# American Institute of Mathematical Sciences

November  2021, 26(11): 5941-5964. doi: 10.3934/dcdsb.2021117

## Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure

 1 Institute of Mathematics NASU, 3 Tereshchenkivska Str., 01601 Kyiv, Ukraine 2 Department of Economics, University of Bamberg, Feldkirchenstrasse 21, 96045 Bamberg, Germany

* Corresponding author: anastasiia.panchuk@gmail.com

Received  October 2020 Revised  February 2021 Published  November 2021 Early access  April 2021

We study a simple financial market model with interacting chartists and fundamentalists that may give rise to multiband chaotic attractors. In particular, asset prices fluctuate erratically around their fundamental values, displaying a significant bull and bear market behavior. An in-depth analytical and numerical study of our model furthermore reveals the emergence of a new bifurcation structure, a phenomenon that we call a bandcount accretion bifurcation structure. The latter consists of regions associated with chaotic dynamics only, the boundaries of which are not defined by homoclinic bifurcations, but mainly by contact bifurcations of particular type where two distinct critical points of certain ranks coincide.

Citation: Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5941-5964. doi: 10.3934/dcdsb.2021117
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(a) Schematic representation of the $( \varepsilon, \mu)$ parameter plane; (b)–(g) Sample plots of function $f$ and its absorbing intervals
(a) 2D bifurcation diagram in the $( \varepsilon, \mu)$ parameter plane of $f$, with $a = 1.25$. (b) Close up of the rectangular area marked in (a)
1D bifurcation diagram corresponding to the blue arrow in Figure 2(a), with $\varepsilon = 0.72$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively
Plot of function $f$ and formation of gaps in the upper part $B^R$ of the chaotic attractor $\mathcal{Q}$
Plot of function $f$ and formation of gaps in the lower part $B^L$ of the chaotic attractor $\mathcal{Q}$
(a) 2D bifurcation diagram in the $( \varepsilon, \mu)$ parameter plane of $f$, with $a = 1.25$, related to a triangular shaped "jut" of chaoticity region $\mathcal{C}_{2+1}$. (b) Close up of the rectangular area marked in (a)
1D bifurcation diagram corresponding to the blue arrow marked "1" in Figure 6(b), with $\varepsilon = 1.37$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively
Plot of function $f$, with $a = 1.25$, $\varepsilon = 1.37$ and $\mu = \tilde{\mu} = -3.44$. Pink and dark-red lines show the related sequences of preimages of zero
Plot of function $f$, with $a = 1.25$, $\varepsilon = 1.37$ and $\mu = \tilde{\mu} = -3.454$. Pink and dark-red lines show the related sequences of preimages of zero
1D bifurcation diagram corresponding to the blue arrow marked "2" in Figure 6(b), with $\varepsilon = 1.34$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively
1D bifurcation diagram corresponding to the blue arrow marked "3" in Figure 6(b), with $\varepsilon = 1.32$ and $a = 1.25$. The critical points $\ell_{{i}}$, $\mathit{m}^{-}_{{i}}$, $\mathit{m}^{+}_{{i}}$ and $\mathit{r}_{{i}}$ of different ranks are shown by blue, green, orange and red lines, respectively
The functioning of the financial market model. The panel depicts the dynamics of Figure 9 in the time domain. The parameters are $\varepsilon = 1.37$, $a = 1.25$, and $\mu = -3.454$
(a) 2D bifurcation diagram in the $( \varepsilon, \mu)$ parameter plane of $f$, with $a = 1.1$. (b) Close up of the parallelogram area marked by the dark-red line in (a)
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