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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| Format: | Article |
| Language: | English |
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MDPI AG
2016-11-01
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| Series: | Systems |
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| Online Access: | http://www.mdpi.com/2079-8954/4/4/37 |
| _version_ | 1818001217017610240 |
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| author | Geoff Boeing |
| author_facet | Geoff Boeing |
| author_sort | Geoff Boeing |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-14T03:31:40Z |
| format | Article |
| id | doaj.art-2a31faac6baf48c5b43652633ca81f9f |
| institution | Directory Open Access Journal |
| issn | 2079-8954 |
| language | English |
| last_indexed | 2024-04-14T03:31:40Z |
| publishDate | 2016-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Systems |
| spelling | doaj.art-2a31faac6baf48c5b43652633ca81f9f2022-12-22T02:14:56ZengMDPI AGSystems2079-89542016-11-01443710.3390/systems4040037systems4040037Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of PredictionGeoff Boeing0Department of City and Regional Planning, University of California, Berkeley, CA 94720, USANearly 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.http://www.mdpi.com/2079-8954/4/4/37visualizationnonlinear dynamicschaosfractalattractorbifurcationdynamical systemspredictionpythonlogistic map |
| spellingShingle | Geoff Boeing Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction Systems visualization nonlinear dynamics chaos fractal attractor bifurcation dynamical systems prediction python logistic map |
| title | Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction |
| title_full | Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction |
| title_fullStr | Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction |
| title_full_unstemmed | Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction |
| title_short | Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction |
| title_sort | visual analysis of nonlinear dynamical systems chaos fractals self similarity and the limits of prediction |
| topic | visualization nonlinear dynamics chaos fractal attractor bifurcation dynamical systems prediction python logistic map |
| url | http://www.mdpi.com/2079-8954/4/4/37 |
| work_keys_str_mv | AT geoffboeing visualanalysisofnonlineardynamicalsystemschaosfractalsselfsimilarityandthelimitsofprediction |