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Global phase portrait of a degenerate Bogdanov-Takens system with symmetry

  • * Corresponding author: Xingwu Chen(xingwu.chen@hotmail.com)

    * Corresponding author: Xingwu Chen(xingwu.chen@hotmail.com) 
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  • In this paper we study the global phase portrait of the normal form of a degenerate Bogdanov-Takens system with symmetry, i.e., a class of van der Pol-Duffing oscillators. This normal form is two-parametric and its parameters are considered in the whole parameter space, i.e., not viewed as a perturbation of some Hamiltonian system. We discuss the existence of limit cycles and prove its uniqueness if it exists. Moreover, by constructing a distance function we not only give the necessary and sufficient condition for the existence of heteroclinic loops connecting two saddles, but also prove its monotonicity and smoothness. Finally, we obtain a complete classification on the global phase portraits in the Poincaré disc as well as the complete global bifurcation diagram in the parameter space and find more plentiful phase portraits than the case that parameters are just sufficiently small.

    Mathematics Subject Classification: Primary:34C07, 34C23, 34C37, 34K18.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  The equilibria at infinity

    Figure 2.  Orbits changing under Briot-Bouquet transformations

    Figure 3.  Discussion about the uniqueness of limit cycle for (c4)

    Figure 4.  $y^u(0)+y^s(0)>0$ for $b=0$

    Figure 5.  $y^u(0)+y^s(0)>0$ for sufficiently small $b$

    Figure 6.  Poincaré-Bendixson Annular region

    Figure 7.  The bifurcation diagram of system (5)

    Figure 8.  Global phase portraits of system (5)

    Figure 9.  A heteroclinic loop for $(a, b)=(-1, 0.642054499)$

    Figure 10.  A limit cycle surrounding a node for $(a, b)=(-16, -8)$

    Figure 11.  A limit cycle crossing small neighborhoods of $E_L$ and $E_R$ for $(a, b)=(-8.87,-5.9554)$

    Table 1.  Qualitative properties of equilibria

    possibilities of (a, b) positions of equilibria types and Stability
    a ≥ 0 E0 E0 is a saddle
    a < 0 $b<-2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable bidirectional node
    $b=-2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable unidirectional node
    $-2\sqrt{-a}<b<0$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable rough focus
    $b=0$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable weak focus of order $1$
    $\!\!0<b<2\sqrt{-a}\!\!$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable rough focus
    $b=2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable unidirectional node
    $b>2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable bidirectional node
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