Skip to main content
Log in

Analytic Invariant Curves for an Iterative Equation Related to Ricker-type Second-order Equation

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we consider the existence of analytic invariant curves of an iterative equation

$$f(f(x)) = x{{\rm{e}}^{a - x - f(x)}}$$

which arises from Ricker-type second-order equation. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the parameter at resonance, i.e., at a root of the unity, but also the parameter near resonance under the Brjuno condition.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amleh, A. M., Cmouzis, E., Ladas, G.: On second order rational difference equations, I. J. Differ. Equ. Appl., 13, 969–1004 (2007)

    Article  MathSciNet  Google Scholar 

  2. Brjuno, A. D.: Analytic form of differential equations. Trans. Moscow Math. Soc., 25, 131–288 (1971)

    MathSciNet  Google Scholar 

  3. Carleson, L., Gamelin, T.: Complex Dynamics, Tracts in Mathematics, Springer-Verlag, New York, 1996

    Google Scholar 

  4. Carletti, T., Marmi, S.: Linearization of Analytic and Non-Analytic Germs of Diffeomorphisms of (ℂ, 0). Bull. Soc. Math. France, 128, 69–85 (2000)

    Article  MathSciNet  Google Scholar 

  5. Camouzis, E., Ladas, G.: When does local stability imply global attractivity in rational equations?. J. Differ. Equ. Appl., 12, 863–885 (2006)

    Article  MathSciNet  Google Scholar 

  6. Davie, A. M.: The critical function for the semistandard map. Nonlinearity, 7, 219–229 (1994)

    Article  MathSciNet  Google Scholar 

  7. Grove, E. A., Janowski, E. J., Kent, C. M., et al.: On the rational recursive sequence \({x_{n + 1}} = {{(\alpha {x_n} + \beta )} \over {\gamma {x_n} + \delta }}\). Commun. Appl. Nonlinear Anal., 1, 61–72 (1994)

    Google Scholar 

  8. Kulenović, M. R. S., Ladas, G.: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall, Boca Raton, FL, 2002

    MATH  Google Scholar 

  9. Lazaryan, N., Sedaghat, H.: Extinction, periodicity and multistability in a Ricker model of stage-structured populations. J. Differ. Equ. Appl., 22, 519–544 (2016)

    Article  MathSciNet  Google Scholar 

  10. Lazaryan, N., Sedaghat, H.: Periodic and non-periodic solutions of a Ricker-type second-order equation with periodic parameters. J. Differ. Equ. Appl., 22, 1199–1223 (2016)

    Article  MathSciNet  Google Scholar 

  11. Liz, E., Pilarczyk, P.: Global dynamics in a stage-structured discrete-time population model with harvesting. J. Theor. Biol., 297, 148–165 (2012)

    Article  MathSciNet  Google Scholar 

  12. Luis, R., Elaydi, S., Oliveira, H.: Stability of a Ricker-type competition model and the competitive exclusion principle. J. Biol. Dyn., 5, 636–660 (2011)

    Article  MathSciNet  Google Scholar 

  13. Marmi, S.: An introduction to small divisors problems, Quaderni del Dottorato Di Ricerca, University of Pisa, Istituti Editoriali Poligrafici Internazionali, 2000

  14. Marmi, S., Moussa, P., Yoccoz, J. C.: The Brjuno functions and their regularity properties. Comm. Math. Phys., 186, 265–293 (1997)

    Article  MathSciNet  Google Scholar 

  15. Ng, C. T., Zhang, W. N.: Invariant curves for planar mappings. J. Differ. Equ. Appl., 3, 147–168 (1997)

    Article  MathSciNet  Google Scholar 

  16. Ng, C. T., Zhang, W. N.: Quadratic invariant curves for a planar mapping. J. Differ. Equ. Appl., 6, 147–163 (2000)

    Article  MathSciNet  Google Scholar 

  17. Ricker, W. E.: Stock and recruitment. Journal of Fish Research Board of Canada, 11, 559–623 (1954)

    Article  Google Scholar 

  18. Sacker, R.: A note on periodic Ricker maps. J. Differ. Equ. Appl., 13, 89–92 (2007)

    Article  MathSciNet  Google Scholar 

  19. Sedaghat, H.: A note: All homogeneous second order difference equations of degree one have semiconjugate factorizations. J. Differ. Equ. Appl., 13, 453–456 (2007)

    Article  MathSciNet  Google Scholar 

  20. Si, J. G., Li, X. L.: Small divisor problem in dynamical systems and analytic solutions of the Shabat equation. J. Math. Anal. Appl., 367, 287–295 (2010)

    Article  MathSciNet  Google Scholar 

  21. Si, J. G., Zhang, W. N.: Analytic solutions of a functional equations for invariant curves. J. Math. Anal. Appl., 259, 83–93 (2001)

    Article  MathSciNet  Google Scholar 

  22. Wang, W.: Analytic invariant curves of a nonlinear second order difference equation. Acta Math. Sci., 29, 415–426 (2009)

    Article  MathSciNet  Google Scholar 

  23. Zhao, H. Y.: Analytic invariant curves for an iterative equation related to Pielou’s equation. J. Differ. Equ. Appl., 19, 1082–1092 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hou Yu Zhao.

Additional information

Partially supported by the National Natural Science Foundation of China (Grant Nos. 11671061, 11971081), the Natural Science Foundation of Chongqing (Grant No. cstc2020jcyj-msxmX0857), Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201800502, KJQN201900525), Foundation of youth talent of Chongqing Normal University (02030307-00039), the (Grant Nos. VEGA-MS 1/0358/20, VEGA-SAV 2/0127/20) and by the Slovak Research and Development Agency under the contract (Grant No. APVV-18-0308)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, H.Y., Fečkan, M. Analytic Invariant Curves for an Iterative Equation Related to Ricker-type Second-order Equation. Acta. Math. Sin.-English Ser. 37, 1041–1052 (2021). https://doi.org/10.1007/s10114-021-8530-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-021-8530-x

Keywords

MR(2010) Subject Classification

Navigation