American Institute of Mathematical Sciences

August  2016, 9(4): 1039-1068. doi: 10.3934/dcdss.2016041

Piecewise smooth systems near a co-dimension 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, I-70125, Bari, Italy

Received  August 2015 Revised  March 2016 Published  August 2016

In this work we attempt to understand what behavior one should expect of a solution trajectory near $\Sigma$ when $\Sigma$ is attractive, what to expect when $\Sigma$ ceases to be attractive (at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature, whereby the piecewise smooth system is replaced by a smooth differential system.
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.
Citation: Luca Dieci, Cinzia Elia. Piecewise smooth systems near a co-dimension 2 discontinuity manifold: Can one say what should happen?. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1039-1068. doi: 10.3934/dcdss.2016041
References:
 [1] J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, I: Blending., Houston J. Math., 24 (1998), 545. Google Scholar [2] J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, II: Hysteresis., Houston J. Math., 25 (1999), 185. Google Scholar [3] Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits,, Mathematics Bohemica, 127 (2002), 139. Google Scholar [4] J. Cortes, Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability,, IEEE Control Systems Magazine, 28 (2008), 36. doi: 10.1109/MCS.2008.919306. Google Scholar [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), 1270. doi: 10.1137/130950483. Google Scholar [6] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications., Applied Mathematical Sciences 163. Springer-Verlag, (2008). Google Scholar [7] L. Dieci, Sliding motion on the intersection of two manifolds: Spirally attractive case,, Communications in Nonlinear Science and Numerical Simulation, 26 (2015), 65. doi: 10.1016/j.cnsns.2015.02.002. Google Scholar [8] L. Dieci and F. Difonzo, A Comparison of Filippov sliding vector fields in co-dimension $2$,, Journal of Computational and Applied Mathematics, 262 (2014), 161. doi: 10.1016/j.cam.2013.10.055. Google Scholar [9] L. Dieci and F. Difonzo, The Moments sliding vector field on the intersection of two manifolds,, Journal of Dynamics and Differential Equations, (2015), 1. doi: 10.1007/s10884-015-9439-9. Google Scholar [10] L. Dieci, C. Elia and L. Lopez, A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis,, J. Differential Equations, 254 (2013), 1800. doi: 10.1016/j.jde.2012.11.007. Google Scholar [11] L. Dieci, C. Elia and L. Lopez, Sharp sufficient attractivity conditions for sliding on a co-dimension 2 discontinuity surface,, Mathematics and Computers in Simulations, 110 (2015), 3. doi: 10.1016/j.matcom.2013.12.005. Google Scholar [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), 1453. doi: 10.1007/s00332-015-9265-6. Google Scholar [13] L. Dieci and N. Guglielmi, Regularizing piecewise smooth differential systems: Co-dimension 2 discontinuity surface,, J. Dynamics and Differential Equations, 25 (2013), 71. doi: 10.1007/s10884-013-9287-4. Google Scholar [14] A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey,, SIAM REVIEW, 34 (1992), 263. doi: 10.1137/1034050. Google Scholar [15] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides,, Mathematics and Its Applications, (1988). doi: 10.1007/978-94-015-7793-9. Google Scholar [16] N. Guglielmi and E. Hairer, Classification of hidden dynamics in discontinuous dynamical systems,, SIADS, 14 (2015), 1454. doi: 10.1137/15100326X. Google Scholar [17] M. Jeffrey, Dynamics at a switching intersection: Hierarchy, isonomy, and multiple sliding,, SIAM J. Applied Dyn. Systems, 13 (2014), 1082. doi: 10.1137/13093368X. Google Scholar [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), 309. doi: 10.1007/s10884-006-9057-7. Google Scholar [19] A. Machina, R. Edwards and P. van den Driessche, Singular dynamics in gene network models,, SIAM J. Appl. Dyn. Syst., 12 (2013), 95. doi: 10.1137/120872747. Google Scholar [20] E. Plahte and S. Kjóglum, Analysis and generic properties of gene regulatory networks with graded response functions,, Physica D, 201 (2005), 150. doi: 10.1016/j.physd.2004.11.014. Google Scholar [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), 511. doi: 10.1016/j.jtbi.2009.07.040. Google Scholar [22] T. Seidman, Some limit results for relays,, Proc.s of World Congress of Nonlinear Analysts, 1 (1996), 787. Google Scholar [23] T. Seidman, The residue of model reduction. The residue of model reduction,, In Hybrid Systems III. Verification and Control, (1996), 201. Google Scholar [24] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector field,, In International Conference on Differential Equations, (1998), 207. Google Scholar [25] V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems., MIR Publisher, (1978). Google Scholar [26] V. I. Utkin, Sliding Mode in Control and Optimization,, Springer, (1992). doi: 10.1007/978-3-642-84379-2. Google Scholar

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References:
 [1] J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, I: Blending., Houston J. Math., 24 (1998), 545. Google Scholar [2] J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, II: Hysteresis., Houston J. Math., 25 (1999), 185. Google Scholar [3] Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits,, Mathematics Bohemica, 127 (2002), 139. Google Scholar [4] J. Cortes, Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability,, IEEE Control Systems Magazine, 28 (2008), 36. doi: 10.1109/MCS.2008.919306. Google Scholar [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), 1270. doi: 10.1137/130950483. Google Scholar [6] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications., Applied Mathematical Sciences 163. Springer-Verlag, (2008). Google Scholar [7] L. Dieci, Sliding motion on the intersection of two manifolds: Spirally attractive case,, Communications in Nonlinear Science and Numerical Simulation, 26 (2015), 65. doi: 10.1016/j.cnsns.2015.02.002. Google Scholar [8] L. Dieci and F. Difonzo, A Comparison of Filippov sliding vector fields in co-dimension $2$,, Journal of Computational and Applied Mathematics, 262 (2014), 161. doi: 10.1016/j.cam.2013.10.055. Google Scholar [9] L. Dieci and F. Difonzo, The Moments sliding vector field on the intersection of two manifolds,, Journal of Dynamics and Differential Equations, (2015), 1. doi: 10.1007/s10884-015-9439-9. Google Scholar [10] L. Dieci, C. Elia and L. Lopez, A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis,, J. Differential Equations, 254 (2013), 1800. doi: 10.1016/j.jde.2012.11.007. Google Scholar [11] L. Dieci, C. Elia and L. Lopez, Sharp sufficient attractivity conditions for sliding on a co-dimension 2 discontinuity surface,, Mathematics and Computers in Simulations, 110 (2015), 3. doi: 10.1016/j.matcom.2013.12.005. Google Scholar [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), 1453. doi: 10.1007/s00332-015-9265-6. Google Scholar [13] L. Dieci and N. Guglielmi, Regularizing piecewise smooth differential systems: Co-dimension 2 discontinuity surface,, J. Dynamics and Differential Equations, 25 (2013), 71. doi: 10.1007/s10884-013-9287-4. Google Scholar [14] A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey,, SIAM REVIEW, 34 (1992), 263. doi: 10.1137/1034050. Google Scholar [15] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides,, Mathematics and Its Applications, (1988). doi: 10.1007/978-94-015-7793-9. Google Scholar [16] N. Guglielmi and E. Hairer, Classification of hidden dynamics in discontinuous dynamical systems,, SIADS, 14 (2015), 1454. doi: 10.1137/15100326X. Google Scholar [17] M. Jeffrey, Dynamics at a switching intersection: Hierarchy, isonomy, and multiple sliding,, SIAM J. Applied Dyn. Systems, 13 (2014), 1082. doi: 10.1137/13093368X. Google Scholar [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), 309. doi: 10.1007/s10884-006-9057-7. Google Scholar [19] A. Machina, R. Edwards and P. van den Driessche, Singular dynamics in gene network models,, SIAM J. Appl. Dyn. Syst., 12 (2013), 95. doi: 10.1137/120872747. Google Scholar [20] E. Plahte and S. Kjóglum, Analysis and generic properties of gene regulatory networks with graded response functions,, Physica D, 201 (2005), 150. doi: 10.1016/j.physd.2004.11.014. Google Scholar [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), 511. doi: 10.1016/j.jtbi.2009.07.040. Google Scholar [22] T. Seidman, Some limit results for relays,, Proc.s of World Congress of Nonlinear Analysts, 1 (1996), 787. Google Scholar [23] T. Seidman, The residue of model reduction. The residue of model reduction,, In Hybrid Systems III. Verification and Control, (1996), 201. Google Scholar [24] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector field,, In International Conference on Differential Equations, (1998), 207. Google Scholar [25] V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems., MIR Publisher, (1978). Google Scholar [26] V. I. Utkin, Sliding Mode in Control and Optimization,, Springer, (1992). doi: 10.1007/978-3-642-84379-2. Google Scholar
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