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Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension
Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two
Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Fujian 362021, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic Publishers Group, Dordrecht, 1988.
doi: 10.1007/978-94-015-7793-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[37] |
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, Inc., New York, 1987. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic Publishers Group, Dordrecht, 1988.
doi: 10.1007/978-94-015-7793-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[37] |
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, Inc., New York, 1987. |
[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. |
[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. |


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