Multi-scroll Systems Synchronization on Strongly Connected Digraphs

Year 2022, Volume 4, Issue 4, 205 - 211, 31.12.2022

Eber J. ÁVİLA MARTÍNEZ J. L. ECHENAUSÍA-MONROY Adriana RUİZ-SİLVA

https://doi.org/10.51537/chaos.1185892

Abstract

In this paper, we study the synchronization problem in complex dynamic networks of Piece Wise Linear (PWL) systems. PWL systems exhibit multi-scrolls and belong to a special class of Unstable Dissipative Systems (UDS). We consider strongly connected digraphs and linear diffusive couplings. The synchronization regions are computed using the concept of disagreement vectors, generalized algebraic connectivity of the network topology, and Lyapunov functions, which provide lower bounds on the coupling gain of the network. Then, different combinations of linear diffusive coupling are explored by changing the observed and measured variables to illustrate the contribution of our results. The theoretical results are validated by numerical simulations.

Keywords

Synchronization, Complex Networks, Digraphs, Multi-Scroll Attractors, Unstable Dissipative Systems

References

• Allaire, G. and S. M. Kaber, 2007 Numerical Linear Algebra, volume 55 of Texts in Applied Mathematics. Springer-Verlag New York.
• Anzo-Hernández, A., E. Campos-Cantón, and M. Nicol, 2019 Itinerary synchronization between pwl systems coupled with unidirectional links. Communications in Nonlinear Science and Numerical Simulation 70: 102–124.
• Anzo-Hernández, A., H. E. Gilardi-Velázquez, and E. Campos-Cantón, 2018 On multistability behavior of unstable dissipative systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 28: 033613.
• Arenas, A., A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, 2008 Synchronization in complex networks. Physics reports 469: 93–153.
• Ávila-Martínez, E. J., 2022 Obstacle avoidance flocking motion in multi-agent systems with limited sensing radius and heterogeneous input constraints. Advanced Robotics.
• Ávila-Martínez, E. J. and J. G. Barajas-Ramírez, 2018 Distributed control for consensus on leader-followers proximity graphs. In IFAC-PapersOnLine, volume 51, pp. 240–245, Guadalajara, Jalisco, México.
• Ávila-Martínez, E. J. and J. G. Barajas-Ramírez, 2021 Flocking motion in swarms with limited sensing radius and heterogeneous input constraints. Journal of The Franklin Institute 358: 2346–2366.
• Boccaletti, S., J. Kurths, G. Osipov, D. Valladares, and C. Zhou, 2002 The synchronization of chaotic systems. Physics reports 366: 1–101.
• Boccaletti, S., V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, 2006 Complex networks: Structure and dynamics. Physics reports 424: 175–308.
• Campos-Cantón, E., 2016 Chaotic attractors based on unstable dissipative systems via third-order differential equation. International Journal of Modern Physics C 27: 1650008.
• Campos-Cantón, E., J. G. Barajas-Ramirez, G. Solis-Perales, and R. Femat, 2010 Multiscroll attractors by switching systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 20: 013116.
• Campos-Cantón, E., R. Femat, and G. Chen, 2012 Attractors generated from switching unstable dissipative systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 22: 033121.
• Chen, G., X. Wang, and X. Li, 2014 Fundamentals of Complex Networks. John Wiley & Sons Singapore Pte. Ltd, Singapore.
• Echenausía-Monroy, J., J. García-López, R. Jaimes-Reátegui, and G. Huerta-Cuéllar, 2020 Parametric control for multiscroll generation: Electronic implementation and equilibrium analysis. Nonlinear Analysis: Hybrid Systems 38: 100929.
• Echenausía-Monroy, J., L. Ontañón-García, and J. P. Ramirez, 2021 On synchronization of unidirectionally coupled multi-scroll systems: Dynamic vs static interconnections. IFAC-PapersOnLine 54: 53–58.
• Echenausía-Monroy, J. L., J. H. García-López, R. Jaimes-Reátegui, D. López-Mancilla, and G. Huerta-Cuellar, 2018 Family of bistable attractors contained in an unstable dissipative switching system associated to a SNLF. Complexity 2018.
• Echenausía-Monroy, J. L. and G. Huerta-Cuellar, 2020 A novel approach to generate attractors with a high number of scrolls. Nonlinear Analysis: Hybrid Systems 35: 100822.
• Gilardi-Velázquez, H. E., L. Ontañón-García, D. G. Hurtado-Rodriguez, and E. Campos-Cantón, 2017 Multistability in piecewise linear systems versus eigenspectra variation and round function. International Journal of Bifurcation and Chaos 27: 1730031.
• Huang, L., Q. Chen, Y.-C. Lai, and L. M. Pecora, 2009 Generic behavior of master-stability functions in coupled nonlinear dynamical systems. Physical Review E 80: 036204.
• Li, Z., 2015 Cooperative Control of Multi-agent Systems: A Consensus Region Approach. CRC Press, Boca Raton, Florida, first edition.
• Li, Z., Z. Duan, G. Chen, and L. Huang, 2010 Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans. Circuits Syst. I Regul. Pap. 57:213–224.
• Mishra, A. K., S. Das, and V. K. Yadav, 2022 Finite-time synchronization of multi-scroll chaotic systems with sigmoid non-linearity and uncertain terms. Chinese Journal of Physics 75: 235–245.
• Muñoz-Pacheco, J. M., E. Zambrano-Serrano, O. Félix-Beltrán, L. C. Gómez-Pavón, and A. Luis-Ramos, 2012 Synchronization of pwl function-based 2d and 3d multi-scroll chaotic systems. Nonlinear Dynamics 70: 1633–1643.
• Ontañón-García, L., I. C. Cantón, and J. P. Ramirez, 2021 Dynamic behavior in a pair of lorenz systems interacting via positivenegative coupling. Chaos, Solitons & Fractals 145: 110808.
• Pecora, L. M. and T. L. Carroll, 1998 Master stability functions for synchronized coupled systems. Physical review letters 80: 2109.
• Pikovsky, A., M. Rosenblum, and J. Kurths, 2002 Synchronization: a universal concept in nonlinear science.
• Posadas-Castillo, C., E. Garza-González, D. Diaz-Romero, E. Alcorta-García, and C. Cruz-Hernández, 2014 Synchronization of irregular complex networks with chaotic oscillators: Hamiltonian systems approach. Journal of applied research and technology 12: 782–791.
• R. Olfati-Saber and R. M. Murray, 2004 Consensus Problems in Networks of Agents with Switching Topology and Time-Delays. IEEE Trans. Automat. Contr. 49(9): 1520–1533.
• Ruiz-Silva, A., B. Cassal-Quiroga, G. Huerta-Cuellar, and H. Gilardi-Velázquez, 2022 On the behavior of bidirectionally coupled multistable systems. The European Physical Journal Special Topics pp. 1–11.
• Ruiz-Silva, A., H. Gilardi-Velázquez, and E. Campos, 2021 Emergence of synchronous behavior in a network with chaotic multistable systems. Chaos, Solitons & Fractals 151: 111263.
• Soriano-Sánchez, A. G., C. Posadas-Castillo, M. A. Platas-Garza, C. Cruz-Hernández, and R. M. López-Gutiérrez, 2016 Coupling strength computation for chaotic synchronization of complex networks with multi-scroll attractors. Applied Mathematics and Computation 275: 305–316.
• Wu, C. W., 2007 Synchronization in Complex Networks of Nonlinear Dynamical Systems.World Scientific, Singapore.
• Yu, W., G. Chen, M. Cao, and J. Kurths, 2010 Second-Order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans. Syst. Man, Cybern. Part B Cybern. 40: 881–891.