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

  • * Corresponding author: anastasiia.panchuk@gmail.com

    * Corresponding author: anastasiia.panchuk@gmail.com 
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  • 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.

    Mathematics Subject Classification: Primary: 37E05, 37G35; Secondary: 37N40, 39A33.

    Citation:

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  • Figure 1.  (a) Schematic representation of the $ ( \varepsilon, \mu) $ parameter plane; (b)–(g) Sample plots of function $ f $ and its absorbing intervals

    Figure 2.  (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)

    Figure 3.  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

    Figure 4.  Plot of function $ f $ and formation of gaps in the upper part $ B^R $ of the chaotic attractor $ \mathcal{Q} $

    Figure 5.  Plot of function $ f $ and formation of gaps in the lower part $ B^L $ of the chaotic attractor $ \mathcal{Q} $

    Figure 6.  (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)

    Figure 7.  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

    Figure 8.  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

    Figure 9.  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

    Figure 10.  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

    Figure 11.  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

    Figure 12.  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 $

    Figure 13.  (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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