The structure of <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets of asymptotically non-autonomous discrete dynamical systems

Emma D'Aniello; Saber Elaydi

doi:10.3934/dcdsb.2019195

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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 March 2020 Early access 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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.
[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.
[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.
[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.
[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.
[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.
[15]	J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass, 1966.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	R. Kempf, On $\Omega$-limit sets of discrete-time dynamical systems, J. Diff. Equ. Appl., 8 (2002), 1121-1131. doi: 10.1080/10236190290029024.
[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.
[25]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, AMS, Providence, RI, 2011. doi: 10.1090/surv/176.
[26]	S. F. Kolyada, On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems, 12 (1992), 749-768. doi: 10.1017/S0143385700007082.
[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.
[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.
[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.
[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.
[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.
[32]	L. Silva, Periodic attractors of nonautonomous flat-topped tent systems, Discrete Contin. Dyn. Syst. B, 24 (2019), 1867-1874.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755316.
[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.
[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.
[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.
[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.
[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.
[15]	J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass, 1966.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[23]	R. Kempf, On $\Omega$-limit sets of discrete-time dynamical systems, J. Diff. Equ. Appl., 8 (2002), 1121-1131. doi: 10.1080/10236190290029024.
[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.
[25]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, AMS, Providence, RI, 2011. doi: 10.1090/surv/176.
[26]	S. F. Kolyada, On dynamics of triangular maps of the square, Ergodic Theory Dynam. Systems, 12 (1992), 749-768. doi: 10.1017/S0143385700007082.
[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.
[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.
[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.
[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.
[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.
[32]	L. Silva, Periodic attractors of nonautonomous flat-topped tent systems, Discrete Contin. Dyn. Syst. B, 24 (2019), 1867-1874.

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

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