American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment
  • DCDS-B Home
  • This Issue
  • Next Article
    Global dynamics of a reaction-diffusion system with intraguild predation and internal storage
March  2020, 25(3): 903-915. doi: 10.3934/dcdsb.2019195

The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems

Emma D'Aniello 1, and  Saber Elaydi 2,,

1. 

Dipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli", Viale Lincoln n.5, 81100 Caserta, Italia
2. 

Department of Mathematics, Trinity University, San Antonio, TX 78212-7200, USA

* Corresponding author: Saber Elaydi

Saber Elaydi acknowledges the hospitality of the Department of Mathematics and Physics of the Universit`a degli Studi della Campania “Luigi Vanvitelli”

Revised  December 2018 Published  September 2019

We consider a discrete non-autonomous semi-dynamical system generated by a family of continuous maps defined on a locally compact metric space. It is assumed that this family of maps uniformly converges to a continuous map. Such a non-autonomous system is called an asymptotically autonomous system. We extend the dynamical system to the metric one-point compactification of the phase space. This is done via the construction of an associated skew-product dynamical system. We prove, among other things, that the omega limit sets are invariant and invariantly connected. We apply our results to two populations models, the Ricker model with no Allee effect and Elaydi-Sacker model with the Allee effect, where it is assumed that the reproduction rate changes with time due to habitat fluctuation.

Keywords: Dynamical system, skew product, attractor.
Mathematics Subject Classification: Primary: 54H20, 37C70; Secondary: 39A05.
Citation: Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
References:
[1]

W. C. Allee, The Social Life of Animals, 3rd Edition, William Heineman Ltd, London and Toronto, 1941. Google Scholar

[2]

L. Assas, S. Elaydi, E. Kwessi, G. Livadiotis and D. Ribble, Hierarchical competition models with the Allee effect, J. Biological Dynamics, 9 (2015), 32-44.  doi: 10.1080/17513758.2014.923118.  Google Scholar

[3]

L. Assas, B. Dennis, S. Elaydi, E. Kwessi and G. Livadiotis, Hierarchical competition models with the Allee effect II: The case of immigration, J. Biological Dynamics, 9 (2015), 288-316.  doi: 10.1080/17513758.2015.1077999.  Google Scholar

[4]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Diff. Equ. Appl., 7 (2001), 895-913.  doi: 10.1080/10236190108808310.  Google Scholar

[5]

B. Aulbach and T. Wanner, Invariant foliations and decoupling of non-autonomous difference equations, J. Diff. Equ. Appl., 9 (2003), 459-472.  doi: 10.1080/1023619031000076524.  Google Scholar

[6]

B. Aulbach and T. Wanner, Topological simplification of non-autonomous difference equations, J. Diff. Equ. Appl., 12 (2006), 283-296.  doi: 10.1080/10236190500489384.  Google Scholar

[7]

E. Cabral Balreira, S. Elaydi and R. Luís, Global dynamics of triangular maps, Nonlinear Analysis, Theory, Methods and Appl., Ser. A, 104 (2014), 75-83.  doi: 10.1016/j.na.2014.03.019.  Google Scholar

[8]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762.  Google Scholar

[9] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755316.  Google Scholar
[10]

J. S. Cánovas, On $\omega$-limit sets of non-autonomous discrete systems, J. Diff. Equ. Appl., 12 (2006), 95-100.  doi: 10.1080/10236190500424274.  Google Scholar

[11]

E. D'Aniello and H. Oliveira, Pitchfork bifurcation for non-autonomous interval maps, J. Diff. Equ. Appl., 15 (2009), 291-302.  doi: 10.1080/10236190802258669.  Google Scholar

[12]

E. D'Aniello and T. H. Steele, The $\omega$-limit sets of alternating systems, J. Diff. Equ. Appl., 17 (2011), 1793-1799.  doi: 10.1080/10236198.2010.488227.  Google Scholar

[13]

E. D'Aniello and T. H. Steele, Stability in the family of $\omega$-limit sets of alternating systems, J. Math. Anal. Appl., 389 (2012), 1191-1203.  doi: 10.1016/j.jmaa.2011.12.056.  Google Scholar

[14]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc., 4 (1954), 168-176.  doi: 10.1112/plms/s3-4.1.168.  Google Scholar

[15]

J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass, 1966.  Google Scholar

[16]

J. Dvořáková, N. Neumärker and M. Štefánková, On $\omega$-limit sets of non-autonomous dynamical systems with a uniform limit of type $2^{\infty}$, J. Diff. Equ. Appl., 22 (2016), 636-644.  doi: 10.1080/10236198.2015.1123706.  Google Scholar

[17]

S. Elaydi, E. Kwessi and G. Livadiotis, Hierarchical competition models with the Allee effect III: Multispecies, J. Biological Dynamics, 12 (2018), 271-287.  doi: 10.1080/17513758.2018.1439537.  Google Scholar

[18]

S. Elaydi and R. J. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Diff. Equ. Appl., 208 (2005), 258-273.  doi: 10.1016/j.jde.2003.10.024.  Google Scholar

[19]

S. Elaydi and R. Sacker, Skew-Product Dynamical Systems: Applications to Difference Equations, 2004. Available from: http://www-bcf.usc.edu/ rsacker/pubs/UAE.pdf. Google Scholar

[20]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biological Dynamics, 4 (2010), 397-408.  doi: 10.1080/17513750903377434.  Google Scholar

[21]

N. Franco, L. Silva and P. Simões, Symbolic dynamics and renormalization of non-autonomous $k$-periodic dynamical systems, J. Diff. Equ. Appl., 19 (2013), 27-38.  doi: 10.1080/10236198.2011.611804.  Google Scholar

[22]

J. E. Franke and A.-A. Yakubu, Population models with periodic recruitment functions and survival rates, J. Diff. Equ. Appl., 11 (2005), 1169-1184.  doi: 10.1080/10236190500386275.  Google Scholar

[23]

R. Kempf, On $\Omega$-limit sets of discrete-time dynamical systems, J. Diff. Equ. Appl., 8 (2002), 1121-1131.  doi: 10.1080/10236190290029024.  Google Scholar

[24]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[25]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, AMS, Providence, RI, 2011. doi: 10.1090/surv/176.  Google Scholar

[26]

S. F. Kolyada, On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems, 12 (1992), 749-768.  doi: 10.1017/S0143385700007082.  Google Scholar

[27]

J. P. La Salle, The stability of dynamical systems, in CBMS-NSf Regional Conference Series in Applied Mathematics, Siam, (1976). doi: 10.1137/1.9781611970432.  Google Scholar

[28]

V. Jiménez López and J. Smítal, $\omega$-limit sets for triangular mappings, Fundamenta Math., 167 (2001), 1-15.  doi: 10.4064/fm167-1-1.  Google Scholar

[29]

M. Mandelkern, Metrization of the one-point compactification, Proc. Amer. Math. Soc., 107 (1989), 1111-1115.  doi: 10.1090/S0002-9939-1989-0991703-4.  Google Scholar

[30]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Note in Mathematics, 2002. Springer-Verlag, Berlin 2010. doi: 10.1007/978-3-642-14258-1.  Google Scholar

[31]

G. Rangel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[32]

L. Silva, Periodic attractors of nonautonomous flat-topped tent systems, Discrete Contin. Dyn. Syst. B, 24 (2019), 1867-1874.   Google Scholar

show all references

Saber Elaydi acknowledges the hospitality of the Department of Mathematics and Physics of the Universit`a degli Studi della Campania “Luigi Vanvitelli”

References:
[1]

W. C. Allee, The Social Life of Animals, 3rd Edition, William Heineman Ltd, London and Toronto, 1941. Google Scholar

[2]

L. Assas, S. Elaydi, E. Kwessi, G. Livadiotis and D. Ribble, Hierarchical competition models with the Allee effect, J. Biological Dynamics, 9 (2015), 32-44.  doi: 10.1080/17513758.2014.923118.  Google Scholar

[3]

L. Assas, B. Dennis, S. Elaydi, E. Kwessi and G. Livadiotis, Hierarchical competition models with the Allee effect II: The case of immigration, J. Biological Dynamics, 9 (2015), 288-316.  doi: 10.1080/17513758.2015.1077999.  Google Scholar

[4]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Diff. Equ. Appl., 7 (2001), 895-913.  doi: 10.1080/10236190108808310.  Google Scholar

[5]

B. Aulbach and T. Wanner, Invariant foliations and decoupling of non-autonomous difference equations, J. Diff. Equ. Appl., 9 (2003), 459-472.  doi: 10.1080/1023619031000076524.  Google Scholar

[6]

B. Aulbach and T. Wanner, Topological simplification of non-autonomous difference equations, J. Diff. Equ. Appl., 12 (2006), 283-296.  doi: 10.1080/10236190500489384.  Google Scholar

[7]

E. Cabral Balreira, S. Elaydi and R. Luís, Global dynamics of triangular maps, Nonlinear Analysis, Theory, Methods and Appl., Ser. A, 104 (2014), 75-83.  doi: 10.1016/j.na.2014.03.019.  Google Scholar

[8]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762.  Google Scholar

[9] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755316.  Google Scholar
[10]

J. S. Cánovas, On $\omega$-limit sets of non-autonomous discrete systems, J. Diff. Equ. Appl., 12 (2006), 95-100.  doi: 10.1080/10236190500424274.  Google Scholar

[11]

E. D'Aniello and H. Oliveira, Pitchfork bifurcation for non-autonomous interval maps, J. Diff. Equ. Appl., 15 (2009), 291-302.  doi: 10.1080/10236190802258669.  Google Scholar

[12]

E. D'Aniello and T. H. Steele, The $\omega$-limit sets of alternating systems, J. Diff. Equ. Appl., 17 (2011), 1793-1799.  doi: 10.1080/10236198.2010.488227.  Google Scholar

[13]

E. D'Aniello and T. H. Steele, Stability in the family of $\omega$-limit sets of alternating systems, J. Math. Anal. Appl., 389 (2012), 1191-1203.  doi: 10.1016/j.jmaa.2011.12.056.  Google Scholar

[14]

Y. N. Dowker and F. G. Friedlander, On limit sets in dynamical systems, Proc. London Math. Soc., 4 (1954), 168-176.  doi: 10.1112/plms/s3-4.1.168.  Google Scholar

[15]

J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass, 1966.  Google Scholar

[16]

J. Dvořáková, N. Neumärker and M. Štefánková, On $\omega$-limit sets of non-autonomous dynamical systems with a uniform limit of type $2^{\infty}$, J. Diff. Equ. Appl., 22 (2016), 636-644.  doi: 10.1080/10236198.2015.1123706.  Google Scholar

[17]

S. Elaydi, E. Kwessi and G. Livadiotis, Hierarchical competition models with the Allee effect III: Multispecies, J. Biological Dynamics, 12 (2018), 271-287.  doi: 10.1080/17513758.2018.1439537.  Google Scholar

[18]

S. Elaydi and R. J. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Diff. Equ. Appl., 208 (2005), 258-273.  doi: 10.1016/j.jde.2003.10.024.  Google Scholar

[19]

S. Elaydi and R. Sacker, Skew-Product Dynamical Systems: Applications to Difference Equations, 2004. Available from: http://www-bcf.usc.edu/ rsacker/pubs/UAE.pdf. Google Scholar

[20]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biological Dynamics, 4 (2010), 397-408.  doi: 10.1080/17513750903377434.  Google Scholar

[21]

N. Franco, L. Silva and P. Simões, Symbolic dynamics and renormalization of non-autonomous $k$-periodic dynamical systems, J. Diff. Equ. Appl., 19 (2013), 27-38.  doi: 10.1080/10236198.2011.611804.  Google Scholar

[22]

J. E. Franke and A.-A. Yakubu, Population models with periodic recruitment functions and survival rates, J. Diff. Equ. Appl., 11 (2005), 1169-1184.  doi: 10.1080/10236190500386275.  Google Scholar

[23]

R. Kempf, On $\Omega$-limit sets of discrete-time dynamical systems, J. Diff. Equ. Appl., 8 (2002), 1121-1131.  doi: 10.1080/10236190290029024.  Google Scholar

[24]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[25]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, AMS, Providence, RI, 2011. doi: 10.1090/surv/176.  Google Scholar

[26]

S. F. Kolyada, On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems, 12 (1992), 749-768.  doi: 10.1017/S0143385700007082.  Google Scholar

[27]

J. P. La Salle, The stability of dynamical systems, in CBMS-NSf Regional Conference Series in Applied Mathematics, Siam, (1976). doi: 10.1137/1.9781611970432.  Google Scholar

[28]

V. Jiménez López and J. Smítal, $\omega$-limit sets for triangular mappings, Fundamenta Math., 167 (2001), 1-15.  doi: 10.4064/fm167-1-1.  Google Scholar

[29]

M. Mandelkern, Metrization of the one-point compactification, Proc. Amer. Math. Soc., 107 (1989), 1111-1115.  doi: 10.1090/S0002-9939-1989-0991703-4.  Google Scholar

[30]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Note in Mathematics, 2002. Springer-Verlag, Berlin 2010. doi: 10.1007/978-3-642-14258-1.  Google Scholar

[31]

G. Rangel, Global attractors in partial differential equations, in Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885–982. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[32]

L. Silva, Periodic attractors of nonautonomous flat-topped tent systems, Discrete Contin. Dyn. Syst. B, 24 (2019), 1867-1874.   Google Scholar

Figure 1.  The space $ \widehat {\cal F} = \left\{ {{f_n}:n = 0,1,2, \ldots } \right\} \cup \left\{ f \right\} $, where $ f_{n} \rightarrow f $, uniformly, as $ n \rightarrow \infty $. If $ x_{0} $ is on the fiber $ {\mathcal X}_{0} $, then $ f_{0}(x_{0}) = x_{1} $ is in the fiber $ {\mathcal X}_{1} $, and $ f_{1}(x_{1}) = x_{2} $ is on the fiber $ {\mathcal X}_{2} $, etc
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  This commuting diagram illustrates the notion of a skew product discrete semidynamical system. Here $ p $ is the projection map, $ p: X \times {\mathcal F} \rightarrow {\mathcal F} $, that is $ p(x,g) = g $, for each $ (x, g) \in X \times {\mathcal F} $, $ i_{d} $ is the identity map and $ \sigma $ is the shift map, where $ \sigma(f_{i}, n) = f_{i+n} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Beverton-Holt maps with $ {B}_{0} $, $ {B}_{1} $, $ {B}_{2} $, $ {B}_{3} $, $ {B}_{4} $, $ \dots $, with time-dependent $ {r}_{n} $ converging to the map $ B $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The phase space diagram of the 2-species hierarchical model with four interior fixed points and five fixed points on the axes
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[2]

Wen Huang, Jianya Liu, Ke Wang. Möbius disjointness for skew products on a circle and a nilmanifold. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3531-3553. doi: 10.3934/dcds.2021006
[3]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399
[4]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[5]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031
[6]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043
[7]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022
[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]

Ru Li, Guolin Yu. Strict efficiency of a multi-product supply-demand network equilibrium model. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2203-2215. doi: 10.3934/jimo.2020065
[10]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91
[11]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[12]

Serena Brianzoni, Giovanni Campisi. Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021116
[13]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012
[14]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013
[15]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006
[16]

Zhisong Chen, Shong-Iee Ivan Su. Assembly system with omnichannel coordination. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021047
[17]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[18]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[19]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
[20]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (295)
  • HTML views (292)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences