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February  2022, 27(2): 1055-1073. doi: 10.3934/dcdsb.2021080

Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two

Dingheng Pi ,

Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Fujian 362021, China

* Corresponding author: Dingheng Pi

Received  November 2019 Revised  January 2021 Published  February 2022 Early access  March 2021

Fund Project: This work was partially supported by NNSF of China grant 11671040, Cultivation Program for Outstanding Young Scientific talents of Fujian Province in 2017, Program for Innovative Research Team in Science and Technology in Fujian Province University, Quanzhou High-Level Talents Support Plan under Grant 2017ZT012 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 $ \Sigma $ which is an intersection of two hyperplanes $ \Sigma_1 $ and $ \Sigma_2 $. We investigate the relation between periodic orbit of PWS system and periodic orbit of its double regularized system. If this PWS system has an asymptotically stable sliding periodic orbit(including type Ⅰ and type Ⅱ), we establish conditions to ensure that also a double regularization of the given system has a unique, asymptotically stable, periodic orbit in a neighbourhood of $ \gamma $, converging to $ \gamma $ as both of the two regularization parameters go to $ 0 $ by applying implicit function theorem and geometric singular perturbation theory.

Keywords: Piecewise smooth systems, regularization, limit cycle, codimension two switching manifold, asymptotic stability.
Mathematics Subject Classification: Primary: 34A36.
Citation: Dingheng Pi. Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 1055-1073. doi: 10.3934/dcdsb.2021080
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]

M. Antali and G. Stepan, Sliding and crossing dynamics in extended Filippov systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 823-858.  doi: 10.1137/17M1110328.  Google Scholar

[3]

J. Awrejcewicz, M. Fe$\breve{c}$kan and P. Olejnik, On continuous approximation of discontnuous systems, Nonlinear Anal., 62 (2005), 1317-1331.  doi: 10.1016/j.na.2005.04.033.  Google Scholar

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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

[5]

C. Bonet-Reves Reves, J. Larrosa and T. M-Seara, Regularization around a generic codimension one fold-fold singularity, J. Differential Equations, 265 (2018), 1761-1838.  doi: 10.1016/j.jde.2018.04.047.  Google Scholar

[6]

C. Bonet-Revés and T. M-Seara, Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems, Discrete Contin. Dyn. Syst., 36 (2016), 3545-3601.  doi: 10.3934/dcds.2016.36.3545.  Google Scholar

[7]

C. A. Buzzi, T. Carvalho and R. D. Euzébio, On Poincaré-Bendixson Theorem and nontrivial minimal sets in planar nonsmooth vector fields, Publ. Mat., 62 (2018), 113-131.  doi: 10.5565/PUBLMAT6211806.  Google Scholar

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C. A. Buzzi, T. 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

[9]

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

[10]

L. Dieci, T. Eirola and C. Elia, Periodic orbits of planar discontinuous system under discretization, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2743-2762.  doi: 10.3934/dcdsb.2018103.  Google Scholar

[11]

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

[12]

L. Dieci and C. Elia, Piecewise smooth systems near a codimension 2 discontinuity manifold: Can we say what should happen?, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1039-1068.  doi: 10.3934/dcdss.2016041.  Google Scholar

[13]

L. Dieci and C. Elia, Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2935-2950.  doi: 10.3934/dcdsb.2018112.  Google Scholar

[14]

L. Dieci, C. Elia and L. Lopez, A Filippov sliding vector field on an attracting codimension 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

[15]

L. Dieci, C. 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

[16]

L. Dieci, C. Elia and D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3091-3112.  doi: 10.3934/dcdsb.2017165.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov Systems, J. Differential Equations, 250 (2011), 1967-2023.  doi: 10.1016/j.jde.2010.11.016.  Google Scholar

[21]

N. Gugliemi and E. Hairer, Classification of hidden dynamics in discontinuous dynamical systems, SIAM J. Appl. Dyn. Syst., 14 (2015), 1454-1477.  doi: 10.1137/15100326X.  Google Scholar

[22]

N. Gugliemi and E. Hairer, Solutions leaving a codimension-2 sliding, Nolinear. Dyn., 88 (2017), 1427-1439.  doi: 10.1007/s11071-016-3320-1.  Google Scholar

[23]

M. R. Jeffrey, Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking, Chaos, 26 (2016), 033108, 19 pp. doi: 10.1063/1.4943386.  Google Scholar

[24]

K. U. Kristiansen and S. J. Hogan, Regularization of two-fold bifurations in planar piecewise smooth systems using blowup, SIAM J. Appl. Dyn. Syst., 14 (2015), 1731-1786.  doi: 10.1137/15M1009731.  Google Scholar

[25]

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

[26]

J. Llibre, P. R. 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

[27]

J. Llibre, P. R. da Silva and M. A. Teixeira, Sliding vector fields via slow-fast systems, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 851-869.  doi: 10.36045/bbms/1228486412.  Google Scholar

[28]

J. Llibre, P. 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

[29]

D. Panazzolo and P. R. da Silva, Regularization of discontinuous foliations: blowing up and sliding conditions via Fenichel theory, J. Differential Equations, 263 (2017), 8362-8390.  doi: 10.1016/j.jde.2017.08.042.  Google Scholar

[30]

D. Pi, Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two, Discrete Contin. Dyn. Syst.Ser. B, 24 (2019), 881-905.  doi: 10.3934/dcdsb.2018211.  Google Scholar

[31]

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

[32]

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

[33]

H. Schiller and M. Arnold, Convergence of continuous approximations for discontinuous ODEs, Appl. Numer. Math., 62 (2012), 1503-1514.  doi: 10.1016/j.apnum.2012.06.021.  Google Scholar

[34]

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

[35]

J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), World Sci. Publ., River Edge, NJ, 1998,207–223.  Google Scholar

[36]

S. Tang, J. Liang, Y. Xiao and R. A. Cheke, 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

[37]

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, Inc., New York, 1987.  Google Scholar

[38]

J. Yang and L. Zhao, Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations, J. Differential Equations, 264 (2018), 5734-5757.  doi: 10.1016/j.jde.2018.01.017.  Google Scholar

[39]

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]

M. Antali and G. Stepan, Sliding and crossing dynamics in extended Filippov systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 823-858.  doi: 10.1137/17M1110328.  Google Scholar

[3]

J. Awrejcewicz, M. Fe$\breve{c}$kan and P. Olejnik, On continuous approximation of discontnuous systems, Nonlinear Anal., 62 (2005), 1317-1331.  doi: 10.1016/j.na.2005.04.033.  Google Scholar

[4]

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

[5]

C. Bonet-Reves Reves, J. Larrosa and T. M-Seara, Regularization around a generic codimension one fold-fold singularity, J. Differential Equations, 265 (2018), 1761-1838.  doi: 10.1016/j.jde.2018.04.047.  Google Scholar

[6]

C. Bonet-Revés and T. M-Seara, Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems, Discrete Contin. Dyn. Syst., 36 (2016), 3545-3601.  doi: 10.3934/dcds.2016.36.3545.  Google Scholar

[7]

C. A. Buzzi, T. Carvalho and R. D. Euzébio, On Poincaré-Bendixson Theorem and nontrivial minimal sets in planar nonsmooth vector fields, Publ. Mat., 62 (2018), 113-131.  doi: 10.5565/PUBLMAT6211806.  Google Scholar

[8]

C. A. Buzzi, T. 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

[9]

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

[10]

L. Dieci, T. Eirola and C. Elia, Periodic orbits of planar discontinuous system under discretization, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2743-2762.  doi: 10.3934/dcdsb.2018103.  Google Scholar

[11]

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

[12]

L. Dieci and C. Elia, Piecewise smooth systems near a codimension 2 discontinuity manifold: Can we say what should happen?, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1039-1068.  doi: 10.3934/dcdss.2016041.  Google Scholar

[13]

L. Dieci and C. Elia, Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2935-2950.  doi: 10.3934/dcdsb.2018112.  Google Scholar

[14]

L. Dieci, C. Elia and L. Lopez, A Filippov sliding vector field on an attracting codimension 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

[15]

L. Dieci, C. 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

[16]

L. Dieci, C. Elia and D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3091-3112.  doi: 10.3934/dcdsb.2017165.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov Systems, J. Differential Equations, 250 (2011), 1967-2023.  doi: 10.1016/j.jde.2010.11.016.  Google Scholar

[21]

N. Gugliemi and E. Hairer, Classification of hidden dynamics in discontinuous dynamical systems, SIAM J. Appl. Dyn. Syst., 14 (2015), 1454-1477.  doi: 10.1137/15100326X.  Google Scholar

[22]

N. Gugliemi and E. Hairer, Solutions leaving a codimension-2 sliding, Nolinear. Dyn., 88 (2017), 1427-1439.  doi: 10.1007/s11071-016-3320-1.  Google Scholar

[23]

M. R. Jeffrey, Exit from sliding in piecewise-smooth flows: Deterministic vs. determinacy-breaking, Chaos, 26 (2016), 033108, 19 pp. doi: 10.1063/1.4943386.  Google Scholar

[24]

K. U. Kristiansen and S. J. Hogan, Regularization of two-fold bifurations in planar piecewise smooth systems using blowup, SIAM J. Appl. Dyn. Syst., 14 (2015), 1731-1786.  doi: 10.1137/15M1009731.  Google Scholar

[25]

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

[26]

J. Llibre, P. R. 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

[27]

J. Llibre, P. R. da Silva and M. A. Teixeira, Sliding vector fields via slow-fast systems, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 851-869.  doi: 10.36045/bbms/1228486412.  Google Scholar

[28]

J. Llibre, P. 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

[29]

D. Panazzolo and P. R. da Silva, Regularization of discontinuous foliations: blowing up and sliding conditions via Fenichel theory, J. Differential Equations, 263 (2017), 8362-8390.  doi: 10.1016/j.jde.2017.08.042.  Google Scholar

[30]

D. Pi, Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two, Discrete Contin. Dyn. Syst.Ser. B, 24 (2019), 881-905.  doi: 10.3934/dcdsb.2018211.  Google Scholar

[31]

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

[32]

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

[33]

H. Schiller and M. Arnold, Convergence of continuous approximations for discontinuous ODEs, Appl. Numer. Math., 62 (2012), 1503-1514.  doi: 10.1016/j.apnum.2012.06.021.  Google Scholar

[34]

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

[35]

J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), World Sci. Publ., River Edge, NJ, 1998,207–223.  Google Scholar

[36]

S. Tang, J. Liang, Y. Xiao and R. A. Cheke, 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

[37]

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, Inc., New York, 1987.  Google Scholar

[38]

J. Yang and L. Zhao, Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations, J. Differential Equations, 264 (2018), 5734-5757.  doi: 10.1016/j.jde.2018.01.017.  Google Scholar

[39]

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 1.  Sliding periodic orbit of type Ⅰ
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Figure 2.  Sliding periodic orbit of type Ⅱ
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