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June  2018, 23(4): 1581-1600. doi: 10.3934/dcdsb.2018062

Two codimension-two bifurcations of a second-order difference equation from macroeconomics

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

* Corresponding author: sf_deng@sohu.com, sfdeng@vt.edu

Received  May 2017 Revised  August 2017 Published  February 2018

Fund Project: The paper was supported by the National Natural Science Foundation of China (No. 11371314 and No. 11771197), the Guangdong Natural Science Foundation of China (No. 2017A030313030), the High-Level Talent Project of Colleges and Universities in Guangdong Province (No. QBS201501), the Startup Foundation for Doctors of Lingnan Normal University (No. ZL1605), and the Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (Grant No. 2015QYJ06).

In this paper we mainly investigate two codimension-two bifurcations of a second-order difference equation from macroeconomics. Applying the center manifold theorem and the normal form analysis, we firstly give the parameter conditions for the generalized flip bifurcation, and prove that the system does not produce a strong resonance. Then, we compute the normal forms to obtain the parameter conditions for the Neimark-Sacker bifurcation, from which we present the conditions for the Chenciner bifurcation. In order to verify the correctness of our results, we also numerically simulate a half stable invariant circle and two invariant circles, one stable and one unstable, arising from the Chenciner bifurcation.

Citation: Jiyu Zhong, Shengfu Deng. Two codimension-two bifurcations of a second-order difference equation from macroeconomics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1581-1600. doi: 10.3934/dcdsb.2018062
References:
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D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University, Cambridge, 1990.  Google Scholar

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J. Carr, Application of Center Manifold Theory, Springer, New York, 1981. doi: 10.1007/978-1-4612-5929-9.  Google Scholar

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S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, Cambridge, 1994.  Google Scholar

[4]

S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005. doi: 10.1007/978-1-4757-9168-6.  Google Scholar

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H. A. El-Morshedy, On the global attractivity and oscillations in a class of second-order difference equations from macroeconomics, J. Differ. Equ. Appl., 17 (2011), 1643-1650.  doi: 10.1080/10236191003730548.  Google Scholar

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J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors, Springer, New York, 1983.  Google Scholar

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G. Iooss, Bifurcation of Maps and Applications, Mathematical Studies, 36, North Holland, Amsterdam, 1979.  Google Scholar

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Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

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S. Li and W. Zhang, Bifurcations in a second-order difference equation from macroeconomics, J. Differ. Equ. Appl., 14 (2008), 91-104.  doi: 10.1080/10236190701483145.  Google Scholar

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J. LiuZ. Yu and W. Zhang, Invariant curves for a second-order difference equation modelled from macroeconomics, J. Differ. Equ. Appl., 21 (2015), 757-773.  doi: 10.1080/10236198.2015.1040008.  Google Scholar

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H. Sedaghat, A class of nonlinear second-order difference equations from macroeconomics, Nonlinear Anal., 29 (1997), 593-603.  doi: 10.1016/S0362-546X(96)00054-5.  Google Scholar

[14]

H. Sedaghat, Regarding the equation $x_{n+1} = cx_n+f(x_n-x_{n-1})$, J. Differ. Equ. Appl., 8 (2002), 667-671.  doi: 10.1080/10236190290032525.  Google Scholar

[15]

H. Sedaghat, Global attractivity, oscillations and chaos in a class of nonlinear, second order difference equations, Cubo, 7 (2005), 89-110.   Google Scholar

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I. Sushko, T. Puu and L. Gardini, A Goodwin-type model with cubic investment function, in Business cycle dynamics: models and tools (eds. T. Puu and I. Suchko), Springer, (2006), 299-316. doi: 10.1007/3-540-32168-3_12.  Google Scholar

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W. Wang, Analytic invariant curves of nonlinear second order equation, Acta Mathematica Scientia, 29 (2009), 415-426.  doi: 10.1016/S0252-9602(09)60041-2.  Google Scholar

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S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. doi: 10.1007/b97481.  Google Scholar

show all references

References:
[1]

D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University, Cambridge, 1990.  Google Scholar

[2]

J. Carr, Application of Center Manifold Theory, Springer, New York, 1981. doi: 10.1007/978-1-4612-5929-9.  Google Scholar

[3]

S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, Cambridge, 1994.  Google Scholar

[4]

S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005. doi: 10.1007/978-1-4757-9168-6.  Google Scholar

[5]

H. A. El-Morshedy, On the global attractivity and oscillations in a class of second-order difference equations from macroeconomics, J. Differ. Equ. Appl., 17 (2011), 1643-1650.  doi: 10.1080/10236191003730548.  Google Scholar

[6]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors, Springer, New York, 1983.  Google Scholar

[7]

G. Iooss, Bifurcation of Maps and Applications, Mathematical Studies, 36, North Holland, Amsterdam, 1979.  Google Scholar

[8]

C. M. Kent and H. Sedaghat, Global stability and boundedness in $x_{n+1} = cx_n+f(x_n-x_{n-1})$, J. Differ. Equ. Appl., 10 (2004), 1215-1227.  doi: 10.1080/10236190410001652829.  Google Scholar

[9]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[10]

S. Li and W. Zhang, Bifurcations in a second-order difference equation from macroeconomics, J. Differ. Equ. Appl., 14 (2008), 91-104.  doi: 10.1080/10236190701483145.  Google Scholar

[11]

J. LiuZ. Yu and W. Zhang, Invariant curves for a second-order difference equation modelled from macroeconomics, J. Differ. Equ. Appl., 21 (2015), 757-773.  doi: 10.1080/10236198.2015.1040008.  Google Scholar

[12]

P. A. Samuelson, Interaction between themultiplier analysis and the principle of acceleration, Rev. Econ. Stat., 21 (1939), 75-78.  doi: 10.2307/1927758.  Google Scholar

[13]

H. Sedaghat, A class of nonlinear second-order difference equations from macroeconomics, Nonlinear Anal., 29 (1997), 593-603.  doi: 10.1016/S0362-546X(96)00054-5.  Google Scholar

[14]

H. Sedaghat, Regarding the equation $x_{n+1} = cx_n+f(x_n-x_{n-1})$, J. Differ. Equ. Appl., 8 (2002), 667-671.  doi: 10.1080/10236190290032525.  Google Scholar

[15]

H. Sedaghat, Global attractivity, oscillations and chaos in a class of nonlinear, second order difference equations, Cubo, 7 (2005), 89-110.   Google Scholar

[16]

I. Sushko, T. Puu and L. Gardini, A Goodwin-type model with cubic investment function, in Business cycle dynamics: models and tools (eds. T. Puu and I. Suchko), Springer, (2006), 299-316. doi: 10.1007/3-540-32168-3_12.  Google Scholar

[17]

W. Wang, Analytic invariant curves of nonlinear second order equation, Acta Mathematica Scientia, 29 (2009), 415-426.  doi: 10.1016/S0252-9602(09)60041-2.  Google Scholar

[18]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. doi: 10.1007/b97481.  Google Scholar

Figure 1.  Bifurcation diagram for $q_5>0$
Figure 2.  Bifurcation diagram of system (20)
Figure 3.  Bifurcation diagram for $q_5<0$
Figure 4.  Bifurcation diagram for $\mathcal{L}>0$
Figure 5.  Chenciner bifurcation of system (44) in the case $c_5(0)<0$
Figure 6.  Chenciner bifurcation of system (3) in the case $\mathcal{L}<0$
Figure 7.  Invariant circles arising from Chenciner bifurcation
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