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  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 & 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.   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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