Depart. of Applied Mathematics, Holon Institute of Technology, Holon, Israel |
We perform a geometric study of the equilibrium locus of the flow that models the diffusion process over a circular network of cells. We prove that when considering the set of all possible values of the parameters, the equilibrium locus is a smooth manifold with corners, while for a given value of the parameters, it is an embedded smooth and connected curve. For different values of the parameters, the curves are all isomorphic.
Moreover, we show how to build a homotopy between different curves obtained for different values of the parameter set. This procedure allows the efficient computation of the equilibrium point for each value of some first integral of the system. This point would have been otherwise difficult to be computed for higher dimensions. We illustrate this construction by some numerical experiments.
Eventually, we show that when considering the parameters as inputs, one can easily bring the system asymptotically to any equilibrium point in the reachable set, which we also easily characterize.
| [1] |
C. Ehresmann,
Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles 1950, Paris, (1951), 29-55.
|
| [2] |
D. Lazard and F. Rouillier,
Solving parametric polynomial systems, Journal of Symbolic Computations, 42 (2007), 636-667.
doi: 10.1016/j.jsc.2007.01.007. |
| [3] |
J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$, Springer, 2013. |
| [4] |
T. Y. Li,
Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta Numerica, 6 (1997), 399-436.
doi: 10.1017/S0962492900002749. |
| [5] |
J. Mather,
Notes on topological stability, Bulletin of the American Mathematical Society, 49 (2012), 475-506.
doi: 10.1090/S0273-0979-2012-01383-6. |
| [6] |
A. Raveh, Y. Zarai, M. Margaliot and T. Ruller,
Ribosome flow model on a ring, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 12 (2015), 1429-1439.
doi: 10.1109/TCBB.2015.2418782. |
show all references
| [1] |
C. Ehresmann,
Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles 1950, Paris, (1951), 29-55.
|
| [2] |
D. Lazard and F. Rouillier,
Solving parametric polynomial systems, Journal of Symbolic Computations, 42 (2007), 636-667.
doi: 10.1016/j.jsc.2007.01.007. |
| [3] |
J. Lee, Introduction to Smooth Manifolds, 2$^{nd}$, Springer, 2013. |
| [4] |
T. Y. Li,
Numerical solution of multivariate polynomial systems by homotopy continuation methods, Acta Numerica, 6 (1997), 399-436.
doi: 10.1017/S0962492900002749. |
| [5] |
J. Mather,
Notes on topological stability, Bulletin of the American Mathematical Society, 49 (2012), 475-506.
doi: 10.1090/S0273-0979-2012-01383-6. |
| [6] |
A. Raveh, Y. Zarai, M. Margaliot and T. Ruller,
Ribosome flow model on a ring, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 12 (2015), 1429-1439.
doi: 10.1109/TCBB.2015.2418782. |
| [1] |
Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 |
| [2] |
Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 |
| [3] |
Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021068 |
| [4] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
| [5] |
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
| [6] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
| [7] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
| [8] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
| [9] |
Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 |
2019 Impact Factor: 1.233
[Back to Top]
