IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v137y2020ics096007792030309x.html
   My bibliography  Save this article

Taxonomy of commodities assets via complexity-entropy causality plane

Author

Listed:
  • Fernandes, Leonardo H.S.
  • Araújo, Fernando H.A.

Abstract

This paper promotes insights based on an empirical analysis of the complex dynamics of 12 significant assets of the commodity market. We applied the Complexity-entropy causality plane (CECP) to quantify the permutation entropy and appropriate statistical complexity measures consider time series of monthly spot and futures prices of these records, which made it possible to map the commodities assets and their respective locations along the CECP. The results obtained from our procedure fitting show that the price dynamics behavior varies widely along the plane from the lower-right region, characterized by high entropy and low complexity to the middle region of the plane, marked by more complex and less entropic. It suggests that the commodities that are located in the lower-right region are closer to their fundamental prices, so they are more efficient, while the commodities start pricing lies significantly farther from the right corner are more susceptible to speculative activities and present a low degree of efficiency. We also used the Bandt-Pompe permutation entropy and the Jensen-Shannon statistical complexity to elaborate the taxonomy of this market based on the complexity hierarchy, which expands the understanding inherent of the phenomenology of this important economic aggregate and can support the work of policymakers. Additionally, we proposed a novel Macrophysics policy that can be adopted as an alternative complementary of the classical fundamentals of macroeconomics used by policymakers to develop more efficient policies.

Suggested Citation

  • Fernandes, Leonardo H.S. & Araújo, Fernando H.A., 2020. "Taxonomy of commodities assets via complexity-entropy causality plane," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
  • Handle: RePEc:eee:chsofr:v:137:y:2020:i:c:s096007792030309x
    DOI: 10.1016/j.chaos.2020.109909
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007792030309X
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2020.109909?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Carmen M. Reinhart & Vincent Reinhart & Christoph Trebesch, 2016. "Global Cycles: Capital Flows, Commodities, and Sovereign Defaults, 1815-2015," American Economic Review, American Economic Association, vol. 106(5), pages 574-580, May.
    2. Kilian, Lutz & Lee, Thomas K., 2014. "Quantifying the speculative component in the real price of oil: The role of global oil inventories," Journal of International Money and Finance, Elsevier, vol. 42(C), pages 71-87.
    3. Ron Alquist & Olivier Gervais, 2013. "The Role of Financial Speculation in Driving the Price of Crude Oil," The Energy Journal, , vol. 34(3), pages 35-54, July.
    4. Ing-Haw Cheng & Wei Xiong, 2014. "Financialization of Commodity Markets," Annual Review of Financial Economics, Annual Reviews, vol. 6(1), pages 419-441, December.
    5. Robert Ready & Nikolai Roussanov & Colin Ward, 2017. "Commodity Trade and the Carry Trade: A Tale of Two Countries," Journal of Finance, American Finance Association, vol. 72(6), pages 2629-2684, December.
    6. Cesar A. Hidalgo & Ricardo Hausmann, 2009. "The Building Blocks of Economic Complexity," Papers 0909.3890, arXiv.org.
    7. Marshall Burke & Lauren Falcao Bergquist & Edward Miguel, 2019. "Sell Low and Buy High: Arbitrage and Local Price Effects in Kenyan Markets," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 134(2), pages 785-842.
    8. Ajanovic, Amela, 2011. "Biofuels versus food production: Does biofuels production increase food prices?," Energy, Elsevier, vol. 36(4), pages 2070-2076.
    9. Aurelio F. Bariviera & M. Belen Guercio & Lisana B. Martinez & Osvaldo A. Rosso, 2015. "A permutation Information Theory tour through different interest rate maturities: the Libor case," Papers 1509.00217, arXiv.org.
    10. Jesse M. Shapiro, 2013. "Fungibility and Consumer Choice: Evidence from Commodity Price Shocks," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 128(4), pages 1449-1498.
    11. Grilli, Enzo R & Yang, Maw Cheng, 1988. "Primary Commodity Prices, Manufactured Goods Prices, and the Terms of Trade of Developing Countries: What the Long Run Shows," The World Bank Economic Review, World Bank, vol. 2(1), pages 1-47, January.
    12. Christiane Baumeister & Lutz Kilian, 2016. "Forty Years of Oil Price Fluctuations: Why the Price of Oil May Still Surprise Us," Journal of Economic Perspectives, American Economic Association, vol. 30(1), pages 139-160, Winter.
    13. Kristoufek, Ladislav & Vosvrda, Miloslav, 2014. "Commodity futures and market efficiency," Energy Economics, Elsevier, vol. 42(C), pages 50-57.
    14. James D. Hamilton, 1985. "Historical Causes of Postwar Oil Shocks and Recessions," The Energy Journal, International Association for Energy Economics, vol. 0(Number 1), pages 97-116.
    15. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2010. "Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(9), pages 1891-1901.
    16. Herrera, Rodrigo & Rodriguez, Alejandro & Pino, Gabriel, 2017. "Modeling and forecasting extreme commodity prices: A Markov-Switching based extreme value model," Energy Economics, Elsevier, vol. 63(C), pages 129-143.
    17. He, Xue-Zhong & Westerhoff, Frank H., 2005. "Commodity markets, price limiters and speculative price dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 29(9), pages 1577-1596, September.
    18. Cifarelli, Giulio & Paladino, Giovanna, 2010. "Oil price dynamics and speculation: A multivariate financial approach," Energy Economics, Elsevier, vol. 32(2), pages 363-372, March.
    19. Gu, Rongbao & Zhang, Bing, 2016. "Is efficiency of crude oil market affected by multifractality? Evidence from the WTI crude oil market," Energy Economics, Elsevier, vol. 53(C), pages 151-158.
    20. Kristoufek, Ladislav & Vosvrda, Miloslav, 2016. "Gold, currencies and market efficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 449(C), pages 27-34.
    21. Michael Sockin & Wei Xiong, 2015. "Informational Frictions and Commodity Markets," Journal of Finance, American Finance Association, vol. 70(5), pages 2063-2098, October.
    22. Lin, Min & Wang, Gang-Jin & Xie, Chi & Stanley, H. Eugene, 2018. "Cross-correlations and influence in world gold markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 504-512.
    23. Bodart, Vincent & Candelon, Bertrand & Carpantier, Jean-Francois, 2015. "Real exchanges rates, commodity prices and structural factors in developing countries," Journal of International Money and Finance, Elsevier, vol. 51(C), pages 264-284.
    24. Zhang, Yi-Cheng, 1999. "Toward a theory of marginally efficient markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(1), pages 30-44.
    25. Tiago V. De V. Cavalcanti & Kamiar Mohaddes & Mehdi Raissi, 2015. "Commodity Price Volatility and the Sources of Growth," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 30(6), pages 857-873, September.
    26. Silvennoinen, Annastiina & Thorp, Susan, 2013. "Financialization, crisis and commodity correlation dynamics," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 24(C), pages 42-65.
    27. Liu, Chang & Sun, Xiaolei & Chen, Jianming & Li, Jianping, 2016. "Statistical properties of country risk ratings under oil price volatility: Evidence from selected oil-exporting countries," Energy Policy, Elsevier, vol. 92(C), pages 234-245.
    28. Drechsel, Thomas & Tenreyro, Silvana, 2018. "Commodity booms and busts in emerging economies," Journal of International Economics, Elsevier, vol. 112(C), pages 200-218.
    29. Du, Xiaodong & Yu, Cindy L. & Hayes, Dermot J., 2011. "Speculation and volatility spillover in the crude oil and agricultural commodity markets: A Bayesian analysis," Energy Economics, Elsevier, vol. 33(3), pages 497-503, May.
    30. James D. Hamilton, 2009. "Understanding Crude Oil Prices," The Energy Journal, International Association for Energy Economics, vol. 0(Number 2), pages 179-206.
    31. Zunino, Luciano & Ribeiro, Haroldo V., 2016. "Discriminating image textures with the multiscale two-dimensional complexity-entropy causality plane," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 679-688.
    32. He, Ling-Yun & Chen, Shu-Peng, 2011. "Multifractal Detrended Cross-Correlation Analysis of agricultural futures markets," Chaos, Solitons & Fractals, Elsevier, vol. 44(6), pages 355-361.
    33. Christopher R. Knittel & Robert S. Pindyck, 2016. "The Simple Economics of Commodity Price Speculation," American Economic Journal: Macroeconomics, American Economic Association, vol. 8(2), pages 85-110, April.
    34. Adams, Zeno & Glück, Thorsten, 2015. "Financialization in commodity markets: A passing trend or the new normal?," Journal of Banking & Finance, Elsevier, vol. 60(C), pages 93-111.
    35. Lutz Kilian, 2009. "Not All Oil Price Shocks Are Alike: Disentangling Demand and Supply Shocks in the Crude Oil Market," American Economic Review, American Economic Association, vol. 99(3), pages 1053-1069, June.
    36. Headey, Derek, 2011. "Rethinking the global food crisis: The role of trade shocks," Food Policy, Elsevier, vol. 36(2), pages 136-146, April.
    37. Choi, Kyongwook & Hammoudeh, Shawkat, 2010. "Volatility behavior of oil, industrial commodity and stock markets in a regime-switching environment," Energy Policy, Elsevier, vol. 38(8), pages 4388-4399, August.
    38. Colin A. Carter & Gordon C. Rausser & Aaron Smith, 2011. "Commodity Booms and Busts," Annual Review of Resource Economics, Annual Reviews, vol. 3(1), pages 87-118, October.
    39. Shleifer, Andrei & Vishny, Robert W, 1997. "The Limits of Arbitrage," Journal of Finance, American Finance Association, vol. 52(1), pages 35-55, March.
    40. Fama, Eugene F, 1991. "Efficient Capital Markets: II," Journal of Finance, American Finance Association, vol. 46(5), pages 1575-1617, December.
    41. Dietrich Domanski & Alexandra Heath, 2007. "Financial investors and commodity markets," BIS Quarterly Review, Bank for International Settlements, March.
    42. Delbianco, Fernando & Tohmé, Fernando & Stosic, Tatijana & Stosic, Borko, 2016. "Multifractal behavior of commodity markets: Fuel versus non-fuel products," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 573-580.
    43. Radetzki, Marian, 2006. "The anatomy of three commodity booms," Resources Policy, Elsevier, vol. 31(1), pages 56-64, March.
    44. Suleyman Basak & Anna Pavlova, 2016. "A Model of Financialization of Commodities," Journal of Finance, American Finance Association, vol. 71(4), pages 1511-1556, August.
    45. Westerhoff, Frank, 2003. "Speculative markets and the effectiveness of price limits," Journal of Economic Dynamics and Control, Elsevier, vol. 28(3), pages 493-508, December.
    46. Luciana Juvenal & Ivan Petrella, 2015. "Speculation in the Oil Market," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 30(4), pages 621-649, June.
    47. Farzanegan, Mohammad Reza, 2011. "Oil revenue shocks and government spending behavior in Iran," Energy Economics, Elsevier, vol. 33(6), pages 1055-1069.
    48. Acharya, Viral V. & Lochstoer, Lars A. & Ramadorai, Tarun, 2013. "Limits to arbitrage and hedging: Evidence from commodity markets," Journal of Financial Economics, Elsevier, vol. 109(2), pages 441-465.
    49. Efthymios G. Pavlidis & Ivan Paya & David A. Peel, 2018. "Using Market Expectations to Test for Speculative Bubbles in the Crude Oil Market," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 50(5), pages 833-856, August.
    50. Farzanegan, Mohammad Reza & Markwardt, Gunther, 2009. "The effects of oil price shocks on the Iranian economy," Energy Economics, Elsevier, vol. 31(1), pages 134-151, January.
    51. Yi-Cheng Zhang, 1999. "Toward a Theory of Marginally Efficient Markets," Papers cond-mat/9901243, arXiv.org.
    52. Wu, Chih-Chiang & Chung, Huimin & Chang, Yu-Hsien, 2012. "The economic value of co-movement between oil price and exchange rate using copula-based GARCH models," Energy Economics, Elsevier, vol. 34(1), pages 270-282.
    53. Miller, J. Isaac & Ratti, Ronald A., 2009. "Crude oil and stock markets: Stability, instability, and bubbles," Energy Economics, Elsevier, vol. 31(4), pages 559-568, July.
    54. Scott H. Irwin & Dwight R. Sanders, 2011. "Index Funds, Financialization, and Commodity Futures Markets," Applied Economic Perspectives and Policy, Agricultural and Applied Economics Association, vol. 33(1), pages 1-31.
    55. Ortiz-Cruz, Alejandro & Rodriguez, Eduardo & Ibarra-Valdez, Carlos & Alvarez-Ramirez, Jose, 2012. "Efficiency of crude oil markets: Evidences from informational entropy analysis," Energy Policy, Elsevier, vol. 41(C), pages 365-373.
    56. Ricardo Hausmann & César Hidalgo, 2011. "The network structure of economic output," Journal of Economic Growth, Springer, vol. 16(4), pages 309-342, December.
    57. Stosic, Darko & Stosic, Dusan & Ludermir, Teresa B. & Stosic, Tatijana, 2019. "Exploring disorder and complexity in the cryptocurrency space," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 548-556.
    58. Fama, Eugene F, 1970. "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance, American Finance Association, vol. 25(2), pages 383-417, May.
    59. Bekiros, Stelios & Nguyen, Duc Khuong & Sandoval Junior, Leonidas & Uddin, Gazi Salah, 2017. "Information diffusion, cluster formation and entropy-based network dynamics in equity and commodity markets," European Journal of Operational Research, Elsevier, vol. 256(3), pages 945-961.
    60. Zunino, Luciano & Fernández Bariviera, Aurelio & Guercio, M. Belén & Martinez, Lisana B. & Rosso, Osvaldo A., 2012. "On the efficiency of sovereign bond markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(18), pages 4342-4349.
    61. Yang, Yan-Hong & Shao, Ying-Hui & Shao, Hao-Lin & Stanley, H. Eugene, 2019. "Revisiting the weak-form efficiency of the EUR/CHF exchange rate market: Evidence from episodes of different Swiss franc regimes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 734-746.
    62. Aurelio Fernandez Bariviera & María Belén Guercio & Lisana B. Martinez & Osvaldo A. Rosso, 2015. "The (in)visible hand in the Libor market: an information theory approach," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 88(8), pages 1-9, August.
    63. Aloui, Chaker & Mabrouk, Samir, 2010. "Value-at-risk estimations of energy commodities via long-memory, asymmetry and fat-tailed GARCH models," Energy Policy, Elsevier, vol. 38(5), pages 2326-2339, May.
    64. Lutz Kilian & Daniel P. Murphy, 2014. "The Role Of Inventories And Speculative Trading In The Global Market For Crude Oil," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 29(3), pages 454-478, April.
    65. Adams, Zeno & Glueck, Thorsten, 2014. "Financialization in Commodity Markets: A Passing Trend or the New Normal?," Working Papers on Finance 1413, University of St. Gallen, School of Finance, revised Aug 2015.
    66. Zunino, Luciano & Tabak, Benjamin M. & Serinaldi, Francesco & Zanin, Massimiliano & Pérez, Darío G. & Rosso, Osvaldo A., 2011. "Commodity predictability analysis with a permutation information theory approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(5), pages 876-890.
    67. Shahzad, Syed Jawad Hussain & Bouri, Elie & Roubaud, David & Kristoufek, Ladislav & Lucey, Brian, 2019. "Is Bitcoin a better safe-haven investment than gold and commodities?," International Review of Financial Analysis, Elsevier, vol. 63(C), pages 322-330.
    68. Nazlioglu, Saban & Soytas, Ugur, 2012. "Oil price, agricultural commodity prices, and the dollar: A panel cointegration and causality analysis," Energy Economics, Elsevier, vol. 34(4), pages 1098-1104.
    69. Wiggins, Seth & Etienne, Xiaoli L., 2017. "Turbulent times: Uncovering the origins of US natural gas price fluctuations since deregulation," Energy Economics, Elsevier, vol. 64(C), pages 196-205.
    70. Ribeiro, Haroldo V. & Zunino, Luciano & Mendes, Renio S. & Lenzi, Ervin K., 2012. "Complexity–entropy causality plane: A useful approach for distinguishing songs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(7), pages 2421-2428.
    71. Lima, L.S. & Miranda, L.L.B., 2018. "Price dynamics of the financial markets using the stochastic differential equation for a potential double well," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 828-833.
    72. Martin, M.T. & Plastino, A. & Rosso, O.A., 2006. "Generalized statistical complexity measures: Geometrical and analytical properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 369(2), pages 439-462.
    73. Dai, Yimei & Zhang, Hesheng & Mao, Xuegeng & Shang, Pengjian, 2018. "Complexity–entropy causality plane based on power spectral entropy for complex time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 501-514.
    Full references (including those not matched with items on IDEAS)

    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. Lang, Korbinian & Auer, Benjamin R., 2020. "The economic and financial properties of crude oil: A review," The North American Journal of Economics and Finance, Elsevier, vol. 52(C).
    2. Fernandes, Leonardo H.S. & de Araujo, Fernando H.A. & Tabak, Benjamin M., 2021. "Insights from the (in)efficiency of Chinese sectoral indices during COVID-19," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 578(C).
    3. Zunino, Luciano & Bariviera, Aurelio F. & Guercio, M. Belén & Martinez, Lisana B. & Rosso, Osvaldo A., 2016. "Monitoring the informational efficiency of European corporate bond markets with dynamical permutation min-entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 456(C), pages 1-9.
    4. Yao, Wei & Alexiou, Constantinos, 2024. "On the transmission mechanism between the inventory arbitrage activity, speculative activity and the commodity price under the US QE policy: Evidence from a TVP-VAR model," International Review of Economics & Finance, Elsevier, vol. 89(PA), pages 1054-1072.
    5. Fernandes, Leonardo H.S. & de Araújo, Fernando H.A. & Silva, Igor E.M., 2020. "The (in)efficiency of NYMEX energy futures: A multifractal analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).
    6. Aiube, Fernando Antonio Lucena & Faquieri, Winicius Botelho, 2019. "Can Gaussian factor models of commodity prices capture the financialization phenomenon?," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    7. Ben Amar, Amine & Goutte, Stéphane & Isleimeyyeh, Mohammad, 2022. "Asymmetric cyclical connectedness on the commodity markets: Further insights from bull and bear markets," The Quarterly Review of Economics and Finance, Elsevier, vol. 85(C), pages 386-400.
    8. Filippo Natoli, 2021. "Financialization Of Commodities Before And After The Great Financial Crisis," Journal of Economic Surveys, Wiley Blackwell, vol. 35(2), pages 488-511, April.
    9. Boyd, Naomi E. & Harris, Jeffrey H. & Li, Bingxin, 2018. "An update on speculation and financialization in commodity markets," Journal of Commodity Markets, Elsevier, vol. 10(C), pages 91-104.
    10. Liu, Li & Wang, Yudong & Wu, Chongfeng & Wu, Wenfeng, 2016. "Disentangling the determinants of real oil prices," Energy Economics, Elsevier, vol. 56(C), pages 363-373.
    11. Wen, Xiaoqian & Xie, Yuxin & Pantelous, Athanasios A., 2022. "Extreme price co-movement of commodity futures and industrial production growth: An empirical evaluation," Energy Economics, Elsevier, vol. 108(C).
    12. Adams, Zeno & Collot, Solène & Kartsakli, Maria, 2020. "Have commodities become a financial asset? Evidence from ten years of Financialization," Energy Economics, Elsevier, vol. 89(C).
    13. Chkir, Imed & Guesmi, Khaled & Brayek, Angham Ben & Naoui, Kamel, 2020. "Modelling the nonlinear relationship between oil prices, stock markets, and exchange rates in oil-exporting and oil-importing countries," Research in International Business and Finance, Elsevier, vol. 54(C).
    14. Argyroudis, George S. & Siokis, Fotios M., 2019. "Spillover effects of Great Recession on Hong-Kong’s Real Estate Market: An analysis based on Causality Plane and Tsallis Curves of Complexity–Entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 576-586.
    15. Haase, Marco & Seiler Zimmermann, Yvonne & Zimmermann, Heinz, 2016. "The impact of speculation on commodity futures markets – A review of the findings of 100 empirical studies," Journal of Commodity Markets, Elsevier, vol. 3(1), pages 1-15.
    16. Duc Huynh, Toan Luu & Burggraf, Tobias & Nasir, Muhammad Ali, 2020. "Financialisation of natural resources & instability caused by risk transfer in commodity markets," Resources Policy, Elsevier, vol. 66(C).
    17. Figuerola-Ferretti, Isabel & McCrorie, J. Roderick & Paraskevopoulos, Ioannis, 2020. "Mild explosivity in recent crude oil prices," Energy Economics, Elsevier, vol. 87(C).
    18. Qadan, Mahmoud & Idilbi-Bayaa, Yasmeen, 2020. "Risk appetite and oil prices," Energy Economics, Elsevier, vol. 85(C).
    19. Siokis, Fotios M., 2018. "Credit market Jitters in the course of the financial crisis: A permutation entropy approach in measuring informational efficiency in financial assets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 266-275.
    20. Xiao, Qin & Yan, Meilan & Zhang, Dalu, 2023. "Commodity market financialization, herding and signals: An asymmetric GARCH R-vine copula approach," International Review of Financial Analysis, Elsevier, vol. 89(C).

    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:eee:chsofr:v:137:y:2020:i:c:s096007792030309x. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.