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Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems

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  • In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi-homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi-homogeneous systems. Moreover, the center is global and non-isochronous, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi-homogeneous but non-homogeneous systems are obtained. Finally we investigate limit cycle bifurcations of the piecewise quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.

    Mathematics Subject Classification: Primary: 37G05; Secondary: 37G10, 34C23, 34C20.

    Citation:

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  • Figure 1.  Existence of closed orbits for system $(I)$

    Figure 2.  The global phase portraits of system $(I)$

    Figure 3.  The global phase portraits of system $(III)$

    Figure 4.  The closed orbit of system $(I)$ and its perturbation

    Table 1.  Parameter conditions of Figure 2

    Figure 2Parameter conditions
    (1) $ b_1>0$, $a_1>0$ and $\tilde{a}_1>0$
    (2) $ b_1>0$, $a_1>0$ and $\tilde{a}_1<0$
    (3) $ b_1>0$, $a_1<0$ and $\tilde{a}_1<0$
    (4) $ b_1>0$, $a_1<0$ and $\tilde{a}_1>0$
    (5) $ b_1<0$, $a_1>0$ and $\tilde{a}_1>0$
    (6) $ b_1<0$, $a_1>0$ and $\tilde{a}_1<0$
    (7) $ b_1<0$, $a_1<0$ and $\tilde{a}_1<0$
    (8) $ b_1<0$, $a_1<0$ and $\tilde{a}_1>0$
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    Table 2.  Parameter conditions of Figure 3

    Figure 3Parameter conditions
    (1) $a_{31}<0$, $ b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$
    (2) $a_{31}<0$, $ b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$
    (3) $a_{31}<0$, $ b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $ \tilde{a}_{31}<0$
    (4) $a_{31}<0$, $ b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}>2$
    (5) $a_{31}<0$, $ b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$
    (6) $a_{31}<0$, $ b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}<0$
    (7) $a_{31}<0$, $ b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}>2$
    (8) $a_{31}<0$, $ b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$
    (9) $a_{31}<0$, $ b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}<0$
    (10) $a_{31}<0$, $ b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $ \tilde{a}_{31}>2$
    (11) $a_{31}<0$, $ b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$
    (12) $a_{31}<0$, $ b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $ \tilde{a}_{31}<0$
    (13) $a_{31}>0$, $ b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $ \tilde{a}_{31}>2$
    (14) $a_{31}>0$, $ b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$
    (15) $a_{31}>0$, $ b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $ \tilde{a}_{31}<0$
    (16) $a_{31}>0$, $ b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}>2$
    (17) $a_{31}>0$, $ b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$
    (18) $a_{31}>0$, $ b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}<0$
    (19) $a_{31}>0$, $ b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $ \tilde{a}_{31}>2$
    (20) $a_{31}>0$, $ b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$
    (21) $a_{31}>0$, $ b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $ \tilde{a}_{31}<0$
    (22) $a_{31}>0$, $ b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}>2$
    (23) $a_{31}>0$, $ b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$
    (24) $a_{31}>0$, $ b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $ \tilde{a}_{31}<0$
    (25) $a_{31}>0$, $ b_3<0$, $a_{32}<0$ and $ \tilde{a}_{31}>2$
    (26) $a_{31}>0$, $ b_3<0$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$
    (27) $a_{31}>0$, $ b_3<0$, $a_{32}<0$ and $ \tilde{a}_{31}<0$
    (28) $a_{31}>0$, $ b_3<0$, $a_{32}>0$ and $ \tilde{a}_{31}>2$
    (29) $a_{31}>0$, $ b_3<0$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$
    (30) $a_{31}>0$, $ b_3<0$, $a_{32}>0$ and $ \tilde{a}_{31}<0$
    (31) $a_{31}<0$, $ b_3>0$, $a_{32}<0$ and $ \tilde{a}_{31}>2$
    (32) $a_{31}<0$, $ b_3>0$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$
    (33) $a_{31}<0$, $ b_3>0$, $a_{32}<0$ and $ \tilde{a}_{31}<0$
    (34) $a_{31}<0$, $ b_3>0$, $a_{32}>0$ and $ \tilde{a}_{31}>2$
    (35) $a_{31}<0$, $ b_3>0$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$
    (36) $a_{31}<0$, $ b_3>0$, $a_{32}>0$ and $ \tilde{a}_{31}<0$
     | Show Table
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