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A nonsmooth van der Pol-Duffing oscillator (I): The sum of indices of equilibria is $ -1 $

  • * Corresponding author: Hebai Chen

    * Corresponding author: Hebai Chen
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  • The paper deals with the bifurcation diagram and all global phase portraits in the Poincaré disc of a nonsmooth van der Pol-Duffing oscillator with the form $ \dot{x} = y $, $ \dot{y} = a_1x+a_2x^3+b_1y+b_2|x|y $, where $ a_i, b_i $ are real and $ a_2b_2\neq0 $, $ i = 1, 2 $. The system is an equivariant system. When the sum of indices of equilibria is $ -1 $, i.e., $ a_2>0 $, it is proven that the bifurcation diagram includes one Hopf bifurcation surface, one pitchfork bifurcation surface and one heteroclinic bifurcation surface. Although the vector field is only $ C^1 $, we still obtain that the heteroclinic bifurcation surface is $ C^{\infty} $ and a generalized Hopf bifurcation occurs. Moreover, we also find that the heteroclinic bifurcation surface and the focus-node surface have exactly one intersection curve.

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

    Citation:

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  • Figure 1.  The slice $ \mu_3 = {\mu_3}^{(0)} $ of the bifurcation diagram and global phase portraits of system (1.3a)

    Figure 2.  The relative positions of $ HL $ and $ FN_1 $

    Figure 3.  Dynamical behaviors near $ D $

    Figure 4.  Dynamical behaviors near $ I_y $

    Figure 5.  Dynamical behaviors near saddles

    Figure 6.  The closed orbit $ \gamma $

    Figure 7.  Annular regions according the positions of $ A $ and $ B $

    Figure 8.  The changes of unstable and stable manifolds

    Figure 9.  Numerical phase portraits with three equilibria

    Figure 10.  Numerical phase portraits with one closed orbit surrounding a focus

    Figure 11.  Numerical phase portraits with one closed orbit surrounding an unidirectional node

    Figure 12.  Numerical phase portraits with one closed orbit surrounding a bidirectional node

    Table 1.  Properties of $ E_0 $, $ E_l $ and $ E_r $

    possibilities of $ (\mu_1, \mu_2) $types and stabilities
    $ \mu_1< 0 $, $ \mu_2<-2\sqrt{-\mu_1} $$ E_l $, $ E_r $ saddles; $ E_0 $ stable bidirectional node
    $ \mu_1< 0 $, $ \mu_2 = -2\sqrt{-\mu_1} $$ E_l $, $ E_r $ saddles; $ E_0 $ stable unidirectional node
    $ \mu_1< 0 $, $ -2\sqrt{-\mu_1}<\mu_2<0 $$ E_l $, $ E_r $ saddles; $ E_0 $ stable rough focus
    $ \mu_1< 0 $, $ \mu_2 = 0 $$ E_l $, $ E_r $ saddles; $ E_0 $ stable weak focus
    $ \mu_1< 0 $, $ 0<\mu_2<2\sqrt{-\mu_1} $$ E_l $, $ E_r $ saddles; $ E_0 $ unstable rough focus
    $ \mu_1< 0 $, $ \mu_2 = 2\sqrt{-\mu_1} $$ E_l $, $ E_r $ saddles; $ E_0 $ unstable unidirectional node
    $ \mu_1< 0 $, $ \mu_2>2\sqrt{-\mu_1} $$ E_l $, $ E_r $ saddles; $ E_0 $ unstable bidirectional node
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