February  2019, 24(2): 881-905. doi: 10.3934/dcdsb.2018211

Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two

School of Mathematical Sciences, Huaqiao University, Fujian 362021, China

* Corresponding author: Dingheng Pi

Received  October 2017 Revised  January 2018 Published  February 2019 Early access  June 2018

Fund Project: This work was partially supported NNSF of China grants 11401228 and 11671040, Cultivation Program for Outstanding Young Scientific talents of Fujian Province in 2017 and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-YX401).

In this paper we consider an $n$ dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold Σ which is an intersection of two co-dimension one switching manifolds Σ1 and Σ2. We investigate the relation of periodic orbit of PWS between periodic orbit of its regularized system. If this PWS system has an asymptotically stable crossing periodic orbit γ or sliding periodic orbit, we establish conditions to ensure that also a regularization of the given system has a unique, asymptotically stable, limit cycle in a neighbourhood of γ, converging to γ as the regularization parameter goes to 0.

Citation: Dingheng Pi. Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 881-905. doi: 10.3934/dcdsb.2018211
References:
[1]

J. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces. I. Blending, Houston J. Math., 24 (1998), 545-569. 

[2]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, UK, 1996.

[3]

M. Di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. 163, Springer-Verlag, London, 2008.

[4]

C. A. Buzzi, T. De Carvalho and R. D. Euzébio, On Poincaré-Bendixson Theorem and nontrivial minimal sets in planar nonsmooth vector fields, Publ. Mat., 62 (2018), 113–131, arXiv: 1307.6825v1 [math. DS]. doi: 10.5565/PUBLMAT6211806.

[5]

C. A. BuzziT. De Carvalho and P. R. Da Silva, Closed Poly-trajectories and Poincaré index of non-smooth vector fields on the plane, J. Dyn. Control. Sys., 19 (2013), 173-193.  doi: 10.1007/s10883-013-9169-4.

[6]

C. A. BuzziT. De Carvalho and M. A. Teixeira, Birth of limit cycles bifurcating from a nonsmooth center, J. Math. Pure. Appl., 102 (2014), 36-47.  doi: 10.1016/j.matpur.2013.10.013.

[7]

L. Dieci and F. Difonzo, A comparison of Filippov sliding vector fields in codimension 2, J. Comput. Appl. Math., 262 (2014), 161-179.  doi: 10.1016/j.cam.2013.10.055.

[8]

L. Dieci and C. Elia, Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can we say what should happen?, DCDS -S, 9 (2016), 1039-1068.  doi: 10.3934/dcdss.2016041.

[9]

L. DieciC. Elia and L. Lopez, A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis, J. Differential Equations, 254 (2013), 1800-1832.  doi: 10.1016/j.jde.2012.11.007.

[10]

L. DieciC. Elia and L. Lopez, Uniqueness of Filippov sliding vector field on the intersection of two surfaces in $\mathbb{R}^3$ and implications for stability of periodic orbits, J. Nonlinear Sci., 25 (2015), 1453-1471.  doi: 10.1007/s00332-015-9265-6.

[11]

L. DieciC. Elia and D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, DCDS-B, 22 (2017), 3091-3112.  doi: 10.3934/dcdsb.2017165.

[12]

L. Dieci and L. Lopez, Fundamental matrix solutions of piecewise smooth differential systems, Math. Comput. Simulation, 81 (2011), 932-953.  doi: 10.1016/j.matcom.2010.10.012.

[13]

L. Dieci and N. Guglielmi, Regularizing piecewise smooth differential systems: Co-dimension 2 discontinuity surface, J. Dynam. Differential Equations, 25 (2013), 71-94.  doi: 10.1007/s10884-013-9287-4.

[14]

Z. Du and Y. Li, Bifurcation of periodic orbits with multiple crossings in a class of planar Filippov systems, Math. Comput. Modelling, 55 (2012), 1072-1082.  doi: 10.1016/j.mcm.2011.09.032.

[15]

A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.

[16]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, 18, Springer-verlag, Berlin, 2004. doi: 10.1007/978-3-540-44398-8.

[17]

J. LlibreP. da Silva and M. A. Teixeira, Regularization of discontinuous vector fields on $\mathbb{R}^3$ via singular perturbation, J. Dynam. Differential Equations, 19 (2007), 309-331.  doi: 10.1007/s10884-006-9057-7.

[18]

J. LlibreP. R. da Silva and M. A. Teixeira, Sliding vector fields via slow-fast systems, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 851-869. 

[19]

J. LlibreP. R. da Silva and M. A. Teixeira, Study of singularities in nonsmooth dynamical systems via singular perturbation, SIAM. J. Applied Dynam. Sys., 8 (2009), 508-526.  doi: 10.1137/080722886.

[20]

P. C. Müller, Calculation of lyapunov exponents for dynamic systems with discontinuities, Chaos, Solitons and Fractals, 5 (1995), 1671-1681.  doi: 10.1016/0960-0779(94)00170-U.

[21]

D. Pi and X. Zhang, The sliding bifurcations in planar piecewise smooth differential systems, J. Dynam. Differential Equations, 25 (2013), 1001-1026.  doi: 10.1007/s10884-013-9327-0.

[22]

L. A. Sanchez, Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems, J. Differential Equations, 246 (2009), 1978-1990.  doi: 10.1016/j.jde.2008.10.015.

[23]

J. Sotomayor and A. L. Machado, Sructurally stable discontinuous vector fields on the plane, Qual. Theory of Dynamical Systems, 3 (2002), 227-250.  doi: 10.1007/BF02969339.

[24]

J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207–223.

[25]

S. TangJ. LiangY. Xiao and A. Robert, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J.Appl.Math., 72 (2012), 1061-1080.  doi: 10.1137/110847020.

[26]

Y. WangM. Han and D. Constantinescu, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos Solitons Fractals, 83 (2016), 158-177. 

[27]

H. R. Zhu and H. L. Smith, Stable periodic orbits for a class of three-dimensional competitive systems, J. Differential Equations, 110 (1994), 143-156.  doi: 10.1006/jdeq.1994.1063.

show all references

References:
[1]

J. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces. I. Blending, Houston J. Math., 24 (1998), 545-569. 

[2]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, UK, 1996.

[3]

M. Di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. 163, Springer-Verlag, London, 2008.

[4]

C. A. Buzzi, T. De Carvalho and R. D. Euzébio, On Poincaré-Bendixson Theorem and nontrivial minimal sets in planar nonsmooth vector fields, Publ. Mat., 62 (2018), 113–131, arXiv: 1307.6825v1 [math. DS]. doi: 10.5565/PUBLMAT6211806.

[5]

C. A. BuzziT. De Carvalho and P. R. Da Silva, Closed Poly-trajectories and Poincaré index of non-smooth vector fields on the plane, J. Dyn. Control. Sys., 19 (2013), 173-193.  doi: 10.1007/s10883-013-9169-4.

[6]

C. A. BuzziT. De Carvalho and M. A. Teixeira, Birth of limit cycles bifurcating from a nonsmooth center, J. Math. Pure. Appl., 102 (2014), 36-47.  doi: 10.1016/j.matpur.2013.10.013.

[7]

L. Dieci and F. Difonzo, A comparison of Filippov sliding vector fields in codimension 2, J. Comput. Appl. Math., 262 (2014), 161-179.  doi: 10.1016/j.cam.2013.10.055.

[8]

L. Dieci and C. Elia, Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can we say what should happen?, DCDS -S, 9 (2016), 1039-1068.  doi: 10.3934/dcdss.2016041.

[9]

L. DieciC. Elia and L. Lopez, A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis, J. Differential Equations, 254 (2013), 1800-1832.  doi: 10.1016/j.jde.2012.11.007.

[10]

L. DieciC. Elia and L. Lopez, Uniqueness of Filippov sliding vector field on the intersection of two surfaces in $\mathbb{R}^3$ and implications for stability of periodic orbits, J. Nonlinear Sci., 25 (2015), 1453-1471.  doi: 10.1007/s00332-015-9265-6.

[11]

L. DieciC. Elia and D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, DCDS-B, 22 (2017), 3091-3112.  doi: 10.3934/dcdsb.2017165.

[12]

L. Dieci and L. Lopez, Fundamental matrix solutions of piecewise smooth differential systems, Math. Comput. Simulation, 81 (2011), 932-953.  doi: 10.1016/j.matcom.2010.10.012.

[13]

L. Dieci and N. Guglielmi, Regularizing piecewise smooth differential systems: Co-dimension 2 discontinuity surface, J. Dynam. Differential Equations, 25 (2013), 71-94.  doi: 10.1007/s10884-013-9287-4.

[14]

Z. Du and Y. Li, Bifurcation of periodic orbits with multiple crossings in a class of planar Filippov systems, Math. Comput. Modelling, 55 (2012), 1072-1082.  doi: 10.1016/j.mcm.2011.09.032.

[15]

A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.

[16]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, 18, Springer-verlag, Berlin, 2004. doi: 10.1007/978-3-540-44398-8.

[17]

J. LlibreP. da Silva and M. A. Teixeira, Regularization of discontinuous vector fields on $\mathbb{R}^3$ via singular perturbation, J. Dynam. Differential Equations, 19 (2007), 309-331.  doi: 10.1007/s10884-006-9057-7.

[18]

J. LlibreP. R. da Silva and M. A. Teixeira, Sliding vector fields via slow-fast systems, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 851-869. 

[19]

J. LlibreP. R. da Silva and M. A. Teixeira, Study of singularities in nonsmooth dynamical systems via singular perturbation, SIAM. J. Applied Dynam. Sys., 8 (2009), 508-526.  doi: 10.1137/080722886.

[20]

P. C. Müller, Calculation of lyapunov exponents for dynamic systems with discontinuities, Chaos, Solitons and Fractals, 5 (1995), 1671-1681.  doi: 10.1016/0960-0779(94)00170-U.

[21]

D. Pi and X. Zhang, The sliding bifurcations in planar piecewise smooth differential systems, J. Dynam. Differential Equations, 25 (2013), 1001-1026.  doi: 10.1007/s10884-013-9327-0.

[22]

L. A. Sanchez, Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems, J. Differential Equations, 246 (2009), 1978-1990.  doi: 10.1016/j.jde.2008.10.015.

[23]

J. Sotomayor and A. L. Machado, Sructurally stable discontinuous vector fields on the plane, Qual. Theory of Dynamical Systems, 3 (2002), 227-250.  doi: 10.1007/BF02969339.

[24]

J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207–223.

[25]

S. TangJ. LiangY. Xiao and A. Robert, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J.Appl.Math., 72 (2012), 1061-1080.  doi: 10.1137/110847020.

[26]

Y. WangM. Han and D. Constantinescu, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos Solitons Fractals, 83 (2016), 158-177. 

[27]

H. R. Zhu and H. L. Smith, Stable periodic orbits for a class of three-dimensional competitive systems, J. Differential Equations, 110 (1994), 143-156.  doi: 10.1006/jdeq.1994.1063.

Figure 2.  Sliding periodic orbit
Figure 1.  Crossing periodic orbit
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