Asymptotic Stability for Block Triangular Maps
DOI:
https://doi.org/10.5644/SJM.18.01.03Keywords:
Attractor, basin of attraction, difference equation, discrete dynamical system, global asymptotic stability, triangular mapAbstract
We prove a result concerning the asymptotic stability and the basin of attraction of fixed points for block triangular maps in $\mathbb{R}^n$. This result is applied to some families of discrete dynamical systems and several types of difference equations.
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References
M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comput., 176 (2006), 768--774.
A.M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, Part I., J. Difference Equations and Applications, 3 (2008), 1--35.
A.M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, Part II, J. Difference Equations and Applications, 3 (2008), 195--225.
A. Andruch-Sobilo, M. Migda, On rational recursive sequence $x_{n+1}=frac{a x_{n-1}}{b+c x_nx_{n-1}}$, Opuscula Math. 26 (2006), 387--394.
I. Bajo, E. Liz, Global behaviour of a second--order nonlinear difference equation, J. Difference Equations and Applications, 17 (2011), 1471--1486.
E. Cabral Balreira, S. Elaydi, R. Luis, Global dynamics of triangular maps, Nonlinear Anal, 104 (2014), 75--83.
A. Cima, A. Gasull, F. Mañosas, A polynomial class of Markus-Yamabe counterexamples, Publ. Mat., 41 (1997), 85--100.
A. Cima, A. Gasull, V. Mañosas, Dynamics of some rational discrete dynamical systems via invariants, Int. J. Bifurcations and Chaos, 16 (2006), 631--645.
A. Cima, A. Gasull, V. Mañosa, Basin of attraction of triangular maps with applications, J. Difference Equations and Applications, 20, 3 (2014), 423--437.
A. Cima, A. Gasull, F. Mañosas, On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition, Publ. Mat., 58 (2014), suppl., 167--178
C. Çinar, On the solutions of the difference equation $x_{n+1}=frac{ x_{n-1}}{-1+a x_nx_{n-1}}$, Appl. Math. Comput., 158 (2004), 793--797.
E. D'Aniello, S. Elaydi, The structure of $omega$-limit sets of asymptotically non-autonomous discrete dynamical systems, Discrete Contin. Dyn. Syst. B, 25 (2020), 903--915.
T.F. Ibrahim, On the third order rational difference equation $x_{n+1}=frac{x_nx_{n-2}}{x_{n-1}(a+bx_nx_{n-2})}$, Int. J. Contemp. Math. Sciences, 4 (2009), 1321--1334.
M. Krstic, I. Kanellakopoulos, P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995.
M.R.S. Kulenović, G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman and Hall/CRC, Boca Raton FL, 2001.
W. Rudin, Principles of Mathematical Analysis, 3rd ed. Mc.Graw--Hill, New York 1976.
J. Smítal, Why it is important to understand dynamics of triangular maps?, Journal of Difference Equations and Applications, 14 (2008), 597--606.
S. Stević, More on a rational recurrence relation, Appl. Math., E-Notes 4 (2004), 80--85.
