November  2019, 24(11): 5885-5901. doi: 10.3934/dcdsb.2019111

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

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

Received  September 2018 Revised  December 2018 Published  November 2019 Early access  June 2019

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.

Citation: Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
References:
[1]

D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dyn., 73 (2003), 1283-1288.  doi: 10.1007/s11071-013-0862-3.

[2]

L. Dieci and C. Elia, Periodic orbits for planar piecewise smooth systems with a line of discontinuity, J. Dyn. Diff. Equat., 26 (2014), 1049-1078.  doi: 10.1007/s10884-014-9380-3.

[3]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin Heidelberg, 2006.

[4]

R. D. Euzébio and J. Llibre, On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line, J. Math. Anal. Appl., 424 (2015), 475-486.  doi: 10.1016/j.jmaa.2014.10.077.

[5]

F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632.  doi: 10.1088/0951-7715/14/6/311.

[6]

E. FreireE. PonceF. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos, 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728.

[7]

E. FreireE. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.  doi: 10.1137/11083928X.

[8]

E. Freire, E. Ponce and F. Torres, Planar Filippov systems with maximal crossing set and piecewise linear focus dynamics, Progrss and Challenges in Dynamical Systems, 221–232, Springer Proc. Math. Stat., 54, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-38830-9_13.

[9]

E. FreireE. Ponce and F. Torres, A general mechanism to generate three limit cycles in planar Filippov systems with two zones, Nonlinear Dyn., 78 (2014), 251-263.  doi: 10.1007/s11071-014-1437-7.

[10]

S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[11]

S. M. Huan and X. S. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics, Nonlinear Anal., 92 (2013), 82-95.  doi: 10.1016/j.na.2013.06.017.

[12]

S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (2014), 340-353.  doi: 10.1016/j.jmaa.2013.08.064.

[13]

J. Karlin and W. J. Studden, T-systems: With Applications in Analysis and Statistics, Pure Appl. Math. Interscience Publishers, NewYork, London, Sidney, 1966.

[14]

L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., 70 (2014), 1-14.  doi: 10.14232/ejqtde.2014.1.70.

[15]

J. Llibre, D. D. Novaes and M. A. Teixeira, Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones, Internat. J. Bifur. Chaos, 25 (2015), 1550144, 11 pp. doi: 10.1142/S0218127415501448.

[16]

J. LlibreD. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn., 82 (2015), 1159-1175.  doi: 10.1007/s11071-015-2223-x.

[17]

J. LlibreM. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in a planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Ser. B: Real World Appl., 14 (2013), 2002-2012.  doi: 10.1016/j.nonrwa.2013.02.004.

[18]

J. Llibre and E. Ponce, Three limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B, 19 (2012), 325-335. 

[19]

J. Llibre and M. A. Teixeira, Piecewise linear differential systems with only centers can create limit cycles?, Nonlinear Dyn., 91 (2018), 249-255.  doi: 10.1007/s11071-017-3866-6.

[20]

J. Llibre, M. A. Teixeira and J. Torregrosa, Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation, Internat. J. Bifur. Chaos, 23 (2013), 1350066, 10 pp. doi: 10.1142/S0218127413500661.

[21]

J. Llibre and X. Zhang, Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center, J. Math. Anal. Appl., 467 (2018), 537-549.  doi: 10.1016/j.jmaa.2018.07.024.

[22]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part I: Simplest case in $\mathrm{R}^2$, University of California at Berkeley, Memorandum UCB/ERL M90/22, 19 (1991), 251-307.  doi: 10.1002/cta.4490190305.

[23]

E. PonceJ. Ros and E. Vela, The boundary focus-saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators, Nonlinear Anal. Ser. B: Real World Appl., 43 (2018), 495-514.  doi: 10.1016/j.nonrwa.2018.03.011.

show all references

References:
[1]

D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dyn., 73 (2003), 1283-1288.  doi: 10.1007/s11071-013-0862-3.

[2]

L. Dieci and C. Elia, Periodic orbits for planar piecewise smooth systems with a line of discontinuity, J. Dyn. Diff. Equat., 26 (2014), 1049-1078.  doi: 10.1007/s10884-014-9380-3.

[3]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin Heidelberg, 2006.

[4]

R. D. Euzébio and J. Llibre, On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line, J. Math. Anal. Appl., 424 (2015), 475-486.  doi: 10.1016/j.jmaa.2014.10.077.

[5]

F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632.  doi: 10.1088/0951-7715/14/6/311.

[6]

E. FreireE. PonceF. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos, 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728.

[7]

E. FreireE. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.  doi: 10.1137/11083928X.

[8]

E. Freire, E. Ponce and F. Torres, Planar Filippov systems with maximal crossing set and piecewise linear focus dynamics, Progrss and Challenges in Dynamical Systems, 221–232, Springer Proc. Math. Stat., 54, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-38830-9_13.

[9]

E. FreireE. Ponce and F. Torres, A general mechanism to generate three limit cycles in planar Filippov systems with two zones, Nonlinear Dyn., 78 (2014), 251-263.  doi: 10.1007/s11071-014-1437-7.

[10]

S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[11]

S. M. Huan and X. S. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics, Nonlinear Anal., 92 (2013), 82-95.  doi: 10.1016/j.na.2013.06.017.

[12]

S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (2014), 340-353.  doi: 10.1016/j.jmaa.2013.08.064.

[13]

J. Karlin and W. J. Studden, T-systems: With Applications in Analysis and Statistics, Pure Appl. Math. Interscience Publishers, NewYork, London, Sidney, 1966.

[14]

L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., 70 (2014), 1-14.  doi: 10.14232/ejqtde.2014.1.70.

[15]

J. Llibre, D. D. Novaes and M. A. Teixeira, Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones, Internat. J. Bifur. Chaos, 25 (2015), 1550144, 11 pp. doi: 10.1142/S0218127415501448.

[16]

J. LlibreD. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn., 82 (2015), 1159-1175.  doi: 10.1007/s11071-015-2223-x.

[17]

J. LlibreM. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in a planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Ser. B: Real World Appl., 14 (2013), 2002-2012.  doi: 10.1016/j.nonrwa.2013.02.004.

[18]

J. Llibre and E. Ponce, Three limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B, 19 (2012), 325-335. 

[19]

J. Llibre and M. A. Teixeira, Piecewise linear differential systems with only centers can create limit cycles?, Nonlinear Dyn., 91 (2018), 249-255.  doi: 10.1007/s11071-017-3866-6.

[20]

J. Llibre, M. A. Teixeira and J. Torregrosa, Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation, Internat. J. Bifur. Chaos, 23 (2013), 1350066, 10 pp. doi: 10.1142/S0218127413500661.

[21]

J. Llibre and X. Zhang, Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center, J. Math. Anal. Appl., 467 (2018), 537-549.  doi: 10.1016/j.jmaa.2018.07.024.

[22]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part I: Simplest case in $\mathrm{R}^2$, University of California at Berkeley, Memorandum UCB/ERL M90/22, 19 (1991), 251-307.  doi: 10.1002/cta.4490190305.

[23]

E. PonceJ. Ros and E. Vela, The boundary focus-saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators, Nonlinear Anal. Ser. B: Real World Appl., 43 (2018), 495-514.  doi: 10.1016/j.nonrwa.2018.03.011.

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) $
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
Figure 2.  The graphic of the function $ f(t_+) $ in the interval $ (0, \pi) $
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
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
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
F S N
F 3 3 3
S 3 2 2
N 3 2 2
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
F C N N$ ^{'} $
F 3 2 1 1
C 1 0 1 1
S 1 1 1 1
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