# American Institute of Mathematical Sciences

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

 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.

Citation: Dingheng Pi. Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two. Discrete and 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. [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. [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.
Sliding periodic orbit of type Ⅰ
Sliding periodic orbit of type Ⅱ
 [1] 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 [2] Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5581-5599. doi: 10.3934/dcdsb.2020368 [3] Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123 [4] Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 [5] Yuan Chang, Yuzhen Bai. Limit cycle bifurcations by perturbing piecewise Hamiltonian systems with a nonregular switching line via multiple parameters. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022090 [6] Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803 [7] Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021264 [8] Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114 [9] Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133 [10] Luca Dieci, Cinzia Elia. Piecewise smooth systems near a co-dimension 2 discontinuity manifold: Can one say what should happen?. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1039-1068. doi: 10.3934/dcdss.2016041 [11] Wenye Liu, Maoan Han. Limit cycle bifurcations of near-Hamiltonian systems with multiple switching curves and applications. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022053 [12] Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172 [13] Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447 [14] Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2825-2851. doi: 10.3934/dcds.2012.32.2825 [15] Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control and Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359 [16] Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381 [17] Simone Creo, Maria Rosaria Lancia, Alexander Nazarov, Paola Vernole. On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 57-64. doi: 10.3934/dcdss.2019004 [18] Yanli Han, Yan Gao. Determining the viability for hybrid control systems on a region with piecewise smooth boundary. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 1-9. doi: 10.3934/naco.2015.5.1 [19] Jianfeng Lv, Yan Gao, Na Zhao. The viability of switched nonlinear systems with piecewise smooth Lyapunov functions. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1825-1843. doi: 10.3934/jimo.2020048 [20] Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150

2020 Impact Factor: 1.327

## Metrics

• PDF downloads (296)
• HTML views (384)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]