American Institute of Mathematical Sciences

Advanced
November  2020, 25(11): 4257-4276. doi: 10.3934/dcdsb.2020096

Extension, embedding and global stability in two dimensional monotone maps

Ahmad Al-Salman 1, , Ziyad AlSharawi 2,, and  Sadok Kallel 2,

1. 

Department of Mathematics, Sultan Qaboos University, P. O. Box 36, PC 123, Al-Khod, Sultanate of Oman
2. 

Department of Mathematics and Statistics, American University of Sharjah, P. O. Box 26666, University City, Sharjah, UAE

* Corresponding author: Ziyad AlSharawi

Received  July 2019 Revised  November 2019 Published  November 2020 Early access  April 2020

We consider the general second order difference equation $ x_{n+1} = F(x_n, x_{n-1}) $ in which $ F $ is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that $ F $ has a semi-convex compact invariant domain, then make an extension of $ F $ on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of $ F. $ Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.

Keywords: Invariant domain, embedding, monotone maps, global stability.
Mathematics Subject Classification: Primary: 39A30, 39A10; Secondary: 37C25.
Citation: Ahmad Al-Salman, Ziyad AlSharawi, Sadok Kallel. Extension, embedding and global stability in two dimensional monotone maps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4257-4276. doi: 10.3934/dcdsb.2020096
References:
[1]

R. Abu-Saris, Z. AlSharawi and M. B. H. Rhouma, The dynamics of some discrete models with delay under the effect of constant yield harvesting, Chaos Solitons Fractals, 54 (2013), 26-38.  doi: 10.1016/j.chaos.2013.05.008.  Google Scholar

[2]

Z. AlSharawi, A global attractor in some discrete contest competition models with delay under the effect of periodic stocking, Abstr. Appl. Anal., (2013), Art. ID 101649, 7 pp. doi: 10.1155/2013/101649.  Google Scholar

[3]

A. M. Amleh, E. Camouzis and G. Ladas, On second-order rational difference equation. Ⅰ, J. Difference Equ. Appl., 13 (2007), 969-1004.  doi: 10.1080/10236190701388492.  Google Scholar

[4]

A. M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation. Ⅱ, Int. J. Difference Equ., 3 (2008), 195-225.   Google Scholar

[5]

E. Camouzis and G. Ladas, Dynamics of Third-order Rational Difference Equations with Open Problems and Conjectures, Advances in Discrete Mathematics and Applications, vol. 5, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[6]

E. Camouzis and G. Ladas, When does local asymptotic stability imply global attractivity in rational equations?, J. Difference Equ. Appl., 12 (2006), 863-885.  doi: 10.1080/10236190600772663.  Google Scholar

[7]

W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.  doi: 10.1017/S030500410002990X.  Google Scholar

[8]

J.-L. Gouzé and K. P. Hadeler, Monotone flows and order intervals, Nonlinear World, 1 (1994), 23-34.   Google Scholar

[9]

E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2005.  Google Scholar

[10]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[11]

M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2002.  Google Scholar

[12]

M. R. S. Kulenović, G. Ladas, L. F. Martins and I. W. Rodrigues, The dynamics of $x_{n+1} = \frac{\alpha+\beta x_n}{A+Bx_n+Cx_{n-1}}$: Facts and conjectures, Comput. Math. Appl., 45 (2003), 1087–1099, 2003. doi: 10.1016/S0898-1221(03)00090-7.  Google Scholar

[13]

M. R. S. Kulenović, G. Ladas and W. S. Sizer, On the recursive sequence $x_{n+1} = (\alpha x_n+\beta x_{n-1})/(\gamma x_n+\delta x_{n-1})$, Math. Sci. Res. Hot-Line, 2 (1998), 1-16.   Google Scholar

[14]

M. R. S. Kulenović and O. Merino, A note on unbounded solutions of a class of second order rational difference equations, J. Difference Equ. Appl., 12 (2006), 777-781.  doi: 10.1080/10236190600734184.  Google Scholar

[15]

M. R. S. Kulenović and O. Merino, Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.  doi: 10.3934/dcdsb.2009.12.133.  Google Scholar

[16]

M. R. S. Kulenović and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2471-2486.  doi: 10.1142/S0218127410027118.  Google Scholar

[17]

W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971.  Google Scholar

[18]

G. Nyerges, A note on a generalization of Pielou's equation, J. Difference Equ. Appl., 14 (2008), 563-565.  doi: 10.1080/10236190801912316.  Google Scholar

[19]

H. Sedaghat, Nonlinear Difference Equations. Theory with Applications to Social Science Models, Mathematical Modelling: Theory and Applications, vol. 15, Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-94-017-0417-5.  Google Scholar

[20]

H. L. Smith, The discrete dynamics of monotonically decomposable maps, J. Math. Biol., 53 (2006), 747-758.  doi: 10.1007/s00285-006-0004-3.  Google Scholar

[21]

H. L. Smith, Global stability for mixed monotone systems, J. Difference Equ. Appl., 14 (2008), 1159-1164.  doi: 10.1080/10236190802332126.  Google Scholar

show all references

References:
[1]

R. Abu-Saris, Z. AlSharawi and M. B. H. Rhouma, The dynamics of some discrete models with delay under the effect of constant yield harvesting, Chaos Solitons Fractals, 54 (2013), 26-38.  doi: 10.1016/j.chaos.2013.05.008.  Google Scholar

[2]

Z. AlSharawi, A global attractor in some discrete contest competition models with delay under the effect of periodic stocking, Abstr. Appl. Anal., (2013), Art. ID 101649, 7 pp. doi: 10.1155/2013/101649.  Google Scholar

[3]

A. M. Amleh, E. Camouzis and G. Ladas, On second-order rational difference equation. Ⅰ, J. Difference Equ. Appl., 13 (2007), 969-1004.  doi: 10.1080/10236190701388492.  Google Scholar

[4]

A. M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation. Ⅱ, Int. J. Difference Equ., 3 (2008), 195-225.   Google Scholar

[5]

E. Camouzis and G. Ladas, Dynamics of Third-order Rational Difference Equations with Open Problems and Conjectures, Advances in Discrete Mathematics and Applications, vol. 5, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[6]

E. Camouzis and G. Ladas, When does local asymptotic stability imply global attractivity in rational equations?, J. Difference Equ. Appl., 12 (2006), 863-885.  doi: 10.1080/10236190600772663.  Google Scholar

[7]

W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.  doi: 10.1017/S030500410002990X.  Google Scholar

[8]

J.-L. Gouzé and K. P. Hadeler, Monotone flows and order intervals, Nonlinear World, 1 (1994), 23-34.   Google Scholar

[9]

E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2005.  Google Scholar

[10]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[11]

M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2002.  Google Scholar

[12]

M. R. S. Kulenović, G. Ladas, L. F. Martins and I. W. Rodrigues, The dynamics of $x_{n+1} = \frac{\alpha+\beta x_n}{A+Bx_n+Cx_{n-1}}$: Facts and conjectures, Comput. Math. Appl., 45 (2003), 1087–1099, 2003. doi: 10.1016/S0898-1221(03)00090-7.  Google Scholar

[13]

M. R. S. Kulenović, G. Ladas and W. S. Sizer, On the recursive sequence $x_{n+1} = (\alpha x_n+\beta x_{n-1})/(\gamma x_n+\delta x_{n-1})$, Math. Sci. Res. Hot-Line, 2 (1998), 1-16.   Google Scholar

[14]

M. R. S. Kulenović and O. Merino, A note on unbounded solutions of a class of second order rational difference equations, J. Difference Equ. Appl., 12 (2006), 777-781.  doi: 10.1080/10236190600734184.  Google Scholar

[15]

M. R. S. Kulenović and O. Merino, Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.  doi: 10.3934/dcdsb.2009.12.133.  Google Scholar

[16]

M. R. S. Kulenović and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2471-2486.  doi: 10.1142/S0218127410027118.  Google Scholar

[17]

W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971.  Google Scholar

[18]

G. Nyerges, A note on a generalization of Pielou's equation, J. Difference Equ. Appl., 14 (2008), 563-565.  doi: 10.1080/10236190801912316.  Google Scholar

[19]

H. Sedaghat, Nonlinear Difference Equations. Theory with Applications to Social Science Models, Mathematical Modelling: Theory and Applications, vol. 15, Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-94-017-0417-5.  Google Scholar

[20]

H. L. Smith, The discrete dynamics of monotonically decomposable maps, J. Math. Biol., 53 (2006), 747-758.  doi: 10.1007/s00285-006-0004-3.  Google Scholar

[21]

H. L. Smith, Global stability for mixed monotone systems, J. Difference Equ. Appl., 14 (2008), 1159-1164.  doi: 10.1080/10236190802332126.  Google Scholar

Figure 2.1.  In part (a) of this figure, we illustrate our notation of projecting a point $(x, y)$ to the boundary $\partial \Omega.$ In part (b), we illustrate the notion of putting the invariant domain $\Omega$ inside an Origami domain
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.2.  This figure shows the possible options for the boundary of a convex $\Omega$ and a possible ray extension. Part (a) is based on the assumption that $f$ is non-decreasing and Part (b) is based on the assumption that $f$ is non-increasing. Note that the missing quarter in Part (a) is due to the fact that we cannot have $f(\uparrow)$ and $F(\downarrow, \uparrow)$ at the same time. Similarly for the missing quarter of Part (b)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.3.  Grafting $\gamma_1$ along $\gamma_2$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.4.  The various cases for the boundary pieces of an origami domain
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.5.  A semi-convex domain is shown in Part (a). Next is shown a piece $ C_1\cup C_2 $ of the boundary of $ \Omega $ oriented positively, with $ f(t) = F(r(t)) $ changing monotonicity from $ C_1 $ to $ C_2 $. Parts (b) and (c) show two different ray extensions, while (d) shows the adjustment to make to these extensions, before filling the sector. As explained in the text, (b) is not a valid extension if monotonicity is to be preserved in the extension
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.6.  Extending from semi-convex domain to the origami domain reduces to extending over sectors, with cases labeled (a) through (d)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.7.  Filling a rectangle where $ F $ is known on three sides. Extending partially by $ \widetilde{F}_1 $ as shown in (b) then extending over the resulting sector
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.1.  Part (a) of this figure shows the invariant region $\Omega, $ while part (b) shows $T(\Omega)$ together with the extension through horizontal projections. The scale is missing to indicate the general form of the region when $0 < h < \min\{p, \frac{1}{2}\}.$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
[2]

M. R. S. Kulenović, Orlando Merino. A global attractivity result for maps with invariant boxes. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 97-110. doi: 10.3934/dcdsb.2006.6.97
[3]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601
[4]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451
[5]

Hans-Otto Walther. Contracting return maps for monotone delayed feedback. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 259-274. doi: 10.3934/dcds.2001.7.259
[6]

Dario Cordero-Erausquin, Alessio Figalli. Regularity of monotone transport maps between unbounded domains. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7101-7112. doi: 10.3934/dcds.2019297
[7]

Siniša Slijepčević. Stability of invariant measures. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345
[8]

Arno Berger, Roland Zweimüller. Invariant measures for general induced maps and towers. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3885-3901. doi: 10.3934/dcds.2013.33.3885
[9]

Augusto Visintin. Weak structural stability of pseudo-monotone equations. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763
[10]

PaweŁ Hitczenko, Georgi S. Medvedev. Stability of equilibria of randomly perturbed maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 369-381. doi: 10.3934/dcdsb.2017017
[11]

Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613
[12]

Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau. Global dynamics for symmetric planar maps. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2241-2251. doi: 10.3934/dcds.2013.33.2241
[13]

Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101
[14]

I. Baldomá, Àlex Haro. One dimensional invariant manifolds of Gevrey type in real-analytic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 295-322. doi: 10.3934/dcdsb.2008.10.295
[15]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[16]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
[17]

Xiangqing Zhao, Bing-Yu Zhang. Global controllability and stabilizability of Kawahara equation on a periodic domain. Mathematical Control & Related Fields, 2015, 5 (2) : 335-358. doi: 10.3934/mcrf.2015.5.335
[18]

Sabri Bensid, Jesús Ildefonso Díaz. On the exact number of monotone solutions of a simplified Budyko climate model and their different stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1033-1047. doi: 10.3934/dcdsb.2019005
[19]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013
[20]

Tibor Krisztin. The unstable set of zero and the global attractor for delayed monotone positive feedback. Conference Publications, 2001, 2001 (Special) : 229-240. doi: 10.3934/proc.2001.2001.229

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (184)
  • HTML views (260)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy