Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Complexity, Chaos, and the Duffing-Oscillator Model: An Analysis of Inventory Fluctuations in Markets

Journal of Applied Nonlinear Dynamics 3(2) (2014) 147--158 | DOI:10.5890/JAND.2014.06.005

Varsha S. Kulkarni

School of Informatics and Computing, Indiana University, Bloomington, IN47408, USA

Download Full Text PDF



Apparently random financial fluctuations often exhibit complexity, chaos. Predictability of limited length time series is hard to infer. Knowledge about the process driving the dynamics facilitates such analysis. This paper shows that quarterly inventory changes of wheat in the global market, during 1974-2012, follow a nonlinear deterministic process. Weakly chaotic behavior alternates with non-chaotic behavior. Cubic dependence of price changes on inventory changes leads to establishment of Duffing Oscillator model as suitable for examining the inventory changes. Endowing parameters with suitable meanings, one may infer temporary speculation changes reflect inventory volatility that drives the transitions between chaotic and non-chaotic behaviors.


The author dedicates this paper to Suresh Kulkarni. The author thanks Raghav Gaiha, Daniel Bromley, Vidyadhar Mudkavi for advice, Prabha Sharma and Pankaj Wahi for sharing their resources for this work, Chris Raphael for discussions. Financial support in the form of fellowships granted by Indian Institute of Technology Kanpur, Santa Fe Institute, Indiana University is gratefully acknowledged.


  1. [1]  Stanley, H.E. et al. (2002), Self-organized complexity in economics and finance, Proceedings of the National Academy of Sciences , 99, 2561-2565.
  2. [2]  Mantegna, R.N. and Stanley, H.E. (1999), Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge.
  3. [3]  Lorenz, H. (1997), Nonlinear dynamical Economics and Chaotic Motion, Springer, Berlin.
  4. [4]  Faggini, M. (2008), Analysis of Economic Fluctuations : A Contribution from Chaos Theory, in Mathematical and Statistical Methods in Insurance and Finance, C. Perna, M. Sibillo, eds., Springer, Milan, 107-112.
  5. [5]  Rosenstein, M.T., Collins, J.J., and De Luca, C.J. (1993), A practical method for calculating largest Lyapunov exponents from small data sets, Physica D, 65, 117-134.
  6. [6]  Wolf, A. et al. (1985), Determining Lyapunov exponents from a time series, Physica D, 16, 285-317.
  7. [7]  Timmer, C.P. (2010), Reflections on food crises past, Food Policy, 35, 1-11.
  8. [8]  Kyrtsou, C., Labys, W., and Terraza, M. (2003), Noisy chaotic dynamics in commodity markets, Empirical Economics, 29, 489-502.
  9. [9]  Cunningham, S. (2011), Random Walks, Chaos, and Volatility, American Institute of Economic Research report, LXXVIII, 1-3.
  10. [10]  Sprott, J.C. (2003), Chaos and Time Series Analysis, Oxford University press, Oxford.
  11. [11]  Novak, S. and Frehlich, R. (1982), Transitions to chaos in the Duffing oscillator,Physical Review A, 26, 3660-3663.
  12. [12]  Zeni, A.R. and Gallas, J.A.C. (1995), Lyapunov exponents for a Duffing oscillator, Physica D, 89, 71-82.
  13. [13]  Bhalekar, S. and Daftardar-Gejji, V. (2012), Numeric-Analytic Solutions of Dynamical Systems with a New Iterative Method, Journal of Applied Nonlinear Dynamics, 1, 141-158.
  14. [14]  Luo, A.C.J. and Huang, J. (2012), Analytical Routes of Period-1 Motions to Chaos in a Periodically Forced Duffing Oscillator with a Twin-well Potential, Journal of Applied Nonlinear Dynamics, 1, 73-108.
  15. [15]  Ramsey, J. (1989), Economic and Financial Data as Nonlinear Processes, in The Stock Market: Bubbles, Volatility, and Chaos, G. Dwyer, R. Hafer eds., Kluwer Academic press, Boston, 81-139.
  16. [16]  Meadows, D.L. (1970), Dynamics of Commodity Production Cycles, Wright-Allen Press, Cambridge, USA.
  17. [17]  Durbin, J. (1969), Tests for Serial Correlation in Regression Analysis Based on the Periodogram of Least- Squares Residuals, Biometrika, 56, 1-15.
  18. [18]  Grassberger, P. and Procaccia, I. (1983), Characterization of Strange Attractors, Physical Review Letters, 50, 346-349.
  19. [19]  Collins, J. and De Luca, C.J. (1994), Random Walking during Quiet Standing, Physical Review Letters, 73, 764-767.
  20. [20]  Tan, C. and Kang, B. (2001), Chaotic Motions of a Duffing Oscillator Subjected to Combined Parametric and Quasiperiodic Excitation, International Journal of Nonlinear Sciences and Numerical Simulation, 2, 353-364.