American Institute of Mathematical Sciences

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September  2018, 23(7): 2911-2934. doi: 10.3934/dcdsb.2018166

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

Alessandro Colombo 1, , Nicoletta Del Buono 2, , Luciano Lopez 2,, and  Alessandro Pugliese 2,

1. 

Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy
2. 

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

* Corresponding author: Luciano Lopez

Received  June 2017 Revised  February 2018 Published  June 2018

Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.

Keywords: Filippov systems, crossing/sliding regions, continuation methods, sets of attraction of crossing/sliding regions, reconstruction of surfaces.
Mathematics Subject Classification: Primary: 65L05; Secondary: 34L99.
Citation: Alessandro Colombo, Nicoletta Del Buono, Luciano Lopez, Alessandro Pugliese. Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2911-2934. doi: 10.3934/dcdsb.2018166
References:
[1]

V. Acary and B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, Berlin, 2008.Google Scholar

[2]

A. Agosti, L. Formaggia and A. Scotti, Analysis of a model for precipitation and dissolution coupled with a Darcy flux, Journal of Mathematical Analysis and Applications, 431 (2015), 752-781. doi: 10.1016/j.jmaa.2015.06.003. Google Scholar

[3]

A. Agosti, B. Giovanardi, L. Formaggia and A. Scotti, Numerical simulation of geochemical compaction with discontinuous reactions, in Coupled Problems 2015 - Proceedings of the 6th International Conference on Coupled Problems in Science and Engineering, 2015, 300-311.Google Scholar

[4]

A. Agosti, B. Giovanardi, L. Formaggia and A. Scotti, A numerical procedure for geochemical compaction in the presence of discontinuous reaction, Advances in Water Resources, 94 (2016), 332-344. doi: 10.1016/j.advwatres.2016.06.001. Google Scholar

[5]

I. Arango and J. Taborda, Numerical analysis of sliding dynamics in three-dimensional Filippov systems using SPTI method, International Journal of Mathematical Models and Method in Applied Sciences, 2 (2008), 342-354. Google Scholar

[6]

I. Arango and J. Taborda, Integration-free analysis of nonsmooth local dynamics in planar Filippov systems, International Journal of Bifurcation and Chaos, 19 (2009), 947-975. doi: 10.1142/S0218127409023391. Google Scholar

[7]

I. Arango and J. Taborda, Topological classification of limit cycles of piecewise smooth dynamical systems and its associated non-standard bifurcations, Entropy, 16 (2014), 1949-1968. doi: 10.3390/e16041949. Google Scholar

[8]

M. Berardi and L. Lopez, On the continuous extension of Adams - Bashforth methods and the event location in discontinuous ODEs, Applied Mathematics Letters, 25 (2012), 995-999. doi: 10.1016/j.aml.2011.11.014. Google Scholar

[9] M.-D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. Google Scholar
[10]

M. Calvo, J. Montijano and L. Rández, On the solution of discontinuous IVPs by adaptive Runge-Kutta codes, Numerical Algorithms, 33 (2003), 163-182. doi: 10.1023/A:1025507920426. Google Scholar

[11]

M. Calvo, J. I. Montijano and L. Rández, Algorithm 968: Disode45: A matlab runge-kutta solver for piecewise smooth ivps of filippov type, ACM Trans. Math. Softw., 43 (2017), Art.25, 14 pp. doi: 10.1145/2907054. Google Scholar

[12]

R. Casey, H. deJong and J. Gouze, Piecewise-linear models of genetics regulatory networks: Equilibria and their stability, Journal Mathematical Biology, 52 (2006), 27-56. doi: 10.1007/s00285-005-0338-2. Google Scholar

[13]

R. Cavoretto, A. DeRossi, E. Perracchione and E. Venturino, Robust approximation algorithms for the detection of attraction basins in dynamical systems, J. Sci. Comput., 68 (2016), 395-415. doi: 10.1007/s10915-015-0143-z. Google Scholar

[14]

A. Colombo and U. Galvanetto, Stable manifolds of saddles in piecewise smooth systems, Computer Modeling in Engineering & Sciences, 53 (2009), 235-254. Google Scholar

[15]

A. Colombo and U. Galvanetto, Computation of the basins of attraction in non-smooth dynamical systems, in Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, held Aberdeen, UK, 27-30 July 2010 (eds. M. Wiercigroch and G. Rega), vol. 32, Springer Science, 2013, 17-29. Google Scholar

[16]

A. Colombo and M.R. Jeffrey, Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows, SIAM Journal on Applied Dynamical Systems, 10 (2011), 423-451. doi: 10.1137/100801846. Google Scholar

[17]

A. Colombo and M.R. Jeffrey, The two-fold singularity of nonsmooth flows: Leading order dynamics in n-dimensions, Physica D, 263 (2013), 1-10. doi: 10.1016/j.physd.2013.07.015. Google Scholar

[18]

N. DelBuono, C. Elia and L. Lopez, On the equivalence between the sigmoidal approach and Utkin's approach for piecewise-linear models of gene regulatory networks, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1270-1292. doi: 10.1137/130950483. Google Scholar

[19]

N. DelBuono and L. Lopez, Direct event location techniques based on Adams multistep methods for discontinuous ODEs, Applied Mathematics Letters, 49 (2015), 152-158. doi: 10.1016/j.aml.2015.05.012. Google Scholar

[20]

F. Dercole and Y.A. Kuznetsov, SlideCont: An Auto97 driver for bifurcation analysis of filippov systems, ACM Trans. Math. Softw., 31 (2005), 95-119. doi: 10.1145/1055531.1055536. Google Scholar

[21]

A. Dhooge, W. Govaerts, Y. A. Kuznetsov, W. Mestrom, A. M. Riet and B. Sautois, MATCONT and CL MATCONT: Continuation toolboxes in Matlab, december 2006 edition, 2006, URL http://www.ricam.oeaw.ac.at/people/page/jameslu/Teaching/MathModelBioSciences_Summer08/EX3/MATCONT_manual.pdf.Google Scholar

[22]

A. Dhooge, W. Govaerts and Y.A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of odes, ACM Trans. Math. Softw., 29 (2003), 141-164. doi: 10.1145/779359.779362. Google Scholar

[23]

L. Dieci, C. Elia and L. Lopez, Uniqueness of Filippov sliding vector field on the intersection of two surfaces in $\mathbb R^3$ and implications for stability of periodic orbits, Journal of Nonlinear Science, 25 (2015), 1453-1471. doi: 10.1007/s00332-015-9265-6. Google Scholar

[24]

L. Dieci and L. Lopez, Fundamental matrix solutions of piecewise smooth differential systems, Mathematics and Computers in Simulation, 81 (2011), 932-953. doi: 10.1016/j.matcom.2010.10.012. Google Scholar

[25]

L. Dieci and L. Lopez, One-sided direct event location techniques in the numerical solution of discontinuous differential systems, BIT Numerical Mathematics, 55 (2015), 987-1003. doi: 10.1007/s10543-014-0538-5. Google Scholar

[26]

C. Erazo, M. E. Homer, P. T. Piiroinen and M Di Bernardo, Dynamic cell mapping algorithm for computing basins of attraction in planar filippov systems, International Journal of Bifurcation and Chaos, 27 (2017), 1730041, 15PP. doi: 10.1142/S0218127417300415. Google Scholar

[27]

G. F. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2007. doi: 10.1142/6437. Google Scholar

[28]

A. Filippov, Differential Equations with Discontinuous Right Hand Side, Kluwer, Dordrecht, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar

[29]

U. Galvanetto, Computation of the separatrix of basins of attraction in a non-smooth dynamical system, Physica D, 237 (2008), 2263-2271. doi: 10.1016/j.physd.2008.02.009. Google Scholar

[30]

M. Gameiro, J.-P. Lessard and A. Pugliese, Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Foundations of Computational Mathematics, 16 (2016), 531-575. doi: 10.1007/s10208-015-9259-7. Google Scholar

[31]

W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719543. Google Scholar

[32]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7. Google Scholar

[33]

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald and W. Stuetzle, Surface reconstruction from unorganized points, SIGGRAPH Comput. Graph., 26 (1992), 71-78. doi: 10.1145/133994.134011. Google Scholar

[34]

L. Lopez and S. Maset, Time transfomations for the event location of discontinuous ODEs, Math. Comp, Published electronically December 26, 2017 doi: 10.1090/mcom/3305. Google Scholar

[35]

J.-M. Melenk and I. Bubuska, The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Eng., 139 (1996), 289-314. doi: 10.1016/S0045-7825(96)01087-0. Google Scholar

[36]

B.-S. Morse, T.-S. Yoo, P. Rheingans, D.-T. Chen and K.-R. Subramanian, Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions, in Proceedings of International Conference on Shape Modeling and Applications. Genova, Italy May 7-11, 2001, IEEE, 2001, 72-89.Google Scholar

[37]

P. T. Piiroinen and Y. A. Kuznetsov, An event-driven method to simulate Filippov systems with accurate computing of sliding motions, ACM Transactions on Mathematical Software, 34 (2008), Art. 13, 24 pp. doi: 10.1145/1356052.1356054. Google Scholar

[38]

E. Plathe and S. Kjoglum, Analysis and genetic properties of gene regulatory networks with graded response functions, Physica D, 201 (2005), 150-176. doi: 10.1016/j.physd.2004.11.014. Google Scholar

[39]

A. Tornambé, Modelling and control of impact in mechanical systems: Theory and experimental results, IEEE Trans. Automat. Control, 44 (1999), 294-309. doi: 10.1109/9.746255. Google Scholar

[40]

G. Turk and J. F. O'Brien, Modelling with implicit surfaces that interpolate, ACM Trans. Graph., 21 (2002), 855-873. doi: 10.1145/1198555.1198640. Google Scholar

[41] H. Wendland, Scattered Data Approximation. Cambridge Monogr. Appl. Comput. Math., Cambridge Univ. Press, Cambridge, 2005. Google Scholar
[42]

H. Wendland, Fast evaluation of radial basis functions: Methods based on partition of unity, in Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, 2002, 473-483. Google Scholar

show all references

References:
[1]

V. Acary and B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, Berlin, 2008.Google Scholar

[2]

A. Agosti, L. Formaggia and A. Scotti, Analysis of a model for precipitation and dissolution coupled with a Darcy flux, Journal of Mathematical Analysis and Applications, 431 (2015), 752-781. doi: 10.1016/j.jmaa.2015.06.003. Google Scholar

[3]

A. Agosti, B. Giovanardi, L. Formaggia and A. Scotti, Numerical simulation of geochemical compaction with discontinuous reactions, in Coupled Problems 2015 - Proceedings of the 6th International Conference on Coupled Problems in Science and Engineering, 2015, 300-311.Google Scholar

[4]

A. Agosti, B. Giovanardi, L. Formaggia and A. Scotti, A numerical procedure for geochemical compaction in the presence of discontinuous reaction, Advances in Water Resources, 94 (2016), 332-344. doi: 10.1016/j.advwatres.2016.06.001. Google Scholar

[5]

I. Arango and J. Taborda, Numerical analysis of sliding dynamics in three-dimensional Filippov systems using SPTI method, International Journal of Mathematical Models and Method in Applied Sciences, 2 (2008), 342-354. Google Scholar

[6]

I. Arango and J. Taborda, Integration-free analysis of nonsmooth local dynamics in planar Filippov systems, International Journal of Bifurcation and Chaos, 19 (2009), 947-975. doi: 10.1142/S0218127409023391. Google Scholar

[7]

I. Arango and J. Taborda, Topological classification of limit cycles of piecewise smooth dynamical systems and its associated non-standard bifurcations, Entropy, 16 (2014), 1949-1968. doi: 10.3390/e16041949. Google Scholar

[8]

M. Berardi and L. Lopez, On the continuous extension of Adams - Bashforth methods and the event location in discontinuous ODEs, Applied Mathematics Letters, 25 (2012), 995-999. doi: 10.1016/j.aml.2011.11.014. Google Scholar

[9] M.-D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. Google Scholar
[10]

M. Calvo, J. Montijano and L. Rández, On the solution of discontinuous IVPs by adaptive Runge-Kutta codes, Numerical Algorithms, 33 (2003), 163-182. doi: 10.1023/A:1025507920426. Google Scholar

[11]

M. Calvo, J. I. Montijano and L. Rández, Algorithm 968: Disode45: A matlab runge-kutta solver for piecewise smooth ivps of filippov type, ACM Trans. Math. Softw., 43 (2017), Art.25, 14 pp. doi: 10.1145/2907054. Google Scholar

[12]

R. Casey, H. deJong and J. Gouze, Piecewise-linear models of genetics regulatory networks: Equilibria and their stability, Journal Mathematical Biology, 52 (2006), 27-56. doi: 10.1007/s00285-005-0338-2. Google Scholar

[13]

R. Cavoretto, A. DeRossi, E. Perracchione and E. Venturino, Robust approximation algorithms for the detection of attraction basins in dynamical systems, J. Sci. Comput., 68 (2016), 395-415. doi: 10.1007/s10915-015-0143-z. Google Scholar

[14]

A. Colombo and U. Galvanetto, Stable manifolds of saddles in piecewise smooth systems, Computer Modeling in Engineering & Sciences, 53 (2009), 235-254. Google Scholar

[15]

A. Colombo and U. Galvanetto, Computation of the basins of attraction in non-smooth dynamical systems, in Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, held Aberdeen, UK, 27-30 July 2010 (eds. M. Wiercigroch and G. Rega), vol. 32, Springer Science, 2013, 17-29. Google Scholar

[16]

A. Colombo and M.R. Jeffrey, Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows, SIAM Journal on Applied Dynamical Systems, 10 (2011), 423-451. doi: 10.1137/100801846. Google Scholar

[17]

A. Colombo and M.R. Jeffrey, The two-fold singularity of nonsmooth flows: Leading order dynamics in n-dimensions, Physica D, 263 (2013), 1-10. doi: 10.1016/j.physd.2013.07.015. Google Scholar

[18]

N. DelBuono, C. Elia and L. Lopez, On the equivalence between the sigmoidal approach and Utkin's approach for piecewise-linear models of gene regulatory networks, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1270-1292. doi: 10.1137/130950483. Google Scholar

[19]

N. DelBuono and L. Lopez, Direct event location techniques based on Adams multistep methods for discontinuous ODEs, Applied Mathematics Letters, 49 (2015), 152-158. doi: 10.1016/j.aml.2015.05.012. Google Scholar

[20]

F. Dercole and Y.A. Kuznetsov, SlideCont: An Auto97 driver for bifurcation analysis of filippov systems, ACM Trans. Math. Softw., 31 (2005), 95-119. doi: 10.1145/1055531.1055536. Google Scholar

[21]

A. Dhooge, W. Govaerts, Y. A. Kuznetsov, W. Mestrom, A. M. Riet and B. Sautois, MATCONT and CL MATCONT: Continuation toolboxes in Matlab, december 2006 edition, 2006, URL http://www.ricam.oeaw.ac.at/people/page/jameslu/Teaching/MathModelBioSciences_Summer08/EX3/MATCONT_manual.pdf.Google Scholar

[22]

A. Dhooge, W. Govaerts and Y.A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of odes, ACM Trans. Math. Softw., 29 (2003), 141-164. doi: 10.1145/779359.779362. Google Scholar

[23]

L. Dieci, C. Elia and L. Lopez, Uniqueness of Filippov sliding vector field on the intersection of two surfaces in $\mathbb R^3$ and implications for stability of periodic orbits, Journal of Nonlinear Science, 25 (2015), 1453-1471. doi: 10.1007/s00332-015-9265-6. Google Scholar

[24]

L. Dieci and L. Lopez, Fundamental matrix solutions of piecewise smooth differential systems, Mathematics and Computers in Simulation, 81 (2011), 932-953. doi: 10.1016/j.matcom.2010.10.012. Google Scholar

[25]

L. Dieci and L. Lopez, One-sided direct event location techniques in the numerical solution of discontinuous differential systems, BIT Numerical Mathematics, 55 (2015), 987-1003. doi: 10.1007/s10543-014-0538-5. Google Scholar

[26]

C. Erazo, M. E. Homer, P. T. Piiroinen and M Di Bernardo, Dynamic cell mapping algorithm for computing basins of attraction in planar filippov systems, International Journal of Bifurcation and Chaos, 27 (2017), 1730041, 15PP. doi: 10.1142/S0218127417300415. Google Scholar

[27]

G. F. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2007. doi: 10.1142/6437. Google Scholar

[28]

A. Filippov, Differential Equations with Discontinuous Right Hand Side, Kluwer, Dordrecht, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar

[29]

U. Galvanetto, Computation of the separatrix of basins of attraction in a non-smooth dynamical system, Physica D, 237 (2008), 2263-2271. doi: 10.1016/j.physd.2008.02.009. Google Scholar

[30]

M. Gameiro, J.-P. Lessard and A. Pugliese, Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions, Foundations of Computational Mathematics, 16 (2016), 531-575. doi: 10.1007/s10208-015-9259-7. Google Scholar

[31]

W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719543. Google Scholar

[32]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7. Google Scholar

[33]

H. Hoppe, T. DeRose, T. Duchamp, J. McDonald and W. Stuetzle, Surface reconstruction from unorganized points, SIGGRAPH Comput. Graph., 26 (1992), 71-78. doi: 10.1145/133994.134011. Google Scholar

[34]

L. Lopez and S. Maset, Time transfomations for the event location of discontinuous ODEs, Math. Comp, Published electronically December 26, 2017 doi: 10.1090/mcom/3305. Google Scholar

[35]

J.-M. Melenk and I. Bubuska, The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Eng., 139 (1996), 289-314. doi: 10.1016/S0045-7825(96)01087-0. Google Scholar

[36]

B.-S. Morse, T.-S. Yoo, P. Rheingans, D.-T. Chen and K.-R. Subramanian, Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions, in Proceedings of International Conference on Shape Modeling and Applications. Genova, Italy May 7-11, 2001, IEEE, 2001, 72-89.Google Scholar

[37]

P. T. Piiroinen and Y. A. Kuznetsov, An event-driven method to simulate Filippov systems with accurate computing of sliding motions, ACM Transactions on Mathematical Software, 34 (2008), Art. 13, 24 pp. doi: 10.1145/1356052.1356054. Google Scholar

[38]

E. Plathe and S. Kjoglum, Analysis and genetic properties of gene regulatory networks with graded response functions, Physica D, 201 (2005), 150-176. doi: 10.1016/j.physd.2004.11.014. Google Scholar

[39]

A. Tornambé, Modelling and control of impact in mechanical systems: Theory and experimental results, IEEE Trans. Automat. Control, 44 (1999), 294-309. doi: 10.1109/9.746255. Google Scholar

[40]

G. Turk and J. F. O'Brien, Modelling with implicit surfaces that interpolate, ACM Trans. Graph., 21 (2002), 855-873. doi: 10.1145/1198555.1198640. Google Scholar

[41] H. Wendland, Scattered Data Approximation. Cambridge Monogr. Appl. Comput. Math., Cambridge Univ. Press, Cambridge, 2005. Google Scholar
[42]

H. Wendland, Fast evaluation of radial basis functions: Methods based on partition of unity, in Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, 2002, 473-483. Google Scholar

Figure 1.  System (13): Crossing and attractive sliding region for $\lambda = -10$ (up) and $\lambda = 15$ (down)
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Figure 2.  System (13). Up: Behaviour of the the singular sets with respect to $\lambda$ space and for $\lambda = -5$ Down: Projection of the singular sets on the $x_1\lambda$ plane and crossing /sliding region
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Figure 3.  System (15). Crossing and sliding regions for $F = -1$ (up) and $F = 1$ (down)
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Figure 4.  System (15). Localization of the singular sets in the $x_1x_2F$ space for $F = 0$ (up) and $F = 1$ (down)
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Figure 5.  2D continuation method: Computation of the first patch
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Figure 6.  Advancing the front: General step. Incomplete patch centered at a frontal point (A) and completed patch (B) are colored in yellow
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Figure 7.  Boundaries of initial sets for Example 2.1
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Figure 8.  Reconstruction of the separatrices surfaces from some collection of points (blue and red dots on the surfaces). The black line illustrates a piece of a trajectory starting from the initial point $[1,1,5]^{\top}$. The chosen initial condition belongs to the subregion of points in $R_2$ starting from which any trajectory (forward in time) reaches the crossing region on the discontinuity surface $x_2 = 0$
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Figure 9.  Projections of the numerical solution obtained solving (15) from the initial point $[1,1,5]^{\top}$. The red and blue curves on the sliding surface $x_2 = 0$ (the first plot) represent the singular curves depicted in the selected region and obtained using the continuation algorithm
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Figure 10.  Reconstruction of some portions of the separatrix surfaces $m_1$ in $R_1$ (red surface) and $m_2$ in $R_2$ (blue surface) obtained interpolating a collection of points (red and blue dots on the surface, respectively) randomly chosen from trajectories obtained integrating -backward in time- the PWS system
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Figure 11.  Separatrix surface $m_2$ in $R_2$ and exit curves on the discontinuity surface $\Sigma$
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Figure 12.  3D example via 2D continuation, first portion
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Figure 13.  3D example via 2D continuation, second portion
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Table 1.  Values of the interpolated separatrix surface on different points in the interpolation domain
SeparatrixPoint Interpolared surface value
$m_1$ $[1,1,5]^{\top}$ $>0$
$m_1$ $[0.5, -0.5, 10]^{\top}$ $>0$
$m_1$ $[0,1,-0,8]^{\top}$ $<0$
$m_2$ $[1,1,5]^{\top}$ $ < 0$
$m_2$ $[0.5, 0.5, 10]^{\top}$ $ < 0$
$m_2$ $[0,1,20]^{\top}$ $>0$
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SeparatrixPoint Interpolared surface value
$m_1$ $[1,1,5]^{\top}$ $>0$
$m_1$ $[0.5, -0.5, 10]^{\top}$ $>0$
$m_1$ $[0,1,-0,8]^{\top}$ $<0$
$m_2$ $[1,1,5]^{\top}$ $ < 0$
$m_2$ $[0.5, 0.5, 10]^{\top}$ $ < 0$
$m_2$ $[0,1,20]^{\top}$ $>0$
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