In this paper we describe an explicit solution semigroup of the quasi-linear Lasota equation. By constructing the relationship of this solution semigroup with the translation semigroup we obtain some sufficient and necessary conditions for the solution semigroup of the quasi-linear Lasota equation to be hypercyclic or chaotic respectively.
@article{bwmeta1.element.doi-10_1515_math-2015-0036,
author = {Cheng-Hung Hung},
title = {Chaotic and hypercyclic properties of the quasi-linear Lasota equation},
journal = {Open Mathematics},
volume = {13},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0036}
}
Cheng-Hung Hung. Chaotic and hypercyclic properties of the quasi-linear Lasota equation. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0036/
[1] Batty C. J. K., Derivations on the line and flow along orbits, Pacific Journal of Mathmatics, Vol 126, No 2, 1987, 209-225. | Zbl 0578.46045
[2] Brunovsky P. and Komornik J., Ergodicity and exactness of the shift on C[0;1) and the semiflow of a first order partial differential equation, J. Math. Anal. Appl., 104 (1984), 235–245. | Zbl 0602.35070
[3] Chang Y.-H. and Hong C.-H., "The chaos of the solution semigroup for the quasi-linear lasota equation," Taiwanese Journal of Mathematics, vol. 16, no. 5, pp. 1707–1717, 2012. | Zbl 1251.35034
[4] Dawidowicz A. L., Haribash N. and Poskrobko A., On the invariant measure for the quasi-linear Lasota equation, Math. Meth. Appl. Sci., 2007, 30, 779-787. [WoS] | Zbl 1116.35043
[5] Desch W., Schappacher W. and Webb G. F., Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynamical Systems, 17 (1997), 793–819. | Zbl 0910.47033
[6] Hung C.-H. and Chang Y.-H. "Frequently Hypercyclic and Chaotic Behavior of Some First-Order Partial Differential Equation, "Abstract and Applied Analysis Volume 2013, Article ID 679839. [WoS]
[7] Lasota A., Stable and chaotic solutions of a first-order partial differential equation, Nonlinear Analysis 1981 5, 1181–1193. | Zbl 0523.35015
[8] Lasota A. and Szarek T., Dimension of measures invariant with respect to the Wa’zewska partial differential equation, J. Differential Equations, 196, 2004, 448-465. | Zbl 1036.35054
[9] Mackey M. C. and Schwegle H., Ensemble and trajectory statistics in a nonlinear partial differential equation, Journal of Statistical Physics, Vol. 70, Nos. 1/2, 1993.
[10] Matsui M. and Takeo F., Chatoic semigroups generated by certain differential operators of order 1, SUT Journal of Mathmatics, Vol 37, 2001, 43-61.
[11] Murillo-Arcila M. and A. Peris, Strong mixing measures for linear operators and frequent hypercyclicity, J. Math. Anal. Appl., 398 (2013), 462–465. [WoS] | Zbl 1288.47009
[12] Rudnicki R., Invariant measures for the flow of a first-order partial differential equation, Ergodic Theory Dynamical Systems, 8 (1985), 437–443. | Zbl 0566.28013
[13] Rudnicki R., Strong ergodic properties of a first-order partial differential equation, J. Math. Anal. Appl., 133 (1988), 14–26. | Zbl 0673.35012
[14] Rudnicki R., Chaos for some infinite-demensional dynamical systems, Math. Meth. Appl. Sci. 2004, 27,723-738. | Zbl 1156.37322
[15] Rudnicki R., Chaoticity of the blood cell production system, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009), 043112, 1–6. | Zbl 1311.92067
[16] Rudnicki R., An ergodic theory approach to chaos, Discrete Contin. Dyn. Syst. A 35 (2015), 757–770. | Zbl 06377819
[17] Takeo F., Chaos and hypercyclicity for solution semigroups to some partial differential equations, Nonlinear Analysis 63, 2005, e1943 - e1953.
[18] Takeo F., Chatoic or hypercyclic semigroups on a function space C0 .I;C/ or Lp .I;C/, SUT Journal of Mathmatics, Vol 41, No 1, 2005, 43-61. | Zbl 1127.47009