Networks of Piecewise smooth Filippov systems and stability of synchronous periodic orbits

Luca Dieci; Cinzia Elia; Luciano Lopez

doi:10.3934/dcdsb.2024146

Article Contents

2025, Volume 30, Issue 6: 1996-2026. Doi: 10.3934/dcdsb.2024146

This issue Previous Article Extracting coherent sets in aperiodically driven flows from generators of Mather semigroups Next Article Piecewise quadratic Lyapunov functions for stochastic differential equations by linear programming

Networks of Piecewise smooth Filippov systems and stability of synchronous periodic orbits

1.
School of Mathematics, Georgia Tech, Atlanta, GA 30332, U.S.A.
2.
Dipartimento di Matematica, Univ. of Bari, I-70100, Bari, Italy

^*Corresponding author: Luca Dieci
^*Corresponding author: Luca Dieci

Received: February 1, 2024

Revised: September 1, 2024

Early access: October 15, 2024

Published online: October 15, 2024

This work has been partially supported by the GNCS-Indam group and the PRIN2017 [2017E844SL].

Abstract / Introduction Full Text(HTML) Figure(10) / Table(2) Related Papers Cited by

Abstract

In this work, we considered networks of piecewise- smooth (PWS) differential equations of Filippov type, which we call PWS Filippov networks, whose single agent has an asymptotically stable periodic orbit, that becomes a synchronous periodic orbit for the network. We investigated the stability of the synchronous periodic orbit by using both the master stability function (MSF) tool and direct integration of the "regularized network'', that is the network obtained by replacing the PWS differential agents by a suitable smooth approximation. We present several new results on the class of Filippov PWS networks that depend on several coupling matrices. (i) We studied an associated MSF that depends on several coupling strengths, and (ii) we employed a one-parameter family of regularized networks, observed that it is equivalent to a network of regularized agents, and showed that its solutions converge to the solutions of the PWS network as the regularization parameter goes to 0. Furthermore, when the synchronous periodic orbit is asymptotically stable and does not slide on the discontinuity surface, (iii) we showed that the regularized network has an asymptotically stable synchronous periodic orbit as well. Finally, (iv) we performed detailed numerical experiments on two different problems, highlighting specific characteristics of PWS networks with a synchronous periodic orbit. More specifically, (a) first, we considered the case of a network of planar PWS oscillators, with the network depending on two different coupling parameters and the synchronous periodic trajectory undergoing sliding regime; then, (b) we considered a network where the single agent obeys 3D PWS dynamics with the synchronous periodic orbit undergoing repeated crossing of two distinct discontinuity planes. On our two problems, we also showed how the convergence behavior to the synchronous subspace can be used to obtain the same rates of attractivity predicted by the MSF.

Keywords:

Mathematics Subject Classification: Primary: 34A36, 34D06; Secondary: 34D08.

Citation:

Full Text(HTML)

Figure 1. Model periodic orbit of (4) with generic events

Download: Full-size image PowerPoint slide

Figure 2. Stick slip coupled oscillators

Download: Full-size image PowerPoint slide

Figure 3. Equation (33). Stability region in function of $ \eta_1 $ and $ \eta_2 $. Dark dots indicate stability

Download: Full-size image PowerPoint slide

Figure 4. Regularization of equation (33), $ \epsilon = 0.01 $. Synchronization of 32 oscillators for viscous coupling

Download: Full-size image PowerPoint slide

Figure 5. Synchronization intervals for (33) with $ E_1 $ only on the left (i.e., $ \sigma_2 = 0 $) and for $ E_1+E_2 $ on the right

Download: Full-size image PowerPoint slide

Figure 6. Equation (33). Numerical investigation of flip bifurcation. Left: invariant curve for two coupled oscillators and $ \sigma_1 = 2.6 $. The two agents are not synchronized but there is a phase shift between the two. Right: invariant curve for two coupled oscillators and $ \sigma_1 = 2.512 $

Download: Full-size image PowerPoint slide

Figure 7. Equation (35). Synchronization region in the $ (\eta_1, \eta_2) $-plane. There is no synchronization in the white part of the figure

Download: Full-size image PowerPoint slide

Figure 8. Equation (35). $ N = 8 $, $ \sigma_1 = 1.95 $, $ \sigma_2 = 0.03 $. Left: Convergence to synchronous periodic orbit. Right: Transient behavior and convergence for the first agent

Download: Full-size image PowerPoint slide

Figure 9. Equation (35), with matrices $ E_1 $ and $ E_2 $ as in (37), Synchronization region in the $ (\eta_1-\eta_2) $-plane. There is no synchronization in the white region on the bottom left of the figure

Download: Full-size image PowerPoint slide

Figure 10. Convergence rates and line of best fit for the errors. Left: Example 3.1. Right: Example 3.2

Download: Full-size image PowerPoint slide

Table 1. Equation (33). Synchronization intervals for $ \sigma_2 = 0 $. Values of $ \eta_1 $ must be in the intervals $ [\alpha_{2i+1}, \alpha_{2(i+1)}] $, $ i = 0, \ldots, 7 $, to ensure stability of synchronous periodic orbit for just elastic coupling $ E_1 $

$ \alpha_{2i+1} $ $ \alpha_{2(i+1)} $

2.0521 2.2022

5.3053 5.5556

9.6096 9.9600

14.9650 15.4154

21.3714 21.9219

28.9289 29.5295

37.4875 38.1882

47.1972 47.9479

| Show Table

DownLoad: CSV

Table 2. $ \sigma_{\min} $ is the minimum value of $ \sigma $ required in order to stably synchronize $ N $ agents coupled with matrix $ E_1+E_2 $

N $ \sigma_{\min} $

4 1.7234

8 6.6349

16 26.2845

32 104.8850

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	J. C. Alexander and T. Seidman, Sliding modes in intersecting switching surfaces, I: Blending, Houston J. Math., 24 (1998), 545-569.
[2]	D. G. Aronson, E. J. Doedel and H. G. Othmer, An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators, Physica D, 25 (1987), 20-104. doi: 10.1016/0167-2789(87)90095-9.
[3]	J. Awrejcewicz, M. Feckan and Pa wel Olejnik, On continuous approximation of discontinuous systems, Nonlinear Analysis, 62 (2005), 1317-1331. doi: 10.1016/j.na.2005.04.033.
[4]	M. Barahona and L. Pecora, Synchronization in small-world systems, Phys. Rev. Lett., 89 (2002), 054101. doi: 10.1103/PhysRevLett.89.054101.
[5]	B. K. Bera, S. Rakshit and D. Ghosh, Intralayer synchronization in neuronal multiplex network, The European Physical Journal Special Topics, 228 (2019), 2441–2454.
[6]	K. A. Blaha, K. Huang, F. Della Rossa, L. Pecora, M. Hossein-Zadeh and F. Sorrentino, Cluster synchronization in multilayer networks: A fully analog experiment with LC oscillators with physically dissimilar coupling, Physical Review Letters, 122 (2019), 014101.
[7]	D. A. Burbano-L., S. Yaghouti, C. Petrarca, M. de Magistris and M. di Bernardo, Synchronization in multiplex networks of Chua's circuits: Theory and experiments, IEEE Transactions on Circuits and Systems I: Regular Papers, 67 (2020), 927–938. doi: 10.1109/TCSI.2019.2955972.
[8]	S. Coombes, Y. M. Lai, M. Sayli and R. Thul, Networks of piecewise linear neural mass models, European Journal of Applied Mathematics, 29 (2018), 869–890.
[9]	S. Coombes and R. Thul, Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function, European Journal of Applied Mathematics, 27 (2016), 904–922.
[10]	M. Coraggio, P. De Lellis and M. di Bernardo, Convergence and synchronization in networks of piecewise-smooth systems via distributed discontinuous coupling, Automatica, 129 (2021), 109596.
[11]	M. Coraggio, P. De Lellis, S. J. Hogan and M. di Bernardo, Synchronization of networks of piecewise-smooth systems, IEEE Control System Letters, 2 (2018), 653–658. doi: 10.1109/LCSYS.2018.2845302.
[12]	C. I. del Genio, J. Gomez-Gardenes, I. Bonamassa and and S. Boccaletti, Synchronization in networks with multiple interaction layers, Science Advances, 2 (2016), 1601679.
[13]	L. Dieci and C. Elia, Master stability function for piecewise smooth systems, Automatica, 152 (2023), 110939.
[14]	L. Dieci and C. Elia, Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1039-1068. doi: 10.3934/dcdss.2016041.
[15]	L. Dieci, C. Elia and L. Lopez, A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited bifurcation analysis, Journal of Differential Equations, 254 (2013), 1800–1832.
[16]	L. Dieci, C. Elia and D. Pi, Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3091-3112. doi: 10.3934/dcdsb.2017165.
[17]	L. Dieci and N. Guglielmi, Regularization of piecewise smooth systems, J. Dynamics and Differential Equations, 25 (2013), 71-94. doi: 10.1007/s10884-013-9287-4.
[18]	B. Fiedler, M. Belhaq and M. Houssni, Basins of attraction in strongly damped coupled mechanical oscillators: A global example, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 50 (1999), 282–300.
[19]	A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic, Netherlands, 1988.
[20]	U. Galvanetto and S. R. Bishop, Computational techniques for nonlinear dynamics in multiple friction oscillators, Comput. Methods Appl. Mech. Engrg., 163 (1998), 373-382. doi: 10.1016/S0045-7825(98)00025-5.
[21]	U. Galvanetto, S. R. Bishop and L. Briseghella, Mechanical stick slip vibrations, Int. Journal of Bifurcation and Chaos, 5 (1995), 637-651.
[22]	M. Itoh and L. O. Chua, Memristor oscillators, International Journal of Bifurcation and Chaos, 18 (2008), 3183-3206.
[23]	J. Ladenbauer, J. Lehnert, H. Rankoohi, T. Dahms, E. Scholl and K Obermayer, Adaptation controls synchrony and cluster states of coupled threshold-model neurons, Physical Review E, 88 (2013), 042713.
[24]	M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. of Differential Equations, 248 (2010), 1–20.
[25]	J. Llibre, P. R. da Silva and M. A. Teixeira, Study of singularities in nonsmooth dynamical systems via singular perturbation, Siam J. Appl. Dynamical Syst., 8 (2009), 508–526. doi: 10.1137/080722886.
[26]	S. Majhi, D. Ghosh and J. Kurths, Emergence of synchronization in multiplex networks of mobile Rössler oscillators, Physical Review E, 99 (2019), 012308.
[27]	L. Pecora and T. Carroll, Master stability functions for synchronized coupled systems, Physical Review Letters, 80 (1997), 2109-1112. doi: 10.1103/PhysRevLett.80.2109.
[28]	R. Sevilla-Escoboza, I. Sendina-Nadal, I. Leyva, R. Gutierrez, J. M. Buldu and S. Boccaletti, Inter-layer synchronization in multiplex networks of identical layers, Chaos, 26 (2016), 065304.
[29]	F. Sorrentino, Synchronization of hypernetworks of coupled dynamical systems, New Journal of Physics, 14 (2012), 033035. doi: 10.1088/1367-2630/14/3/033035.
[30]	J. Sotomayor and A. L. Machado, Structurally stable discontinuous vector fields on the plane, Qual.Theory of Dynamical Systems, 3 (2002), 227-250. doi: 10.1007/BF02969339.
[31]	J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207-223.
[32]	L. Tang, X. Wu, J. Lü, J.-a. Lu and R. M. D'Souza, Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators, Physical Review E, 99 (2019), 012304.
[33]	W. Wasow, Asymptotic expansions for ordinary differential equations, Dover Publications, Inc., New York, 1987.
[34]	C.-W. Wu, Synchronization in dynamical systems coupled via multiple directed networks, IEEE Transactions on Circuits and Systems-II: Express Briefs, 68 (2021), 1660-1664.