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

Jiyu Zhong; Shengfu Deng

doi:10.3934/dcdsb.2018062

Article Contents

2018, Volume 23, Issue 4: 1581-1600. Doi: 10.3934/dcdsb.2018062

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

Jiyu Zhong and
Shengfu Deng^,

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

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

Received: April 30, 2017

Revised: July 31, 2017

Early access: February 2018

Published: April 2018

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).

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Abstract

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.

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Mathematics Subject Classification: Primary: 39A10, 39A28; Secondary: 58K50.

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Figure 1. Bifurcation diagram for $q_5>0$

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Figure 2. Bifurcation diagram of system (20)

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Figure 3. Bifurcation diagram for $q_5<0$

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Figure 4. Bifurcation diagram for $\mathcal{L}>0$

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Figure 5. Chenciner bifurcation of system (44) in the case $c_5(0)<0$

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Figure 6. Chenciner bifurcation of system (3) in the case $\mathcal{L}<0$

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Figure 7. Invariant circles arising from Chenciner bifurcation

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