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Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network

  • * Corresponding author: Weinian Zhang

    * Corresponding author: Weinian Zhang
Supported by NSFC # 11221101, # 11231001, # 11501475 and SPDEF 16ZB0080.
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  • There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

    Mathematics Subject Classification: 34C23, 92C45.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Reaction scheme

    Figure 2.  Bifurcation surfaces projection on the $(a, \kappa)$-plane

    Figure 3.  Bifurcation diagrams of system (4) for the case that $c>1$ and $b=(c+1)^2/4$

    Figure 4.  An attracting limit cycle

    Table 4.  Dynamics of system (4) in various cases of parameter $(a, \kappa)$

    Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram
    $E_0$ $E_1$ $E_2$ $E_*$
    $ \mathcal{D}_{I}$ saddle unstable focus saddle $\mathcal{D}_{I}$
    $ \mathcal{L}$ saddle unstable focus saddle one homoclinic rrbit $\mathcal{L}$
    $\mathcal{D}_{II}$ saddle unstable focus saddle one limit cycle $\mathcal{D}_{II}$
    $\mathcal{H}$ saddle stable focus saddle $\mathcal{H}$
    $\mathcal{D}_{III}$ saddle stable focus saddle $\mathcal{D}_{III}$
    $ \mathcal{SN}^+$ saddle-node $\mathcal{SN}^+$
    $ \mathcal{D}_{IV}$ $\mathcal{D}_{IV}$
    $(a_*, \kappa_*)$ cusp $(a_*, \kappa_*)$
    $ \mathcal{SN}^-$ saddle-node $\mathcal{SN}^-$
     | Show Table
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