# American Institute of Mathematical Sciences

April  2018, 38(4): 2029-2046. doi: 10.3934/dcds.2018082

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

 1 School of Mathematical Sciences, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai 200240, China 2 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, Maribor, SI-2000 Maribor, Slovenia

Received  March 2017 Revised  August 2017 Published  January 2018

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.

Citation: Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082
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##### References:
Existence of closed orbits for system $(I)$
The global phase portraits of system $(I)$
The global phase portraits of system $(III)$
The closed orbit of system $(I)$ and its perturbation
Parameter conditions of Figure 2
 Figure 2 Parameter 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 $\tilde{a}_1>0$
 Figure 2 Parameter 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 $\tilde{a}_1>0$
Parameter conditions of Figure 3
 Figure 3 Parameter 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 $\tilde{a}_{31}<0$
 Figure 3 Parameter 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 $\tilde{a}_{31}<0$
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