Skip to main content
SpringerLink
Log in

Bifurcation and chaotic behavior of a discrete-time Ricardo–Malthus model

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The dynamics of a discrete-time Ricardo–Malthus model obtained by numerical discretization is investigated, where the step size δ is regarded as a bifurcation parameter. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of \(R^{2}_{+}\) by using the theory of center manifold and normal form. Numerical simulations are presented not only to illustrate our theoretical results, but also to exhibit the system’s complex dynamical behavior, such as the cascade of period-doubling bifurcation in orbits of period 2, 4, 8 16, period-11, 22, 28 orbits, quasiperiodic orbits, and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Lotka, A.J.: Elements of Mathematical Biology. Dover, New York (1956)

    MATH  Google Scholar 

  2. Voltera, V.: Opere matematiche, mmemorie e note, vol. V. Acc. Naz. dei Lincei, Roma (1962)

    Google Scholar 

  3. Xia, Y.H., Cao, J.D.: Almost periodicity in an ecological model with M-predators and N-preys by “pure-delay type” system. Nonlinear Dyn. 39, 275–304 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Xu, C.J., Tang, X.H., Liao, M.X., He, X.F.: Bifurcation analysis in a delayed Lotka–Volterra predator–prey model with two delays. Nonlinear Dyn. 66, 169–183 (2011)

    Article  MathSciNet  Google Scholar 

  5. Malthus, T.R.: An Essay on the Theory of Population. Oxford University Press, Oxford (1798)

    Google Scholar 

  6. Ricardo, D.: In: Principles of Political Economy and Taxation. Dent, London (1817)

    Google Scholar 

  7. D’Alessandro, S.: Non-linear dynamics of population and natural resources: the emergence of different patterns of development. Ecol. Econ. 62, 473–481 (2007)

    Article  Google Scholar 

  8. Yu, W.W., Cao, J.D.: Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay. Nonlinear Anal. 62, 141–165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, N., Yuan, H.Q., Sun, H.Y., Zhang, Q.L.: An impulsive multi-delayed feedback control method for stabilizing discrete chaotic systems. Nonlinear Dyn. (2012). doi:10.1007/s11071-012-0434-y

    Google Scholar 

  10. Liu, X., Xiao, D.: Complex dynamic behaviors of a discrete-time predator–prey system. Chaos Solitons Fractals 32, 80–94 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Agiza, H.N., Elabbasy, E.M., Elmetwally, H., Elsadany, A.A.: Chaotic dynamics of a discrete prey–predator model with Holling type II. Nonlinear Anal., Real World Appl. 10, 116–129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yu, W.W., Cao, J.D., Chen, G.R.: Stability and Hopf bifurcation of a general delayed recurrent neural network. IEEE Trans. Neural Netw. 19, 845–854 (2008)

    Article  Google Scholar 

  13. Yuri, A.K.: Elements of Applied Bifurcation Theory. Springer, New York (1995)

    MATH  Google Scholar 

  14. Carr, J.: Application of Center Manifold Theory. Springer, New York (1981)

    Book  Google Scholar 

  15. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)

    MATH  Google Scholar 

  16. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  17. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd edn. CRC Press, Boca Raton (1999)

    MATH  Google Scholar 

  18. Liu, F., Wang, H.O., Guan, Z.H.: Hopf bifurcation control in the XCP for the Internet congestion control system. Nonlinear Anal., Real World Appl. 13, 1466–1479 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, F., Guan, Z.H., Wang, H.O.: Controlling bifurcations and chaos in TCP-UDP-RED. Nonlinear Anal., Real World Appl. 11, 1491–1501 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guan, Z.H., Liu, F., Li, J., Wang, Y.W.: Chaotification of complex networks with impulsive control. Chaos 22, 023137 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the National Nature Science Foundation under Grants 61073026, 61073065, 61170024, 61170031, and 61272069.

Author information

Authors and Affiliations

  1. Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China

    Xiao-Wei Jiang & Zhi-Hong Guan

  2. School of Power and Mechanical Engineering, Wuhan University, Wuhan, 430072, P.R. China

    Li Ding

  3. School of Mathematics and Statistics, Anyang Normal University, Anyang, 455002, P.R. China

    Fu-Shun Yuan

Authors
  1. Xiao-Wei Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhi-Hong Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fu-Shun Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Li Ding or Zhi-Hong Guan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, XW., Ding, L., Guan, ZH. et al. Bifurcation and chaotic behavior of a discrete-time Ricardo–Malthus model. Nonlinear Dyn 71, 437–446 (2013). https://doi.org/10.1007/s11071-012-0670-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0670-1

Keywords

Search

Navigation