Stability of a Certain 2-Dimensional Map with Cobweb Diagram

Abstract

In this paper, we investigate the following discrete-time map:

$$\begin{aligned} \begin{aligned} x_{n+1}&= \phi (y_n), \\ y_{n+1}&= \psi (x_n).\\ \end{aligned} \end{aligned}$$

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.

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

$$\begin{aligned} \begin{aligned} x_{n+1}&= f^{-1}(y_n), \\ y_{n+1}&= g(x_n),\\ \end{aligned} \end{aligned}$$

(⋆)

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 $\mathbb {R}^2$.