IDEAS home Printed from https://ideas.repec.org/p/osf/socarx/c7p43.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://osf.io/download/58d54f3b9ad5a100455be55f/
    Download Restriction: no
    File URL: https://libkey.io/10.31219/osf.io/c7p43?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Boeing, Geoff, 2017. "Methods and Measures for Analyzing Complex Street Networks and Urban Form," SocArXiv 93h82, Center for Open Science.
    5. Boeing, Geoff, 2018. "Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals," SocArXiv g8nsf, Center for Open Science.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Chen, Yanguang, 2009. "Analogies between urban hierarchies and river networks: Fractals, symmetry, and self-organized criticality," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1766-1778.
    2. Myagmartseren Purevtseren & Bazarkhand Tsegmid & Myagmarjav Indra & Munkhnaran Sugar, 2018. "The Fractal Geometry of Urban Land Use: The Case of Ulaanbaatar City, Mongolia," Land, MDPI, vol. 7(2), pages 1-14, May.
    3. Chen, Yanguang, 2013. "Fractal analytical approach of urban form based on spatial correlation function," Chaos, Solitons & Fractals, Elsevier, vol. 49(C), pages 47-60.
    4. Chen, Yanguang & Lin, Jingyi, 2009. "Modeling the self-affine structure and optimization conditions of city systems using the idea from fractals," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 615-629.
    5. Boeing, Geoff, 2017. "Methods and Measures for Analyzing Complex Street Networks and Urban Form," SocArXiv 93h82, Center for Open Science.
    6. Chen, Yanguang, 2013. "A set of formulae on fractal dimension relations and its application to urban form," Chaos, Solitons & Fractals, Elsevier, vol. 54(C), pages 150-158.
    7. Chen, Yanguang, 2011. "Fractal systems of central places based on intermittency of space-filling," Chaos, Solitons & Fractals, Elsevier, vol. 44(8), pages 619-632.
    8. Chen, Yanguang & Huang, Linshan, 2018. "A scaling approach to evaluating the distance exponent of the urban gravity model," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 303-313.
    9. Chen, Yanguang & Wang, Yihan & Li, Xijing, 2019. "Fractal dimensions derived from spatial allometric scaling of urban form," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 122-134.
    10. Ren, Jinfu & Liu, Yang & Liu, Jiming, 2023. "Chaotic behavior learning via information tracking," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    11. Chen, Yanguang, 2014. "Multifractals of central place systems: Models, dimension spectrums, and empirical analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 402(C), pages 266-282.
    12. Wang, Lei & Chen, Yi-Ming, 2020. "Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    13. Chen, Yanguang, 2014. "An allometric scaling relation based on logistic growth of cities," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 65-77.
    14. Pratap, A. & Raja, R. & Cao, J. & Lim, C.P. & Bagdasar, O., 2019. "Stability and pinning synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous activations," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 241-260.
    15. Fendzi Donfack, Emmanuel & Nguenang, Jean Pierre & Nana, Laurent, 2020. "On the traveling waves in nonlinear electrical transmission lines with intrinsic fractional-order using discrete tanh method," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    16. Bo Liu & Desheng Xue & Yiming Tan, 2019. "Deciphering the Manufacturing Production Space in Global City-Regions of Developing Countries—a Case of Pearl River Delta, China," Sustainability, MDPI, vol. 11(23), pages 1-26, December.
    17. Li, Xing-Yu & Wu, Kai-Ning & Liu, Xiao-Zhen, 2023. "Mittag–Leffler stabilization for short memory fractional reaction-diffusion systems via intermittent boundary control," Applied Mathematics and Computation, Elsevier, vol. 449(C).
    18. Hajipour, Ahamad & Hajipour, Mojtaba & Baleanu, Dumitru, 2018. "On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 497(C), pages 139-153.
    19. Zhijun SONG & Linjun YU, 2019. "Multifractal features of spatial variation in construction land in Beijing (1985–2015)," Palgrave Communications, Palgrave Macmillan, vol. 5(1), pages 1-15, December.
    20. Rafael González-Val, 2021. "The Probability Distribution of Worldwide Forest Areas," Sustainability, MDPI, vol. 13(3), pages 1-19, January.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:osf:socarx:c7p43. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: OSF (email available below). General contact details of provider: https://arabixiv.org .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.