On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium

Shimin Li; Jaume Llibre

doi:10.3934/dcdsb.2019111

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2019, Volume 24, Issue 11: 5885-5901. Doi: 10.3934/dcdsb.2019111

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On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium

Shimin Li^1, and
Jaume Llibre^2, ,

1.
School of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou 510320, China
2.
Departament de Matemàtiques, Universitat Auònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

^* Corresponding author: Jaume Llibre
^* Corresponding author: Jaume Llibre

Received: September 2018

Revised: December 2018

Early access: June 2019

Published: August 2019

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Abstract

This paper deals with planar discontinuous piecewise linear differential systems with two zones separated by a vertical straight line $ x = k $. We assume that the left linear differential system ($ x<k $) and the right linear differential system ($ x>k $) share the same equilibrium, which is located at the origin $ O(0, 0) $ without loss of generality.

Our results show that if $ k = 0 $, that is when the unique equilibrium $ O(0, 0) $ is located on the line of discontinuity, then the discontinuous piecewise linear differential systems have no crossing limit cycles. While for the case $ k\neq 0 $ we provide lower and upper bounds for the number of limit cycles of these planar discontinuous piecewise linear differential systems depending on the type of their linear differential systems, i.e. if those systems have foci, centers, saddles or nodes, see Table 2.

Keywords:

Limit cycle,
piecewise discontinuous linear systems.

Mathematics Subject Classification: Primary: 34A36, 34C07; Secondary: 37G15.

Citation:

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Figure 1. Fig 1.1. A periodic orbit of a system (5) with $ k = 0 $. Fig 1.2. The orbit of system (5) with $ k = 1 $ which pass through the point $ (1, y_1) $

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Figure 3. The unique limit cycle of some systems (17). Fig 3.1. Center-Focus type. Fig 3.2. Center-Node (diagonal) type. Fig 3.3. Center-Node (non-diagonal) type

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Figure 2. The graphic of the function $ f(t_+) $ in the interval $ (0, \pi) $

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Figure 4. Limit cycles of some systems (33). Fig 4.1. Saddle-Focus type. Fig 4.2. Saddle-Center type. Fig 4.3. Saddle-Node (diagonal) type. Fig 4.4. Saddle-Node (non-diagonal) type

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Figure 5. Limit cycles of some systems (33). Fig 5.1. Focus-Focus type. Fig 5.2. Focus-Center type. Fig 5.3. Focus-Node (diagonal) type. Fig 5.4. Focus-Node (non-diagonal) type

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Table 1. Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems (3) known up to now. $ F $, $ S $ and $ N $ denote a linear differential systems having a focus or a center, a saddle and a node, respectively. In the column there is the linear differential systems on $ x>0 $, and on the row the linear differential systems in $ x<0 $

	F	S	N
F	3	3	3
S	3	2	2
N	3	2	2

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Table 2. The lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems (5) with $ k>0 $. See Theorem 2

	F	C	N	N$ ^{'} $
F	3	2	1	1
C	1	0	1	1
S	1	1	1	1

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