A note on attractivity for the intersection of two discontinuity manifolds. (English) Zbl 1469.34028
Summary: In piecewise smooth dynamical systems, a co-dimension 2 discontinuity manifold can be attractive either through partial sliding or by spiraling. In this work we prove that both attractivity regimes can be analyzed by means of the moments solution, a spiraling bifurcation parameter and a novel attractivity parameter, which changes sign when attractivity switches from sliding to spiraling attractivity or vice-versa. We also study what happens at what we call attractivity transition points, showing that the spiraling bifurcation parameter is always zero at those points.
MSC:
| 34A36 | Discontinuous ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
Keywords:
piecewise smooth systems; sliding motion; co-dimension 2; discontinuity manifold; attractivityReferences:
| [1] | V. Acary, B. Brogliato,Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Eletronics, Springer-Verlag, 2008. · Zbl 1173.74001 |
| [2] | M. Berardi, M. D’Abbicco,A critical case for the spiral stability for2×2discontinuous systems and an application to recursive neural networks, Mediterr. J. Math.13(2016) · Zbl 1366.34023 |
| [3] | M. di Bernardo, C. Budd, A. Champneys, P. Kowalczyk,Piecewise-smooth Dynamical Systems. Theory and Applications, Applied Mathematical Sciences, vol. 163, · Zbl 1146.37003 |
| [4] | R. Brualdi, B. Shader,Matrices of Sign-Solvable Linear Systems, Cambridge Tracts in Mathematics, vol. 116, Cambridge University Press, Cambridge, 1995. · Zbl 0833.15002 |
| [5] | T. Carvalho, D. Duarte Novaes, L. Gonçalves,Sliding Shilnikov connection in Filippov-type predator-prey model, Nonlinear Dynamics100(2020) 3, 2973-2987. · Zbl 1516.92079 |
| [6] | M. D’Abbicco, N.D. Buono, P. Gena, M. Berardi, G. Calamita, L. Lopez,A model for the hepatic glucose metabolism based on Hill and step functions, J. Comput. Appl. Math. 292(2016), 746-759. · Zbl 1326.92020 |
| [7] | L. Dieci,Sliding motion on the intersection of two manifolds: Spirally attractive case, Commun. Nonlinear Sci. Numer. Simul.26(2015) 1, 65-74. · Zbl 1463.34050 |
| [8] | L. Dieci, F. Difonzo,A comparison of Filippov sliding vector fields in codimension 2, J. Comput. Appl. Math.262(2014), 161-179. · Zbl 1302.34018 |
| [9] | L. Dieci, F. Difonzo,Minimum variation solutions for sliding vector fields on the intersection of two surfaces inR3, J. Comput. Appl. Math.292(2016), 732-745. · Zbl 1335.34035 |
| [10] | L. Dieci, F. Difonzo,The moments sliding vector field on the intersection of two manifolds, J. Dynam. Differential Equations29(2017) 1, 169-201. · Zbl 1366.34024 |
| [11] | L. Dieci, F. Difonzo,On the inverse of some sign matrices and on the moments sliding vector field on the intersection of several manifolds: Nodally attractive case, J. Dynam. · Zbl 1383.34024 |
| [12] | L. Dieci, C. Elia, L. Lopez,A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis, Journal of Differential · Zbl 1275.34018 |
| [13] | L. Dieci, L. Lopez,Fundamental matrix solutions of piecewise smooth differential systems, Math. Comput. Simulation81(2011), 932-953. · Zbl 1215.65121 |
| [14] | L. Dieci, L. Lopez,A survey of numerical methods for ivps of odes with discontinuous right-hand side, J. Comput. Appl. Math.236(2012) 16, 3967-3991. · Zbl 1246.65111 |
| [15] | A. Filippov,Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. · Zbl 0664.34001 |
| [16] | U. Galvanetto,Discontinuous bifurcations in stick-slip mechanical systems, Proceedings of the ASME Design Engineering Technical Conference6(2001), 1315-1322. |
| [17] | H. Hosham,Bifurcation of limit cycles in piecewise-smooth systems with intersecting discontinuity surfaces, Nonlinear Dynamics99(2019), 2049-2063. · Zbl 1516.34034 |
| [18] | M. Jeffrey,Dynamics at a switching intersection: Hierarchy, isonomy, and multiple sliding, SIAM J. Appl. Dyn. Syst.13(2014), 1082-1105. · Zbl 1310.34018 |
| [19] | B. Kacewicz, P. Przybyłowicz,Optimal solution of a class of non-autonomous initial-value problems with unknown singularities, J. Comput. Appl. Math.261(2014), 364-377. · Zbl 1278.65100 |
| [20] | L. Lopez, N. Del Buono, C. Elia,On the equivalence between the sigmoidal approach and Utkin’s approach for piecewise-linear models of gene regulatory networks, SIAM J. Appl. · Zbl 1321.37079 |
| [21] | P. Piiroinen, Y. Kuznetsov,An event-driven method to simulate Filippov systems with accurate computing of sliding motions, ACM Trans. Math. Softw.34(2008) 3, Article 13. · Zbl 1190.65109 |
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