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Bifurcations in an economic model with fractional degree

  • * Corresponding author: Weinian Zhang

    * Corresponding author: Weinian Zhang

Supported by NSFC grants #11771307, #11831012 and #11821001

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  • 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.

    Mathematics Subject Classification: Primary: 34C05, 34C23, 34C60; Secondary: 91B55.


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  • Figure 1.  Parameter plane and global phase portrait

    Figure 2.  Phase portraits of system (1) in a bounded region

    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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