Figure 1.
System (13): Crossing and attractive sliding region for $\lambda = -10$ (up) and $\lambda = 15$ (down)
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Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics
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
| 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 |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
M.-D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511543241.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
A. Colombo and U. Galvanetto,
Stable manifolds of saddles in piecewise smooth systems, Computer Modeling in Engineering & Sciences, 53 (2009), 235-254.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [27] |
G. F. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2007.
doi: 10.1142/6437. |
| [28] |
A. Filippov, Differential Equations with Discontinuous Right Hand Side, Kluwer, Dordrecht, Netherlands, 1988.
doi: 10.1007/978-94-015-7793-9. |
| [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. |
| [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. |
| [31] |
W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000.
doi: 10.1137/1.9780898719543. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [41] |
H. Wendland, Scattered Data Approximation. Cambridge Monogr. Appl. Comput. Math., Cambridge Univ. Press, Cambridge, 2005.
|
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
M.-D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511543241.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
A. Colombo and U. Galvanetto,
Stable manifolds of saddles in piecewise smooth systems, Computer Modeling in Engineering & Sciences, 53 (2009), 235-254.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [27] |
G. F. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2007.
doi: 10.1142/6437. |
| [28] |
A. Filippov, Differential Equations with Discontinuous Right Hand Side, Kluwer, Dordrecht, Netherlands, 1988.
doi: 10.1007/978-94-015-7793-9. |
| [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. |
| [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. |
| [31] |
W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000.
doi: 10.1137/1.9780898719543. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [41] |
H. Wendland, Scattered Data Approximation. Cambridge Monogr. Appl. Comput. Math., Cambridge Univ. Press, Cambridge, 2005.
|
| [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. |
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
Figure 4.
System (15). Localization of the singular sets in the $x_1x_2F$ space for $F = 0$ (up) and $F = 1$ (down)
Figure 5.
2D continuation method: Computation of the first patch
Figure 6.
Advancing the front: General step. Incomplete patch centered at a frontal point (A) and completed patch (B) are colored in yellow
Figure 7.
Boundaries of initial sets for Example 2.1
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 $x_2 = 0$
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
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
Figure 12.
3D example via 2D continuation, first portion
Figure 13.
3D example via 2D continuation, second portion
Table 1.
Values of the interpolated separatrix surface on different points in the interpolation domain
| Separatrix | Point | Interpolared surface value |
| Separatrix | Point | Interpolared surface value |
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