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Piecewise smooth systems near a codimension 2 discontinuity manifold: Can one say what should happen?
1.  School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States 
2.  Dipartimento di Matematica, University of Bari, I70125, Bari, Italy 
Through analysis and experiments in $\mathbb{R}^3$ and $\mathbb{R}^4$, we will confirm some known facts and provide some important insight: (i) when $\Sigma$ is attractive, a solution trajectory remains near $\Sigma$, viz. sliding on $\Sigma$ is an appropriate idealization (though one cannot a priori decide which sliding vector field should be selected); (ii) when $\Sigma$ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of $\Sigma$; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near $\Sigma$ while $\Sigma$ is attractive, and so that it will be leaving (a neighborhood of) $\Sigma$ when $\Sigma$ looses attractivity.
We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near $\Sigma$ should have been taking place.
References:
[1] 
J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, I: Blending. Houston J. Math., 24 (1998), 545569. 
[2] 
J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, II: Hysteresis. Houston J. Math., 25 (1999), 185211. 
[3] 
Z. Artstein, On singularly perturbed ordinary differential equations with measurevalued limits, Mathematics Bohemica, 127 (2002), 139152. 
[4] 
J. Cortes, Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Systems Magazine, 28 (2008), 3673. doi: 10.1109/MCS.2008.919306. 
[5] 
N. Del Buono, C. Elia and L. Lopez, On the equivalence between the sigmoidal approach and Utkin's approach for models of gene regulatory networks, SIAM J. Applied Dynamical Systems, 13 (2014), 12701292. doi: 10.1137/130950483. 
[6] 
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewisesmooth Dynamical Systems. Theory and Applications. Applied Mathematical Sciences 163. SpringerVerlag, Berlin, 2008. 
[7] 
L. Dieci, Sliding motion on the intersection of two manifolds: Spirally attractive case, Communications in Nonlinear Science and Numerical Simulation, 26 (2015), 6574. doi: 10.1016/j.cnsns.2015.02.002. 
[8] 
L. Dieci and F. Difonzo, A Comparison of Filippov sliding vector fields in codimension $2$, Journal of Computational and Applied Mathematics, 262 (2014), 161179. Corrigendum in Journal of Computational and Applied Mathematics, 272 (2014), 273273. doi: 10.1016/j.cam.2013.10.055. 
[9] 
L. Dieci and F. Difonzo, The Moments sliding vector field on the intersection of two manifolds, Journal of Dynamics and Differential Equations, (2015), 133. doi: 10.1007/s1088401594399. 
[10] 
L. Dieci, C. Elia and L. Lopez, A Filippov sliding vector field on an attracting codimension 2 discontinuity surface, and a limited lossofattractivity analysis, J. Differential Equations, 254 (2013), 18001832. doi: 10.1016/j.jde.2012.11.007. 
[11] 
L. Dieci, C. Elia and L. Lopez, Sharp sufficient attractivity conditions for sliding on a codimension 2 discontinuity surface, Mathematics and Computers in Simulations, 110 (2015), 314. doi: 10.1016/j.matcom.2013.12.005. 
[12] 
L. Dieci, C. Elia and L. Lopez, Uniqueness of Filippov sliding vector field on the intersection of two surfaces in $\mathbbR^3$ and implications for stability of periodic orbits, J. Nonlin. Science, 25 (2015), 14531471. doi: 10.1007/s0033201592656. 
[13] 
L. Dieci and N. Guglielmi, Regularizing piecewise smooth differential systems: Codimension 2 discontinuity surface, J. Dynamics and Differential Equations, 25 (2013), 7194. doi: 10.1007/s1088401392874. 
[14] 
A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM REVIEW, 34 (1992), 263294. doi: 10.1137/1034050. 
[15] 
A. F. Filippov, Differential Equations with Discontinuous RightHand Sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/9789401577939. 
[16] 
N. Guglielmi and E. Hairer, Classification of hidden dynamics in discontinuous dynamical systems, SIADS, 14 (2015), 14541477. doi: 10.1137/15100326X. 
[17] 
M. Jeffrey, Dynamics at a switching intersection: Hierarchy, isonomy, and multiple sliding, SIAM J. Applied Dyn. Systems, 13 (2014), 10821105. doi: 10.1137/13093368X. 
[18] 
J. Llibre, P. R. Silva and M. A. Teixeira, Regularization of discontinuous vector fields on $\mathbbR^3$ via singular perturbation, J. Dynam. Differential Equations, 19 (2007), 309331. doi: 10.1007/s1088400690577. 
[19] 
A. Machina, R. Edwards and P. van den Driessche, Singular dynamics in gene network models, SIAM J. Appl. Dyn. Syst., 12 (2013), 95125. doi: 10.1137/120872747. 
[20] 
E. Plahte and S. Kjóglum, Analysis and generic properties of gene regulatory networks with graded response functions, Physica D, 201 (2005), 150176. doi: 10.1016/j.physd.2004.11.014. 
[21] 
A. Polynikis, S. J. Hogan and M. di Bernardo, Comparing different ODE modelling approaches for gene regulatory networks, Journal of Theoretical Biology, 261 (2009), 511530. doi: 10.1016/j.jtbi.2009.07.040. 
[22] 
T. Seidman, Some limit results for relays, Proc.s of World Congress of Nonlinear Analysts, Ed. V. Lakshmikantham, De Gruyter publisher, 1 (1996), 787796. 
[23] 
T. Seidman, The residue of model reduction. The residue of model reduction, In Hybrid Systems III. Verification and Control, Lecture Notes in Comp. Sci. 1066; R. Alur, T.A. Henzinger, E.D. Sontag, eds. SpringerVerlag, Berlin, (1996), 201207. 
[24] 
J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector field, In International Conference on Differential Equations, (1998), 207223. 
[25] 
V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems. MIR Publisher, Moskow, 1978. 
[26] 
V. I. Utkin, Sliding Mode in Control and Optimization, Springer, Berlin, 1992. doi: 10.1007/9783642843792. 
show all references
References:
[1] 
J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, I: Blending. Houston J. Math., 24 (1998), 545569. 
[2] 
J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, II: Hysteresis. Houston J. Math., 25 (1999), 185211. 
[3] 
Z. Artstein, On singularly perturbed ordinary differential equations with measurevalued limits, Mathematics Bohemica, 127 (2002), 139152. 
[4] 
J. Cortes, Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability, IEEE Control Systems Magazine, 28 (2008), 3673. doi: 10.1109/MCS.2008.919306. 
[5] 
N. Del Buono, C. Elia and L. Lopez, On the equivalence between the sigmoidal approach and Utkin's approach for models of gene regulatory networks, SIAM J. Applied Dynamical Systems, 13 (2014), 12701292. doi: 10.1137/130950483. 
[6] 
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewisesmooth Dynamical Systems. Theory and Applications. Applied Mathematical Sciences 163. SpringerVerlag, Berlin, 2008. 
[7] 
L. Dieci, Sliding motion on the intersection of two manifolds: Spirally attractive case, Communications in Nonlinear Science and Numerical Simulation, 26 (2015), 6574. doi: 10.1016/j.cnsns.2015.02.002. 
[8] 
L. Dieci and F. Difonzo, A Comparison of Filippov sliding vector fields in codimension $2$, Journal of Computational and Applied Mathematics, 262 (2014), 161179. Corrigendum in Journal of Computational and Applied Mathematics, 272 (2014), 273273. doi: 10.1016/j.cam.2013.10.055. 
[9] 
L. Dieci and F. Difonzo, The Moments sliding vector field on the intersection of two manifolds, Journal of Dynamics and Differential Equations, (2015), 133. doi: 10.1007/s1088401594399. 
[10] 
L. Dieci, C. Elia and L. Lopez, A Filippov sliding vector field on an attracting codimension 2 discontinuity surface, and a limited lossofattractivity analysis, J. Differential Equations, 254 (2013), 18001832. doi: 10.1016/j.jde.2012.11.007. 
[11] 
L. Dieci, C. Elia and L. Lopez, Sharp sufficient attractivity conditions for sliding on a codimension 2 discontinuity surface, Mathematics and Computers in Simulations, 110 (2015), 314. doi: 10.1016/j.matcom.2013.12.005. 
[12] 
L. Dieci, C. Elia and L. Lopez, Uniqueness of Filippov sliding vector field on the intersection of two surfaces in $\mathbbR^3$ and implications for stability of periodic orbits, J. Nonlin. Science, 25 (2015), 14531471. doi: 10.1007/s0033201592656. 
[13] 
L. Dieci and N. Guglielmi, Regularizing piecewise smooth differential systems: Codimension 2 discontinuity surface, J. Dynamics and Differential Equations, 25 (2013), 7194. doi: 10.1007/s1088401392874. 
[14] 
A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM REVIEW, 34 (1992), 263294. doi: 10.1137/1034050. 
[15] 
A. F. Filippov, Differential Equations with Discontinuous RightHand Sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1988. doi: 10.1007/9789401577939. 
[16] 
N. Guglielmi and E. Hairer, Classification of hidden dynamics in discontinuous dynamical systems, SIADS, 14 (2015), 14541477. doi: 10.1137/15100326X. 
[17] 
M. Jeffrey, Dynamics at a switching intersection: Hierarchy, isonomy, and multiple sliding, SIAM J. Applied Dyn. Systems, 13 (2014), 10821105. doi: 10.1137/13093368X. 
[18] 
J. Llibre, P. R. Silva and M. A. Teixeira, Regularization of discontinuous vector fields on $\mathbbR^3$ via singular perturbation, J. Dynam. Differential Equations, 19 (2007), 309331. doi: 10.1007/s1088400690577. 
[19] 
A. Machina, R. Edwards and P. van den Driessche, Singular dynamics in gene network models, SIAM J. Appl. Dyn. Syst., 12 (2013), 95125. doi: 10.1137/120872747. 
[20] 
E. Plahte and S. Kjóglum, Analysis and generic properties of gene regulatory networks with graded response functions, Physica D, 201 (2005), 150176. doi: 10.1016/j.physd.2004.11.014. 
[21] 
A. Polynikis, S. J. Hogan and M. di Bernardo, Comparing different ODE modelling approaches for gene regulatory networks, Journal of Theoretical Biology, 261 (2009), 511530. doi: 10.1016/j.jtbi.2009.07.040. 
[22] 
T. Seidman, Some limit results for relays, Proc.s of World Congress of Nonlinear Analysts, Ed. V. Lakshmikantham, De Gruyter publisher, 1 (1996), 787796. 
[23] 
T. Seidman, The residue of model reduction. The residue of model reduction, In Hybrid Systems III. Verification and Control, Lecture Notes in Comp. Sci. 1066; R. Alur, T.A. Henzinger, E.D. Sontag, eds. SpringerVerlag, Berlin, (1996), 201207. 
[24] 
J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector field, In International Conference on Differential Equations, (1998), 207223. 
[25] 
V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems. MIR Publisher, Moskow, 1978. 
[26] 
V. I. Utkin, Sliding Mode in Control and Optimization, Springer, Berlin, 1992. doi: 10.1007/9783642843792. 
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