phys. Math. Theor. doi: 10.1088/1751-8113/46/25/254009 – volume: 13 start-page: 281 year: 2014 ident: ref_62 article-title: A Study of Period Doubling in Logistic Map for Shift Parameter publication-title: Int. J. Eng. Trends Technol. doi: 10.14445/22315381/IJETT-V13P256 – volume: 25 start-page: 083101 year: 2015 ident: ref_65 article-title: Classifying and Quantifying Basins of Attraction publication-title: Chaos doi: 10.1063/1.4927643 – ident: ref_10 doi: 10.4324/9780203773895 – volume: 82 start-page: 985 year: 1975 ident: ref_50 article-title: Period Three Implies Chaos publication-title: Am. Math. Mon. doi: 10.1080/00029890.1975.11994008 – volume: 22 start-page: 580 year: 2007 ident: ref_21 article-title: Economics on the Edge of Chaos: Some Pitfalls of Linearizing Complex Systems publication-title: Environ. Model. Softw. doi: 10.1016/j.envsoft.2005.12.018 – ident: ref_25 doi: 10.4324/9780203864302 – volume: 186 start-page: 645 year: 1974 ident: ref_41 article-title: Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos publication-title: Science doi: 10.1126/science.186.4164.645 – volume: 218 start-page: 157 year: 2011 ident: ref_42 article-title: Optimal Harvesting Policy for Stochastic Logistic Population Model publication-title: Appl. Math. Comput. doi: 10.1016/j.amc.2011.05.079 – volume: 27 start-page: 507 year: 2000 ident: ref_14 article-title: When and Where is a City Fractal? publication-title: Environ. Plan. B doi: 10.1068/b2617 – volume: 25 start-page: 097618 year: 2015 ident: ref_70 article-title: Defining Chaos publication-title: Chaos doi: 10.1063/1.4922973 – ident: ref_30 – ident: ref_57 doi: 10.1119/1.13295 – volume: 3 start-page: 271 year: 2004 ident: ref_28 article-title: Dynamical System Visualization and Analysis via Performance Maps publication-title: Inf. Vis. doi: 10.1057/palgrave.ivs.9500082 – volume: 5 start-page: 2 year: 2014 ident: ref_6 article-title: Initiation of Ventricular Fibrillation by a Single Ectopic Beat in Three Dimensional Numerical Models of Ischemic Heart Disease: Abrupt Transition to Chaos publication-title: J. Clin. Exp. Cardiol. doi: 10.4172/2155-9880.1000346 – volume: 57 start-page: 44 year: 1991 ident: ref_24 article-title: Planning and Chaos Theory publication-title: J. Am. Plan. Assoc. doi: 10.1080/01944369108975471 – volume: 35 start-page: 85 year: 2008 ident: ref_16 article-title: Scaling Laws and Indications of Self-Organized Criticality in Urban Systems publication-title: Chaos Solitons Fractals doi: 10.1016/j.chaos.2006.05.018 – ident: ref_40 – volume: 7 start-page: 1055 year: 1990 ident: ref_61 article-title: Estimating Fractal Dimension publication-title: J. Opt. Soc. Am. A doi: 10.1364/JOSAA.7.001055 – volume: 7 start-page: 16 year: 1983 ident: ref_52 article-title: Universal Behavior in Nonlinear Systems publication-title: Phys. Nonlinear Phenom. doi: 10.1016/0167-2789(83)90112-4 – ident: ref_44 – volume: 25 start-page: 1530011 year: 2015 ident: ref_64 article-title: The Many Facets of Chaos publication-title: Int. J. Bifurc. Chaos doi: 10.1142/S0218127415300116 – ident: ref_68 doi: 10.1002/9780471740360 – ident: ref_29 doi: 10.1007/978-81-322-2556-0 – volume: 16 start-page: 285 year: 1985 ident: ref_67 article-title: Determining Lyapunov Exponents from a Time Series publication-title: Phys. Nonlinear Phenom. doi: 10.1016/0167-2789(85)90011-9 – volume: 75 start-page: 283 year: 2014 ident: ref_49 article-title: Discrete Fractional Logistic Map and Its Chaos publication-title: Nonlinear Dyn. doi: 10.1007/s11071-013-1065-7 – volume: 50 start-page: 346 year: 1983 ident: ref_55 article-title: Characterization of Strange Attractors publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.50.346 – ident: ref_11 doi: 10.3998/mpub.14623 – ident: ref_20 doi: 10.3998/mpub.14623 – ident: ref_12 – volume: 11 start-page: 217 year: 2009 ident: ref_22 article-title: Can Chaos Theory Explain Complexity In Urban Fabric? Applications in Traditional Muslim Settlements publication-title: Nexus Netw. J. doi: 10.1007/s00004-008-0088-8 – volume: 24 start-page: 1 year: 1993 ident: ref_1 article-title: Chaos in Ecology: Is Mother Nature a Strange Attractor? publication-title: Annu. Rev. Ecol. Syst. doi: 10.1146/annurev.es.24.110193.000245 – ident: ref_60 doi: 10.1007/BFb0091903 – volume: 177 start-page: 271 year: 2013 ident: ref_5 article-title: Poincaré Plot Indexes of Heart Rate Variability: Relationships with Other Nonlinear Variables publication-title: Auton. Neurosci. doi: 10.1016/j.autneu.2013.05.004 – volume: 25 start-page: 097610 year: 2015 ident: ref_35 article-title: Nonlinear Time-Series Analysis Revisited publication-title: Chaos doi: 10.1063/1.4917289 – volume: 33 start-page: 89 year: 2009 ident: ref_18 article-title: Chaos in Economics and Finance publication-title: Annu. Rev. Control doi: 10.1016/j.arcontrol.2009.01.002 – ident: ref_19 – ident: ref_66 doi: 10.3998/mpub.14623 – volume: 36 start-page: 1305 year: 2008 ident: ref_17 article-title: Nonlinear Dynamics and Chaos in a Fractional-Order Financial System publication-title: Chaos Solitons Fractals doi: 10.1016/j.chaos.2006.07.051 |
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