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

Yilei Tang

doi:10.3934/dcds.2018082

Article Contents

2018, Volume 38, Issue 4: 2029-2046. Doi: 10.3934/dcds.2018082

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

Yilei Tang^1,2,

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: February 28, 2017

Revised: July 31, 2017

Published: June 2018

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Abstract

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.

Keywords:

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

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

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

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

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

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Table 1. 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 $\tilde{a}_1>0$

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Table 2. 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$

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