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August  2021, 26(8): 4407-4431. doi: 10.3934/dcdsb.2020293

Bifurcations in an economic model with fractional degree

Shaowen Shi and  Weinian Zhang ,

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Weinian Zhang

Received  April 2020 Revised  July 2020 Published  August 2021 Early access  October 2020

Fund Project: Supported by NSFC grants #11771307, #11831012 and #11821001

A planar ODE system which models the industrialization of a small open economy is considered. Because fractional powers are involved, its interior equilibria are hardly found by solving a transcendental equation and the routine qualitative analysis is not applicable. We qualitatively discuss the transcendental equation, eliminating the transcendental term to polynomialize the expression of extreme value, so that we can compute polynomials to obtain the number of interior equilibria in all cases and complete their qualitative analysis. Orbits near the origin, at which the system cannot be extended differentiably, are investigated by using the GNS method. Then we display all bifurcations of equilibria such as saddle-node bifurcation, transcritical bifurcation and a codimension 2 bifurcation on a one-dimensional center manifold. Furthermore, we prove nonexistence of closed orbits, homoclinic loops and heteroclinic loops, exhibit global orbital structure of the system and analyze the tendency of the industrialization development.

Keywords: Equilibria, bifurcation, fractional order, global structure, numerical simulation.
Mathematics Subject Classification: Primary: 34C05, 34C23, 34C60; Secondary: 91B55.
Citation: Shaowen Shi, Weinian Zhang. Bifurcations in an economic model with fractional degree. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4407-4431. doi: 10.3934/dcdsb.2020293
References:
[1]

A. Antoci, P. Russu, S. Sordi and E. Ticci, Industrialization and environmental externalities in a Solow-type model, J. Econ. Dynam. Control, 47 (2014), 211-224.  doi: 10.1016/j.jedc.2014.08.009.  Google Scholar

[2]

A. Antoci, P. Russu and E. Ticci, Structural change, economic growth and environmental dynamics with heterogeneous agents, in Nonlinear Dynamics in Economics, Finance and the Social Sciences, (eds. G.I. Bischi et. al.), Springer Berlin, (2010), 13–38. doi: 10.1007/978-3-642-04023-8_2.  Google Scholar

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.  Google Scholar

[4]

W. Easterly, The political economy of growth without development: A case study of Pakistan, in In Search of Prosperity: Analytic Narratives on Economic Growth, (ed. D. Rodrik), Princeton University Press, Princeton, (2013), 439–472. doi: 10.1515/9781400845897-016.  Google Scholar

[5]

M. Frommer, Die intergralkurven einer gewöhnlichen differentialgleichung erster ordnung in der umgebung rationaler unbestimmtheitsstellen, Math. Ann., 99 (1928), 222-272.  doi: 10.1007/BF01459096.  Google Scholar

[6]

B. Gao and W. Zhang, Equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function, IEEE Trans. Neural Netw., 19 (2008), 782-794.  doi: 10.1109/TNN.2007.912321.  Google Scholar

[7]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[8]

X. Hou, R. Yan and W. Zhang, Bifurcations of a polynomial differential system of degree $n$ in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423.  doi: 10.1016/S0898-1221(02)00108-6.  Google Scholar

[9]

P. Kongsamutt, S. Rebelo and D. Xie, Beyong balanced growth, Rev. Econom. Stud., 68 (2001), 869-882.  doi: 10.1111/1467-937X.00193.  Google Scholar

[10]

R. López, Sustainable economic development: On the coexistence of resource-dependent resource-impacting industries, Environ. Dev. Econ., 15 (2010), 687-705.  doi: 10.1017/S1355770X10000331.  Google Scholar

[11]

R. E. López, G. Anríquez and S. Gulati, Structural change and sustainable development, J. Environ. Econ. Manage., 53 (2007), 307-322.  doi: 10.1016/j.jeem.2006.10.003.  Google Scholar

[12]

R. López and M. Schiff, Interactive dynamics between natural and man-made assets: The impact of external shocks, J. Dev. Econ., 104 (2013), 1-15.  doi: 10.1016/j.jdeveco.2013.04.001.  Google Scholar

[13]

K. Matsuyama, Agricultural productivity, comparative advantage, and economic growth, J. Econ. Theory, 58 (1992), 317-334.  doi: 10.3386/w3606.  Google Scholar

[14] J. A. Ocampo, C. Rada and L. Taylor, Growth and Policy in Developing Countries: A Structuralist Approach, Columbia University Press, New York, 2009.  doi: 10.7312/ocam15014.  Google Scholar
[15]

V. R. Reddya and B. Behera, Impact of water pollution on rural communities: An economic analysis, Ecol. Econ., 58 (2006), 520-537.  doi: 10.1016/j.ecolecon.2005.07.025.  Google Scholar

[16] G. Sansone and R. Conti, Non-Linear Differential Equations, Pergamon Press, Oxford, 1964.   Google Scholar
[17]

Y. Tang, D. Huang and W. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898.  doi: 10.1093/imamat/hxr005.  Google Scholar

[18]

Y. Tang and W. Zhang, Generalized normal sectors and orbits in exceptional directions, Nonlinearity, 17 (2004), 1407-1426.  doi: 10.1088/0951-7715/17/4/015.  Google Scholar

[19]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101, Amer. Math. Soc., Providence, 1992. Google Scholar

show all references

References:
[1]

A. Antoci, P. Russu, S. Sordi and E. Ticci, Industrialization and environmental externalities in a Solow-type model, J. Econ. Dynam. Control, 47 (2014), 211-224.  doi: 10.1016/j.jedc.2014.08.009.  Google Scholar

[2]

A. Antoci, P. Russu and E. Ticci, Structural change, economic growth and environmental dynamics with heterogeneous agents, in Nonlinear Dynamics in Economics, Finance and the Social Sciences, (eds. G.I. Bischi et. al.), Springer Berlin, (2010), 13–38. doi: 10.1007/978-3-642-04023-8_2.  Google Scholar

[3]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.  Google Scholar

[4]

W. Easterly, The political economy of growth without development: A case study of Pakistan, in In Search of Prosperity: Analytic Narratives on Economic Growth, (ed. D. Rodrik), Princeton University Press, Princeton, (2013), 439–472. doi: 10.1515/9781400845897-016.  Google Scholar

[5]

M. Frommer, Die intergralkurven einer gewöhnlichen differentialgleichung erster ordnung in der umgebung rationaler unbestimmtheitsstellen, Math. Ann., 99 (1928), 222-272.  doi: 10.1007/BF01459096.  Google Scholar

[6]

B. Gao and W. Zhang, Equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function, IEEE Trans. Neural Netw., 19 (2008), 782-794.  doi: 10.1109/TNN.2007.912321.  Google Scholar

[7]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[8]

X. Hou, R. Yan and W. Zhang, Bifurcations of a polynomial differential system of degree $n$ in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423.  doi: 10.1016/S0898-1221(02)00108-6.  Google Scholar

[9]

P. Kongsamutt, S. Rebelo and D. Xie, Beyong balanced growth, Rev. Econom. Stud., 68 (2001), 869-882.  doi: 10.1111/1467-937X.00193.  Google Scholar

[10]

R. López, Sustainable economic development: On the coexistence of resource-dependent resource-impacting industries, Environ. Dev. Econ., 15 (2010), 687-705.  doi: 10.1017/S1355770X10000331.  Google Scholar

[11]

R. E. López, G. Anríquez and S. Gulati, Structural change and sustainable development, J. Environ. Econ. Manage., 53 (2007), 307-322.  doi: 10.1016/j.jeem.2006.10.003.  Google Scholar

[12]

R. López and M. Schiff, Interactive dynamics between natural and man-made assets: The impact of external shocks, J. Dev. Econ., 104 (2013), 1-15.  doi: 10.1016/j.jdeveco.2013.04.001.  Google Scholar

[13]

K. Matsuyama, Agricultural productivity, comparative advantage, and economic growth, J. Econ. Theory, 58 (1992), 317-334.  doi: 10.3386/w3606.  Google Scholar

[14] J. A. Ocampo, C. Rada and L. Taylor, Growth and Policy in Developing Countries: A Structuralist Approach, Columbia University Press, New York, 2009.  doi: 10.7312/ocam15014.  Google Scholar
[15]

V. R. Reddya and B. Behera, Impact of water pollution on rural communities: An economic analysis, Ecol. Econ., 58 (2006), 520-537.  doi: 10.1016/j.ecolecon.2005.07.025.  Google Scholar

[16] G. Sansone and R. Conti, Non-Linear Differential Equations, Pergamon Press, Oxford, 1964.   Google Scholar
[17]

Y. Tang, D. Huang and W. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898.  doi: 10.1093/imamat/hxr005.  Google Scholar

[18]

Y. Tang and W. Zhang, Generalized normal sectors and orbits in exceptional directions, Nonlinearity, 17 (2004), 1407-1426.  doi: 10.1088/0951-7715/17/4/015.  Google Scholar

[19]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101, Amer. Math. Soc., Providence, 1992. Google Scholar

Figure 1.  Parameter plane and global phase portrait
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Figure 2.  Phase portraits of system (1) in a bounded region
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Table 1.  Conditions obtained in [1]
cases $ \epsilon $ $ \tilde{E} $ interior equilibria
(C1) $ 0<\epsilon<\tilde\epsilon $ $ 0<\tilde{E}<E_2 $ none
(C2) $ E_2\le\tilde{E}\le E_1 $ unknown
(C3) $ E_1<\tilde{E}\le E_M $ $ S_3 $ saddle
$ S_4 $ stable node
(C4) $ \tilde{E}>E_M $ $ S_3 $ saddle
(C5) $ \epsilon\ge\tilde\epsilon $ $ 0<\tilde{E}<E_2 $ none
(C6) $ E_2\le\tilde{E}\le E_M $ unknown
(C7) $ \tilde{E}>E_M $ $ S_3 $ saddle
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cases $ \epsilon $ $ \tilde{E} $ interior equilibria
(C1) $ 0<\epsilon<\tilde\epsilon $ $ 0<\tilde{E}<E_2 $ none
(C2) $ E_2\le\tilde{E}\le E_1 $ unknown
(C3) $ E_1<\tilde{E}\le E_M $ $ S_3 $ saddle
$ S_4 $ stable node
(C4) $ \tilde{E}>E_M $ $ S_3 $ saddle
(C5) $ \epsilon\ge\tilde\epsilon $ $ 0<\tilde{E}<E_2 $ none
(C6) $ E_2\le\tilde{E}\le E_M $ unknown
(C7) $ \tilde{E}>E_M $ $ S_3 $ saddle
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