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Oscillatory finite-time singularities in finance, population and rupture

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  • Ide, Kayo
  • Sornette, Didier

Abstract

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity; (3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing.

Suggested Citation

  • Ide, Kayo & Sornette, Didier, 2002. "Oscillatory finite-time singularities in finance, population and rupture," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 307(1), pages 63-106.
  • Handle: RePEc:eee:phsmap:v:307:y:2002:i:1:p:63-106
    DOI: 10.1016/S0378-4371(01)00585-4
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Zhou, Wei-Xing & Sornette, Didier, 2009. "A case study of speculative financial bubbles in the South African stock market 2003–2006," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(6), pages 869-880.
    2. Jiang, Zhi-Qiang & Zhou, Wei-Xing & Sornette, Didier & Woodard, Ryan & Bastiaensen, Ken & Cauwels, Peter, 2010. "Bubble diagnosis and prediction of the 2005-2007 and 2008-2009 Chinese stock market bubbles," Journal of Economic Behavior & Organization, Elsevier, vol. 74(3), pages 149-162, June.
    3. Sornette, D & Takayasu, H & Zhou, W.-X, 2003. "Finite-time singularity signature of hyperinflation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 325(3), pages 492-506.
    4. Zhou, Wei-Xing & Sornette, Didier, 2004. "Antibubble and prediction of China's stock market and real-estate," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 337(1), pages 243-268.
    5. Li Lin & Didier Sornette, 2016. "A Simple Mechanism for Financial Bubbles: Time-Varying Momentum Horizon," Swiss Finance Institute Research Paper Series 16-61, Swiss Finance Institute.
    6. Alvarez-Ramirez, Jose & Ibarra-Valdez, Carlos & Bernabe, Araceli & Rodriguez, Eduardo, 2005. "Power-law periodicity in the 2003–2004 crude oil price dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(3), pages 625-640.
    7. Li Lin & Didier Sornette, 2018. "“Speculative Influence Network” during financial bubbles: application to Chinese stock markets," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 13(2), pages 385-431, July.
    8. Yan, Wanfeng & Woodard, Ryan & Sornette, Didier, 2012. "Diagnosis and prediction of rebounds in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1361-1380.
    9. Sornette, Didier & Woodard, Ryan & Zhou, Wei-Xing, 2009. "The 2006–2008 oil bubble: Evidence of speculation, and prediction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1571-1576.
    10. Sornette, Didier & Woodard, Ryan & Yan, Wanfeng & Zhou, Wei-Xing, 2013. "Clarifications to questions and criticisms on the Johansen–Ledoit–Sornette financial bubble model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(19), pages 4417-4428.
    11. Zhou, Wei-Xing & Sornette, Didier, 2005. "Testing the stability of the 2000 US stock market “antibubble”," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 348(C), pages 428-452.
    12. Sornette, Didier & Zhou, Wei-Xing, 2006. "Predictability of large future changes in major financial indices," International Journal of Forecasting, Elsevier, vol. 22(1), pages 153-168.
    13. Wanfeng Yan & Ryan Woodard & Didier Sornette, 2010. "Diagnosis and Prediction of Market Rebounds in Financial Markets," Papers 1003.5926, arXiv.org, revised Mar 2011.
    14. Sornette, Didier & Zhou, Wei-Xing, 2006. "Importance of positive feedbacks and overconfidence in a self-fulfilling Ising model of financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(2), pages 704-726.
    15. Zhou, Wei-Xing & Sornette, Didier, 2003. "Evidence of a worldwide stock market log-periodic anti-bubble since mid-2000," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 330(3), pages 543-583.
    16. Zhou, Wei-Xing & Sornette, Didier, 2003. "Renormalization group analysis of the 2000–2002 anti-bubble in the US S&P500 index: explanation of the hierarchy of five crashes and prediction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 330(3), pages 584-604.
    17. Yukalov, V.I. & Sornette, D. & Yukalova, E.P., 2009. "Nonlinear dynamical model of regime switching between conventions and business cycles," Journal of Economic Behavior & Organization, Elsevier, vol. 70(1-2), pages 206-230, May.
    18. Li Lin & Didier Sornette, 2009. "Diagnostics of Rational Expectation Financial Bubbles with Stochastic Mean-Reverting Termination Times," Papers 0911.1921, arXiv.org.
    19. Didier Sornette & Ryan Woodard, 2009. "Financial Bubbles, Real Estate bubbles, Derivative Bubbles, and the Financial and Economic Crisis," Papers 0905.0220, arXiv.org.
    20. Didier Sornette & Wei-Xing Zhou, 2003. "The US 2000-2002 market descent: clarification," Quantitative Finance, Taylor & Francis Journals, vol. 3(3), pages 39-41.
    21. Li Lin & Didier Sornette, 2015. ""Speculative Influence Network" during financial bubbles: application to Chinese Stock Markets," Papers 1510.08162, arXiv.org.
    22. D. Sornette & R. Woodard, "undated". "Financial Bubbles, Real Estate bubbles, Derivative Bubbles, and the Financial and Economic Crisis," Working Papers CCSS-09-003, ETH Zurich, Chair of Systems Design.
    23. Lin, L. & Ren, R.E. & Sornette, D., 2014. "The volatility-confined LPPL model: A consistent model of ‘explosive’ financial bubbles with mean-reverting residuals," International Review of Financial Analysis, Elsevier, vol. 33(C), pages 210-225.

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