Abstract
In this paper, we investigate the following discrete-time map:
We introduce a novel method to determine the stability of the given two-dimensional map by using a one-dimensional map. A cobweb-like diagram is also introduced in order to analyze the stability of the system. We show that the stability of a fixed point in cobweb diagram implies the stability in phase diagram for the given system.
In addition, an application of the system to a non-hyperbolic fixed point is also given.
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References
Elaydi, S.: An Introduction to Difference Equations. Springer, New York (2000)
Elaydi, S.: Discrete Chaos: With Applications in Science and Engineering, 2nd edn. Chapman & Hall/CRC, Boca Raton (2008)
Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, vol. 13046. Addison-Wesley, Reading (1989)
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1995). ISBN 0-387-94418-4
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A Related Lemmas/Theorems
A Related Lemmas/Theorems
Theorem 3
Let z be an attracting fixed point of a continuous map \(f:I\rightarrow \mathbb {R}\), where I is an interval. Then z is stable.
Proof of the theorem can be found in Elaydi (2008).
Lemma 2
Let \(f:\mathbb {R}\rightarrow \mathbb {R}\) and \(g:\mathbb {R}\rightarrow \mathbb {R}\) are continuous functions and \(f(\bar{x})=g(\bar{x})=\bar{y}\) for some \(\bar{x}\in \mathbb {R}\). Assume that one of the following conditions is satisfied for all real \(\alpha > 0\):
-
(1)
\(f(\bar{x}-\alpha )<g(\bar{x}-\alpha )<\bar{y}<g(\bar{x}+\alpha )<f(\bar{x}+\alpha )\)
-
(2)
\(f(\bar{x}+\alpha )<g(\bar{x}+\alpha )<\bar{y}<g(\bar{x}-\alpha )<f(\bar{x}-\alpha )\)
-
(3)
\(f(\bar{x}-\alpha )<g(\bar{x}+\alpha )<\bar{y}<g(\bar{x}-\alpha )<f(\bar{x}+\alpha )\)
-
(4)
\(f(\bar{x}+\alpha )<g(\bar{x}-\alpha )<\bar{y}<g(\bar{x}+\alpha )<f(\bar{x}-\alpha )\)
Then, for any \(x_0\), \((f^{-1}\circ g)^n(x_0)\rightarrow \bar{x}\) as \(n\rightarrow \infty \), provided \(f^{-1}\) exists.
Theorem 4
Given the discrete dynamical system
where \(f,g:\mathbb {R}\rightarrow \mathbb {R}\) are continuous functions. Assume that \((\bar{x},\bar{y})\) is a fixed point of system (\(\star \)) and one of the following conditions is satisfied for all real \(\alpha >0\):
-
(1)
\(f(\bar{x}-\alpha )<g(\bar{x}-\alpha )<\bar{y}<g(\bar{x}+\alpha )<f(\bar{x}+\alpha )\)
-
(2)
\(f(\bar{x}+\alpha )<g(\bar{x}+\alpha )<\bar{y}<g(\bar{x}-\alpha )<f(\bar{x}-\alpha )\)
-
(3)
\(f(\bar{x}-\alpha )<g(\bar{x}+\alpha )<\bar{y}<g(\bar{x}-\alpha )<f(\bar{x}+\alpha )\)
-
(4)
\(f(\bar{x}+\alpha )<g(\bar{x}-\alpha )<\bar{y}<g(\bar{x}+\alpha )<f(\bar{x}-\alpha )\)
Then \((\bar{x},\bar{y})\) is globally asymptotically stable on \(\mathbb {R}^2\).
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Kapçak, S. (2016). Stability of a Certain 2-Dimensional Map with Cobweb Diagram. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2016. ICCSA 2016. Lecture Notes in Computer Science(), vol 9786. Springer, Cham. https://doi.org/10.1007/978-3-319-42085-1_4
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DOI: https://doi.org/10.1007/978-3-319-42085-1_4
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