# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021264
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## Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones

 1 School of Mathematics, Hunan University, Changsha, Hunan 410082, China 2 College of Mathematics and Computer Science, Changsha University, Changsha, Hunan 410022, China 3 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China

* Corresponding author: Lihong Huang

Received  December 2020 Revised  July 2021 Early access November 2021

In this paper, we consider the nonsmooth bifurcation around a class of critical crossing cycles, which are codimension-2 closed orbits composed of tangency singularities and regular orbits, for a two-parameter family of planar piecewise smooth system with two zones. By the construction of suitable displacement function (equivalently, Poincar${\rm\acute{e}}$ map), the stability and the existence of periodic solutions under the variation of the parameters inside this system are characterized. More precisely, we obtain some parameter regions on the existence of crossing cycles and sliding cycles near those loops. As applications, several examples are given to illustrate our main conclusions.

Citation: 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, doi: 10.3934/dcdsb.2021264
##### References:
 [1] K. D. S. Andrade, O. M. L. Gomide and D. D. Novaes, Qualitative analysis of polycycles in Filippov systems, Preprint, arXiv: 1905.11950, 2019. [2] M. D. Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations, International Journal of Bifurcation and Chaos, 11 (2001), 1121-1140.  doi: 10.1142/S0218127401002584. [3] S. Chen and Z. Du, Stability and perturbations of homoclinic loops in a class of piecewise smooth systems, Internat. J. Bifur. Chaos Appl. Sci., 25 (2015), 1550114, 16 pp. doi: 10.1142/S021812741550114X. [4] S. Coombes, R. Thul and K. C. A. Wedgwood, Nonsmooth dynamics in spiking neuron models, Phys. D, 241 (2012), 2042-2057.  doi: 10.1016/j.physd.2011.05.012. [5] F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theoretical Population Biology, 72 (2007), 197-213.  doi: 10.1016/j.tpb.2007.06.003. [6] M. Di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. O. Tost and P. T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50 (2008), 629-701.  doi: 10.1137/050625060. [7] M. Di Bernardo, M. I. Feigin, S. J. Hogan and M. E. Homer, Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems, Chaos Solitons Fractals, 10 (1999), 1881-1908.  doi: 10.1016/S0960-0779(98)00317-8. [8] Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlinear Anal., 69 (2008), 3610-3628.  doi: 10.1016/j.na.2007.09.045. [9] R. D. Euzbio and J. Llibre, On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line, J. Math. Anal. Appl., 424 (2015), 475-486.  doi: 10.1016/j.jmaa.2014.10.077. [10] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. [11] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.  doi: 10.1137/11083928X. [12] E. Freire, E. Ponce and F. Torres, On the critical crossing cycle bifurcation in planar Filippov systems, J. Differential Equations, 259 (2015), 7086-7107.  doi: 10.1016/j.jde.2015.08.013. [13] U. Galvanetto, S. R. Bishop and L. Briseghella, Mechanical stick-slip vibrations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 637-651.  doi: 10.1142/S0218127495000508. [14] 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. [15] S. M. Huan and X. S. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics, Nonlinear Anal., 92 (2013), 82-95.  doi: 10.1016/j.na.2013.06.017. [16] M. Kunze, Non-Smooth Dynamical Systems, Lecture Notes in Mathematics, 1744. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843. [17] Y. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874. [18] S. Lang, Introduction to Differentiable Manifolds, 2$^nd$ edition, Universitext. Springer-Verlag, New York, 2002. [19] T. Li and X. Chen, Degenerate grazing-sliding bifurcations in planar Filippov systems, J. Differential Equations, 269 (2020), 11396-11434.  doi: 10.1016/j.jde.2020.08.037. [20] F. Liang and M. Han, The stability of some kinds of generalized homoclinic loops in planar piecewise smooth systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350027, 15pp. doi: 10.1142/S0218127413500272. [21] F. Liang, M. Han and X. Zhang, Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems, J. Differential Equations, 255 (2013), 4403-4436.  doi: 10.1016/j.jde.2013.08.013. [22] J. Llibre, M. A. Teixeira and J. Torregrosa, Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350066, 10pp. doi: 10.1142/S0218127413500661. [23] D. D. Novaes, M. A. Teixeira and I. O. Zeli, The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems, Nonlinearity, 31 (2018), 2083-2104.  doi: 10.1088/1361-6544/aaaaf7. [24] J. Wang, X. Chen and L. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl., 469 (2019), 405-427.  doi: 10.1016/j.jmaa.2018.09.024. [25] J. Wang, C. Huang and L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. Hybrid Syst., 33 (2019), 162-178.  doi: 10.1016/j.nahs.2019.03.004. [26] Y. Xiong and M. Han, Limit cycle bifurcations near homoclinic and heteroclinic loops via stability-changing of a homoclinic loop, Chaos Solitons Fractals, 78 (2015), 107-117.  doi: 10.1016/j.chaos.2015.07.015.

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##### References:
 [1] K. D. S. Andrade, O. M. L. Gomide and D. D. Novaes, Qualitative analysis of polycycles in Filippov systems, Preprint, arXiv: 1905.11950, 2019. [2] M. D. Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations, International Journal of Bifurcation and Chaos, 11 (2001), 1121-1140.  doi: 10.1142/S0218127401002584. [3] S. Chen and Z. Du, Stability and perturbations of homoclinic loops in a class of piecewise smooth systems, Internat. J. Bifur. Chaos Appl. Sci., 25 (2015), 1550114, 16 pp. doi: 10.1142/S021812741550114X. [4] S. Coombes, R. Thul and K. C. A. Wedgwood, Nonsmooth dynamics in spiking neuron models, Phys. D, 241 (2012), 2042-2057.  doi: 10.1016/j.physd.2011.05.012. [5] F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theoretical Population Biology, 72 (2007), 197-213.  doi: 10.1016/j.tpb.2007.06.003. [6] M. Di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. O. Tost and P. T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50 (2008), 629-701.  doi: 10.1137/050625060. [7] M. Di Bernardo, M. I. Feigin, S. J. Hogan and M. E. Homer, Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems, Chaos Solitons Fractals, 10 (1999), 1881-1908.  doi: 10.1016/S0960-0779(98)00317-8. [8] Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlinear Anal., 69 (2008), 3610-3628.  doi: 10.1016/j.na.2007.09.045. [9] R. D. Euzbio and J. Llibre, On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line, J. Math. Anal. Appl., 424 (2015), 475-486.  doi: 10.1016/j.jmaa.2014.10.077. [10] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. [11] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.  doi: 10.1137/11083928X. [12] E. Freire, E. Ponce and F. Torres, On the critical crossing cycle bifurcation in planar Filippov systems, J. Differential Equations, 259 (2015), 7086-7107.  doi: 10.1016/j.jde.2015.08.013. [13] U. Galvanetto, S. R. Bishop and L. Briseghella, Mechanical stick-slip vibrations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 637-651.  doi: 10.1142/S0218127495000508. [14] 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. [15] S. M. Huan and X. S. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics, Nonlinear Anal., 92 (2013), 82-95.  doi: 10.1016/j.na.2013.06.017. [16] M. Kunze, Non-Smooth Dynamical Systems, Lecture Notes in Mathematics, 1744. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843. [17] Y. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874. [18] S. Lang, Introduction to Differentiable Manifolds, 2$^nd$ edition, Universitext. Springer-Verlag, New York, 2002. [19] T. Li and X. Chen, Degenerate grazing-sliding bifurcations in planar Filippov systems, J. Differential Equations, 269 (2020), 11396-11434.  doi: 10.1016/j.jde.2020.08.037. [20] F. Liang and M. Han, The stability of some kinds of generalized homoclinic loops in planar piecewise smooth systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350027, 15pp. doi: 10.1142/S0218127413500272. [21] F. Liang, M. Han and X. Zhang, Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems, J. Differential Equations, 255 (2013), 4403-4436.  doi: 10.1016/j.jde.2013.08.013. [22] J. Llibre, M. A. Teixeira and J. Torregrosa, Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350066, 10pp. doi: 10.1142/S0218127413500661. [23] D. D. Novaes, M. A. Teixeira and I. O. Zeli, The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems, Nonlinearity, 31 (2018), 2083-2104.  doi: 10.1088/1361-6544/aaaaf7. [24] J. Wang, X. Chen and L. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl., 469 (2019), 405-427.  doi: 10.1016/j.jmaa.2018.09.024. [25] J. Wang, C. Huang and L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. Hybrid Syst., 33 (2019), 162-178.  doi: 10.1016/j.nahs.2019.03.004. [26] Y. Xiong and M. Han, Limit cycle bifurcations near homoclinic and heteroclinic loops via stability-changing of a homoclinic loop, Chaos Solitons Fractals, 78 (2015), 107-117.  doi: 10.1016/j.chaos.2015.07.015.
Two possible situations of counterclockwise periodic orbit depending on the stability of two foci. (a) unstable-stable; (b) unstable-unstable
The critical crossing cycles. (a) Type A; (b) Type B
The half-return map $P^+_{i}(\xi;\alpha), i = A,B$, of the system $(4)$. The dashed lines correspond to the case $\alpha = 0$
The half-return map $P^-_{i}(\xi;\beta)$, $i = A,B$, of the system $(4)$. The dashed lines correspond to the case $\beta = 0$
On the switching line $\Sigma$, the full lines denote the sliding or escaping regions and the dashed lines denote the crossing regions
Space of parameters $(\beta, \alpha)$ and bifurcation curves. The topological structures of trajectories in regions $A_{ij},\; i = 0,1,j = 0,...,9,$ are illustrated in Figure 7
(colour online) The topological structures of trajectories in Figure 6. The purple and red closed curves denote the stable and (outer) unstable (critical) crossing cycles, respectively. The sliding cycle are marked by blue and the separatrix connections are marked by green. The black closed orbit in $A_{02}$ is a semi-stable crossing cycle
Space of parameters $(\beta, \alpha)$ and bifurcation curves. The topological structures of trajectories in regions $B_{ij},\; i = 0,1,j = 0,...,9,$ are illustrated in Figure 9
(colour online) The topological structures of trajectories in Figure 8. The purple and red closed curves denote the stable and (outer) unstable (critical) crossing cycles, respectively. The sliding cycle are marked by blue and the separatrix connections are marked by green. The black closed orbit in $B_{08}$ is a semi-stable crossing cycle
Space of parameters $(\beta, \alpha)$ and bifurcation curves for $m_0<0$. The topological structures of trajectories in regions $C_{ij},\; i = 0,1,j = 0,...,9,$ are illustrated in Figure 11
(colour online) The topological structures of trajectories in Figure 8. The red closed curves denote the (outer) stable (critical) crossing cycles. The sliding cycle are marked by blue and the separatrix connections are marked by green
A critical crossing cycle of Type A in System $(42)$ when $(\alpha,\beta) = (0,0)$
(a) no cycles at $(\alpha,\beta) = (0.01,0.01)\in A_{01}$; (b) a semi-stable crossing cycle at $(\alpha,\beta) = (0.0094,0.01)\in A_{02}$; (c) two crossing cycles: one unstable and one stable, at $(\alpha,\beta) = (0.007,0.01)\in A_{03}$; (d) a crossing cycle and a critical crossing cycle at $(\alpha,\beta) = (0.0053,0.01)\in A_{04}$; (e) a crossing cycle and a sliding cycle $(\alpha,\beta) = (0.005,0.01)\in A_{05}$; (f) a crossing cycle and a homoclinic sliding cycle connecting the pseudo-saddle $p_e\approx(0.00487,0)$ to itself at $(\alpha,\beta) = (0.0047,0.01)\in A_{06}$; (g) a crossing cycle at $(\alpha,\beta) = (0.0043,0.01)\in A_{07}$; (h) a crossing cycle and a heteroclinic connection at $(\alpha,\beta) = (0.004,0.01)\in A_{08}$; (i) a crossing cycle at $(\alpha,\beta) = (0,0.01)\in A_{09}$
(a) no cycles at $(\alpha,\beta) = (0,-0.01)\in A_{13}$; (b) a heteroclinic connection to two visible fold $(0,0)$ and $(-0.01,0)$ at $(\alpha,\beta) = (-0.00581,-0.01)\in A_{14}$; (c) a sliding heteroclonic connection at $(\alpha,\beta) = (-0.00607,-0.01)\in A_{15}$; (d) a sliding homoclinic cycle connecting the pseudo-saddle $p_s\approx(-0.0053,0)$ to itself at $(\alpha,\beta) = (-0.00714,-0.01)\in A_{16}$; (e) a sliding cycle at $(\alpha,\beta) = (-0.00716,-0.01)\in A_{17}$; (f) a critical crossing cycle at $(\alpha,\beta) = (-0.0073,-0.01)\in A_{18}$; (g) a crossing cycle at $(\alpha,\beta) = (-0.008,-0.01)\in A_{19}$
A critical crossing cycle of Type B in System $(44)$ when $(\alpha,\beta) = (0,0)$
A sliding critical cycle bifurcation and a crossing-sliding bifurcation. (a) a sliding cycle with two segments at $(\alpha,\beta) = (0.045,-0.1)\in C_{01}$; (b) a sliding cycle with one segment at $(\alpha,\beta) = (0.03,-0.1)\in C_{02}$; (c) a sliding cycle with one segment at $(\alpha,\beta) = (-0.0035,-0.1)\in C_{03}$; (d) a critical crossing cycle at $(\alpha,\beta) = (-0.0136,-0.1)\in C_{04}$; (e) a crossing cycle at $(\alpha,\beta) = (-0.0141,-0.1)\in C_{05}$
A crossing-sliding bifurcation and a sliding heteroclinic bifurcation. (a) a sliding cycle at $(\alpha,\beta) = (0,0.1)\in C_{13}$; (b) a critical crossing cycle at $(\alpha,\beta) = (-0.015,0.1)\in C_{12}$; (c) a crossing cycle at $(\alpha,\beta) = (-0.025,0.1)\in C_{11}$; (d) a crossing cycle at $(\alpha,\beta) = (-0.03,0.1)\in C_{10}$; (e) a crossing cycle at $(\alpha,\beta) = (-0.036,0.1)\in C_{09}$
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