Abstract
In this paper, we consider the class of smooth sliding Filippov vector fields in \(\mathbb {R}^3\) on the intersection \(\Sigma \) of two smooth surfaces: \(\Sigma =\Sigma _1\cap \Sigma _2\), where \(\Sigma _i=\{x:\ h_i(x)=0\}\), and \(h_i:\ \mathbb {R}^3\rightarrow \mathbb {R}\), \(i=1,2\), are smooth functions with linearly independent normals. Although, in general, there is no unique Filippov sliding vector field on \(\Sigma \), here we prove that—under natural conditions—all Filippov sliding vector fields are orbitally equivalent to \(\Sigma \). In other words, the aforementioned ambiguity has no meaningful dynamical impact. We also examine the implication of this result in the important case of a periodic orbit a portion of which slides on \(\Sigma \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The simplest case is that of nodal attractivity, when there is sliding motion toward \(\Sigma \), on each of \(\Sigma _{1,2}^\pm \).
References
Aizerman, M.A., Gantmacher, F.R.: On the stability of periodic motion. J. Appl. Math. 22, 1065–1078 (1958)
Alexander, J.C., Seidman, T.: Sliding modes in intersecting switching surfaces, I: blending. Houst. J. Math. 24, 545–569 (1998)
Chicone, C.: Ordinary Differential Equations with Applications. Texts in Applied Mathematics, 2nd edn. Springer, New York (2006)
Dieci, L.: Sliding motion on the intersection of two manifolds: spirally attractive case. Commun. Nonlinear Sci. Numer. Simul. 26, 65–74 (2015)
Dieci, L., Difonzo, F.: A comparison of Filippov sliding vector fields in co-dimension 2. J. Comput. Appl. Math. 262, 161–179. Corrigendum in J. Comput. Appl. Math. 272(2014), 273–273 (2014)
Dieci, L., Difonzo, F.: The moments sliding vector field on the intersection of two manifolds. J. Dyn. Differ. Equ. (2015). doi:10.1007/s10884-015-9439-9
Dieci, L., Elia, C., Lopez, L.: A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis. J. Differ. Equ. 254, 1800–1832 (2013)
Dieci, L., Lopez, L.: Fundamental matrix solutions of Piecewise smooth differential systems. Math. Comput. Simul. 81, 932–953 (2011)
Dieci, L., Lopez, L.: A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side. J. Comput. Appl. Math. 236(16), 3967–3991 (2012)
Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and Its Applications. Kluwer Academic, Dordrecht (1988)
Ivanov, A.P.: The stability of periodic solutions of discontinuous systems that intersect several surfaces of discontinuity. J. Appl. Math. Mech. 62, 677–685 (1998)
Jeffrey, M.R.: Dynamics at a switching intersection: hierarchy, isonomy, and multiple-sliding. SIAM J. Appl. Dyn. Syst. 13(3), 1082–1105 (2014)
Kucucka, P.: Jumps of the fundamental matrix solutions in discontinuous systems and applications. Nonlinear Anal. 66, 2529–2546 (2007)
Llibre, J., Silva, P.R., Teixeira, M.A.: Regularization of discontinuous vector fields on \(R^3\) via singular perturbation. J. Dyn. Differ. Equ. 19, 309–331 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Alan R. Champneys.
This work was begun while the last two authors were visiting the School of Mathematics of Georgia Tech, whose hospitality is gratefully acknowledged. The first author also gratefully acknowledges the support provided by a Tao Aoqing Visiting Professorship at Jilin University, Changchun (China).
Rights and permissions
About this article
Cite this article
Dieci, L., Elia, C. & Lopez, L. 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, 1453–1471 (2015). https://doi.org/10.1007/s00332-015-9265-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-015-9265-6