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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

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