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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:
 [1] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University, Cambridge, 1990. [2] J. Carr, Application of Center Manifold Theory, Springer, New York, 1981. doi: 10.1007/978-1-4612-5929-9. [3] S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, Cambridge, 1994. [4] S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005. doi: 10.1007/978-1-4757-9168-6. [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. [6] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors, Springer, New York, 1983. [7] G. Iooss, Bifurcation of Maps and Applications, Mathematical Studies, 36, North Holland, Amsterdam, 1979. [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. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4757-2421-9. [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. [11] J. Liu, Z. 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. [12] P. A. Samuelson, Interaction between themultiplier analysis and the principle of acceleration, Rev. Econ. Stat., 21 (1939), 75-78. doi: 10.2307/1927758. [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. [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. [15] H. Sedaghat, Global attractivity, oscillations and chaos in a class of nonlinear, second order difference equations, Cubo, 7 (2005), 89-110. [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. [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. [18] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. doi: 10.1007/b97481.

show all references

##### References:
 [1] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University, Cambridge, 1990. [2] J. Carr, Application of Center Manifold Theory, Springer, New York, 1981. doi: 10.1007/978-1-4612-5929-9. [3] S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University, Cambridge, 1994. [4] S. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005. doi: 10.1007/978-1-4757-9168-6. [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. [6] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors, Springer, New York, 1983. [7] G. Iooss, Bifurcation of Maps and Applications, Mathematical Studies, 36, North Holland, Amsterdam, 1979. [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. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 1998. doi: 10.1007/978-1-4757-2421-9. [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. [11] J. Liu, Z. 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. [12] P. A. Samuelson, Interaction between themultiplier analysis and the principle of acceleration, Rev. Econ. Stat., 21 (1939), 75-78. doi: 10.2307/1927758. [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. [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. [15] H. Sedaghat, Global attractivity, oscillations and chaos in a class of nonlinear, second order difference equations, Cubo, 7 (2005), 89-110. [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. [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. [18] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. doi: 10.1007/b97481.
Bifurcation diagram for $q_5>0$
Bifurcation diagram of system (20)
Bifurcation diagram for $q_5<0$
Bifurcation diagram for $\mathcal{L}>0$
Chenciner bifurcation of system (44) in the case $c_5(0)<0$
Chenciner bifurcation of system (3) in the case $\mathcal{L}<0$
Invariant circles arising from Chenciner bifurcation
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