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2018 | OriginalPaper | Buchkapitel

2. One-Dimensional Nonlinear Cobweb Model

verfasst von : Tamotsu Onozaki

Erschienen in: Nonlinearity, Bounded Rationality, and Heterogeneity

Verlag: Springer Japan

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Abstract

This chapter provides an introductory exposition of nonlinear discrete dynamics, which takes readers into the vast ocean outside the equilibrium paradigm, by presenting a one-dimensional nonlinear cobweb model. The model is a simple extension of the standard cobweb model. The new ingredients are an iso-elastic inverse demand function (i.e., an inverse demand function with constant price elasticity) and boundedly rational producers who gradually adjust their production toward the target levels based upon naive price expectations. Thus, the key parameters of the model are the price elasticity of demand and the production adjustment speed of producers. With the aid of mathematical analysis and numerical simulations, it is shown that for a large set of parameter values, the cobweb market exhibits observable chaos (a strange attractor) as well as topological chaos (a horseshoe) associated with homoclinic points.

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The content of this chapter is mainly based upon Onozaki et al. (2000) and Onozaki and Sawada (2001) although we present a new proof of Proposition 2.1 rather than the old one in Onozaki and Sawada (2001). The usual way in which to introduce a nonlinear discrete dynamical system depends upon the famous logistic map $f(x) = \mu \;\!x\,(1-x\;\!)$, where $x \in [0,1]$ and $\mu \in (0,4]$. On the contrary, we use our original model in this monograph. See, for example, Devaney (1989), Lorenz (1993), Alligood et al. (1996), Holmgren (1996), Elaydi (2007), and Hommes (2013) for an exposition of discrete nonlinear dynamics with the aid of the logistic map.

On the early development of the cobweb model, see Ezekiel (1938).

Although the model under consideration is one-dimensional, trajectories in Fig. 2.1 are drawn on a plane for the convenience of better understanding.

Although the spelling “repeller” is more frequently used than “repellor”, we use the latter throughout this book to emphasize the resemblance to the spelling of the antonym “attractor.”

For proofs of the theorems in this subsection, see Devaney (1989), Alligood et al. (1996), Martelli (1999), or Elaydi (2007).

The Taylor series is an approximative representation of an infinitely differentiable continuous function f(x) at a point a such that

$$\begin{aligned} f(x) \approx f(a) + \frac{f^{\prime }(a)}{1!}(x-a) + \frac{f^{\prime \prime }(a)}{2!}(x-a)^{2} + \frac{f^{\prime \prime \prime }(a)}{3!}(x-a)^{3} + \cdots . \end{aligned}$$

The cobweb model in economics and the cobweb diagram in dynamical analysis happen to have the same term “cobweb” because of the appearance of the graphs used in both.

See Finkenstädt (1995) for the chaotic price movements of eggs, potatoes, and pigs in northern Germany.

For a survey of adaptive behaviors, see Day (2001).

The model becomes static (i.e., all initial points are stationary) if $\phi =0$ and reduces to (2.7) if $\phi =1$. Thus, we assume $\phi \in \left( 0,1\right) $.

The proof of this theorem is given by substituting $g=f^{k}$ and applying the chain rule to Theorem 2.1.

With the aid of the Schwarzian derivative, we can establish an upper bound of the number of attracting periodic cycles that a certain map may have. Regarding the map (2.15), we can calculate

$$\begin{aligned} Sf(x)=-\frac{\ \phi \eta (1+\eta )\left( \phi \eta (\eta -1)+2(1-\phi )(2+\eta )x^{1+\eta } \right) \ }{2\left( (\phi -1)x^{2+\eta }+\phi \eta x\right) ^{2}}, \end{aligned}$$

which is negative if (2.18) is satisfied. Therefore, the map (2.15) has at most one periodic attractor. For more detail, see Devaney (1989), pp. 69ff.

For a detailed exposition of bifurcation theory, see, for example, Kuznetsov (2004).

For a proof of this theorem, see Devaney (1989) or Elaydi (2007).

For a proof of this theorem, see, for example, Elaydi (2007).

See, for example, Hommes (2013) for simple examples of saddle-node bifurcation and two more types of bifurcation.

The other name, saddle-node bifurcation, comes from the fact that in two- or higher-dimensional cases, a “saddle” and a stable “node” are created by this type of bifurcation. See, for example, Kuznetsov (2004) for a detailed exposition of this fact. Since one-dimensional systems do not exhibit this creation but only create a pair of stable and unstable periodic cycles, it could be better to restrict the use of the term “tangent bifurcation” to one-dimensional cases. However, the mathematical conditions that characterize this type of bifurcation are essentially the same. Hence, we also use the term “saddle-node bifurcation” for one-dimensional maps in this book.

The notion of crisis was introduced by Grebogi et al. (1982). There are two other types of crises, namely boundary crisis, and attractor merging crisis (Ott (2002)). Boundary crisis is discussed in Sect. 3.3.2 with respect to Neimark–Sacker bifurcation. For a detailed exposition of crises, see, for example, Grebogi et al. (1982, 1983, 1987), Nayfeh and Balachandran (1995), Alligood et al. (1996), Hilborn (2000), and Ott (2002).

The following explanation of how crisis occurs follows Komuro (2005).

In Part I of this book, we introduce the mathematical notions necessary to understand nonlinear dynamical systems but not the basic notions of topology such as uncountable sets, closed sets, and compact sets. See, for example, Rudin (1976), Holmgren (1996), and Martelli (1999) for such basic notions.

For a proof of this theorem, see Li and Yorke (1975) or Elaydi (2007).

This extension is reasonable. For example, the map (2.15) with $\eta =2$ exhibits chaotic behavior even though it does not have a periodic point with period 3 but rather has a periodic point with period 5 around $\phi \approx 0.9691$. Note that five is the second highest number in the Sharkovskii ordering.

Although Li–Yorke’s theorem is only applicable to one-dimensional maps, it also applies to higher-dimensional systems such as

$$\begin{aligned} x_{t+1}= & {} f(x_{t}), \\ y_{t+1}= & {} g(x_{t}, y_{t}), \end{aligned}$$

where f is chaotic in the Li–Yorke sense.

Except in special cases, the topological entropy cannot be computed easily. Collet et al. (1983) and Block et al. (1989) propose computing topological entropy for the special case of unimodal maps of the interval. Here, we use the algorithm presented by Block et al. (1989). Hunt and Ott (2015) propose the notion of expansion entropy to offer a computationally feasible definition of chaos. This is equivalent to topological entropy under appropriate conditions; however, it can only be defined for smooth dynamical systems.

The notion of strange attractors is defined in Sect. 2.4.3.

In a similar fashion, we can construct the Cantor middle-$\alpha $ set for $0< \alpha < 1$.

For a proof of this theorem, see, for example, Holmgren (1996).

See, for example, Elaydi (2007) for detailed exposition of fractal.

For a proof of the statement that $(\sum _{2}^{+}, d)$ is a Cantor set, see, for example, Block and Coppel (1992). For a proof of the statement that $\sigma :\sum _{2}^{+} \rightarrow \sum _{2}^{+}$ is continuous, see, for example, Devaney (1989), Holmgren (1996), or Elaydi (2007).

For proofs of these statements, see, for example, Devaney (1989), Holmgren (1996), or Elaydi (2007).

See Footnote 23.

Although the results presented in this subsection are also applicable to the case where a map has a periodic point with slight modifications, we concentrate on the case where a map has a fixed point.

For the proofs of this and the next propositions, see Devaney (1989).

We discuss snap-back repellors in Sect. 3.4.8.

For a proof of this theorem, see, for example, Elaydi (2007).

For the proof of this theorem, see Devaney (1989) (Chap. 1, Theorem 16.5).

The notion of a horseshoe in higher-dimensional general case is introduced in Sect. 3.4.6.

Similar figures showing the homoclinic bifurcation of the logistic map are found in Panchuk (2016).

For more details and the proof of this theorem, see, for example, Mora and Viana (1993) (Theorem C).

In mathematics, generic properties are properties that hold for “typical” examples.

Titel: One-Dimensional Nonlinear Cobweb Model
verfasst von: Tamotsu Onozaki
Verlag: Springer Japan
Buch: Nonlinearity, Bounded Rationality, and Heterogeneity
Print ISBN: 978-4-431-54970-3

Electronic ISBN: 978-4-431-54971-0

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-4-431-54971-0_2

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