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Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction

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  • Boeing, Geoff

    (Northeastern University)

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.

Suggested Citation

  • Boeing, Geoff, 2017. "Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction," SocArXiv c7p43, Center for Open Science.
    • Handle: RePEc:osf:socarx:c7p43
    DOI: 10.31219/osf.io/c7p43
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    References listed on IDEAS

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    1. Lucien Benguigui & Daniel Czamanski & Maria Marinov & Yuval Portugali, 2000. "When and Where is a City Fractal?," Environment and Planning B, , vol. 27(4), pages 507-519, August.
    2. Chen, Yanguang & Zhou, Yixing, 2008. "Scaling laws and indications of self-organized criticality in urban systems," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 85-98.
    3. Michael J Ostwald, 2013. "The Fractal Analysis of Architecture: Calibrating the Box-Counting Method Using Scaling Coefficient and Grid Disposition Variables," Environment and Planning B, , vol. 40(4), pages 644-663, August.
    4. Ian Stewart, 2000. "The Lorenz attractor exists," Nature, Nature, vol. 406(6799), pages 948-949, August.
    5. Chen, Wei-Ching, 2008. "Nonlinear dynamics and chaos in a fractional-order financial system," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1305-1314.
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    Cited by:

    1. Boeing, Geoff, 2018. "Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals," SocArXiv g8nsf, Center for Open Science.
    2. Mishelle Doorasamy & Prince Kwasi Sarpong, 2018. "Fractal Market Hypothesis and Markov Regime Switching Model: A Possible Synthesis and Integration," International Journal of Economics and Financial Issues, Econjournals, vol. 8(1), pages 93-100.
    3. Boeing, Geoff, 2018. "The Effects of Inequality, Density, and Heterogeneous Residential Preferences on Urban Displacement and Metropolitan Structure: An Agent-Based Model," SocArXiv mkq7d, Center for Open Science.
    4. Chi Zhang & Zhengning Pu & Jiasha Fu, 2018. "The Recurrence Interval Difference of Power Load in Heavy/Light Industries of China," Energies, MDPI, vol. 11(1), pages 1-20, January.
    5. Boeing, Geoff, 2017. "Methods and Measures for Analyzing Complex Street Networks and Urban Form," SocArXiv 93h82, Center for Open Science.

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