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On discrete-time polynomial dynamical systems on hypergraphs (2403.03416v2)

Published 6 Mar 2024 in eess.SY and cs.SY

Abstract: This paper studies the stability of discrete-time polynomial dynamical systems on hypergraphs by utilizing the Perron-Frobenius theorem for nonnegative tensors with respect to the tensors Z-eigenvalues and Z-eigenvectors. Firstly, for a multilinear polynomial system on a uniform hypergraph, we study the stability of the origin of the corresponding systems. Next, we extend our results to non-homogeneous polynomial systems on non-uniform hypergraphs. We confirm that the local stability of any discrete-time polynomial system is in general dominated by pairwise terms. Assuming that the origin is locally stable, we construct a conservative (but explicit) region of attraction from the system parameters. Finally, we validate our results via some numerical examples.

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References (38)
  1. P. E. Paré, J. Liu, C. L. Beck, B. E. Kirwan, and T. Başar, “Analysis, estimation, and validation of discrete-time epidemic processes,” IEEE Transactions on Control Systems Technology, vol. 28, no. 1, pp. 79–93, 2018.
  2. F. Liu, C. Shaoxuan, X. Li, and M. Buss, “On the stability of the endemic equilibrium of a discrete-time networked epidemic model,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 2576–2581, 2020.
  3. S. Cui, F. Liu, H. Jardón-Kojakhmetov, and M. Cao, “Discrete-time layered-network epidemics model with time-varying transition rates and multiple resources,” arXiv preprint arXiv:2206.07425, 2022.
  4. P. Duarte, R. L. Fernandes, and W. M. Oliva, “Dynamics of the attractor in the lotka–volterra equations,” journal of differential equations, vol. 149, no. 1, pp. 143–189, 1998.
  5. A. Slavík, “Lotka–volterra competition model on graphs,” SIAM Journal on Applied Dynamical Systems, vol. 19, no. 2, pp. 725–762, 2020.
  6. B. Goh, “Global stability in two species interactions,” Journal of Mathematical Biology, vol. 3, no. 3, pp. 313–318, 1976.
  7. ——, “Stability in models of mutualism,” The American Naturalist, vol. 113, no. 2, pp. 261–275, 1979.
  8. P. Donnell and M. Banaji, “Local and global stability of equilibria for a class of chemical reaction networks,” SIAM Journal on Applied Dynamical Systems, vol. 12, no. 2, pp. 899–920, 2013.
  9. G. Craciun, “Polynomial dynamical systems, reaction networks, and toric differential inclusions,” SIAM Journal on Applied Algebra and Geometry, vol. 3, no. 1, pp. 87–106, 2019.
  10. D. Angeli, “A tutorial on chemical reaction networks dynamics,” in 2009 European Control Conference (ECC).   IEEE, 2009, pp. 649–657.
  11. W. Ji and S. Deng, “Autonomous discovery of unknown reaction pathways from data by chemical reaction neural network,” The Journal of Physical Chemistry A, vol. 125, no. 4, pp. 1082–1092, 2021.
  12. C. Chen, A. Surana, A. M. Bloch, and I. Rajapakse, “Controllability of hypergraphs,” IEEE Transactions on Network Science and Engineering, vol. 8, no. 2, pp. 1646–1657, 2021.
  13. C. Chen and I. Rajapakse, “Tensor entropy for uniform hypergraphs,” IEEE Transactions on Network Science and Engineering, vol. 7, no. 4, pp. 2889–2900, 2020.
  14. M. M. Wolf, A. M. Klinvex, and D. M. Dunlavy, “Advantages to modeling relational data using hypergraphs versus graphs,” in 2016 IEEE High Performance Extreme Computing Conference (HPEC).   IEEE, 2016, pp. 1–7.
  15. C. Bick, E. Gross, H. A. Harrington, and M. T. Schaub, “What are higher-order networks?” SIAM Review, vol. 65, no. 3, pp. 686–731, 2023.
  16. Y. Wang, Y. Wei, G. Zhang, and S. Y. Chang, “Algebraic riccati tensor equations with applications in multilinear control systems,” arXiv preprint arXiv:2402.13491, 2024.
  17. I. Iacopini, G. Petri, A. Barrat, and V. Latora, “Simplicial models of social contagion,” Nature communications, vol. 10, no. 1, p. 2485, 2019.
  18. P. Cisneros-Velarde and F. Bullo, “Multigroup sis epidemics with simplicial and higher order interactions,” IEEE Transactions on Control of Network Systems, vol. 9, no. 2, pp. 695–705, 2021.
  19. S. Cui, F. Liu, H. Jardón-Kojakhmetov, and M. Cao, “General sis diffusion process with indirect spreading pathways on a hypergraph,” arXiv preprint arXiv:2306.00619, 2023.
  20. A. D. Letten and D. B. Stouffer, “The mechanistic basis for higher-order interactions and non-additivity in competitive communities,” Ecology letters, vol. 22, no. 3, pp. 423–436, 2019.
  21. S. Cui, Q. Zhao, H. Jardon-Kojakhmetov, and M. Cao, “Species coexistence and extinction resulting from higher-order lotka-volterra two-faction competition,” in 2023 62nd IEEE Conference on Decision and Control (CDC).   IEEE, 2023, pp. 467–472.
  22. M. M. Mayfield and D. B. Stouffer, “Higher-order interactions capture unexplained complexity in diverse communities,” Nature ecology & evolution, vol. 1, no. 3, pp. 1–7, 2017.
  23. P. A. Abrams, “Arguments in favor of higher order interactions,” The American Naturalist, vol. 121, no. 6, pp. 887–891, 1983.
  24. K. Tanaka, H. Yoshida, H. Ohtake, and H. O. Wang, “A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems,” IEEE Transactions on Fuzzy systems, vol. 17, no. 4, pp. 911–922, 2008.
  25. R. Mtar, M. Belhaouane, H. B. Ayadi, and N. B. Braiek, “An lmi criterion for the global stability analysis of nonlinear polynomial systems,” Nonlinear Dynamics and Systems Theory, vol. 9, no. 2, pp. 171–183, 2009.
  26. M. M. Belhaouane, M. F. Ghariani, H. Belkhiria Ayadi, N. Benhadj Braiek et al., “Improved results on robust stability analysis and stabilization for a class of uncertain nonlinear systems,” Mathematical Problems in Engineering, vol. 2010, 2010.
  27. D. Cheng and H. Qi, “Global stability and stabilization of polynomial systems,” in 2007 46th IEEE Conference on Decision and Control.   IEEE, 2007, pp. 1746–1751.
  28. K. Chang, L. Qi, and T. Zhang, “A survey on the spectral theory of nonnegative tensors,” Numerical Linear Algebra with Applications, vol. 20, no. 6, pp. 891–912, 2013.
  29. K.-C. Chang, K. Pearson, and T. Zhang, “Perron-frobenius theorem for nonnegative tensors,” Communications in Mathematical Sciences, vol. 6, no. 2, pp. 507–520, 2008.
  30. Y. Yang and Q. Yang, “Further results for perron–frobenius theorem for nonnegative tensors,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 5, pp. 2517–2530, 2010.
  31. Q. Yang and Y. Yang, “Further results for perron–frobenius theorem for nonnegative tensors ii,” SIAM Journal on Matrix Analysis and Applications, vol. 32, no. 4, pp. 1236–1250, 2011.
  32. G. Gallo, G. Longo, S. Pallottino, and S. Nguyen, “Directed hypergraphs and applications,” Discrete applied mathematics, vol. 42, no. 2-3, pp. 177–201, 1993.
  33. C. Chen, “On the stability of multilinear dynamical systems,” arXiv preprint arXiv:2105.01041, 2021.
  34. S. Cui, G. Zhang, H. Jardón-Kojakhmetov, and M. Cao, “On metzler positive systems on hypergraphs,” arXiv preprint arXiv:2401.03652, 2024.
  35. L. Qi, “Eigenvalues of a real supersymmetric tensor,” Journal of Symbolic Computation, vol. 40, no. 6, pp. 1302–1324, 2005.
  36. M. Zhang, G. Ni, and G. Zhang, “Iterative methods for computing u-eigenvalues of non-symmetric complex tensors with application in quantum entanglement,” Computational Optimization and Applications, vol. 75, pp. 779–798, 2020.
  37. C. Mo and Y. Wei, “On nonnegative solution of multi-linear system with strong mz-tensors,” Numer Math Theor Meth Appl, vol. 14, pp. 176–193, 2021.
  38. J. Xie and L. Qi, “Spectral directed hypergraph theory via tensors,” Linear and Multilinear Algebra, vol. 64, no. 4, pp. 780–794, 2016.
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