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Transitions between metastable long-run consumption behaviors in a stochastic peer-driven consumer network

  • * Corresponding author: Jochen Jungeilges

    * Corresponding author: Jochen Jungeilges 

The authors want to thank two anonymous referees for there work.

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  • We study behavioral change - as a transition between coexisting attractors - in the context of a stochastic, non-linear consumption model with interdependent agents. Relying on the indirect approach to the analysis of a stochastic dynamic system, and employing a mix of analytical, numerical and graphical techniques, we identify conditions under which such transitions are likely to occur. The stochastic analysis depends crucially on the stochastic sensitivity function technique as it can be applied to the stochastic analoga of closed invariant curves [14], [1]. We find that in a moderate noise environment increased peer influence actually reduces the complexity of observable long-run consumer behavior.

    Mathematics Subject Classification: Primary: 37G35, 37H20; Secondary: 37N40.

    Citation:

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  • Figure 1.  Bird's eye view of the parameter plane $ D $, where remaining parameters have been fixed at $ (p_x, p_y) = \left(\frac{1}{4}, 1\right), \ $$ (b_1, b_2) = (10,20), \ \alpha_1 = 0.0002, \ \alpha_2 = 0.00052$

    Figure 2.  Bifurcation diagram for $ D^N $ with $ (p_x, p_y) = \left(\frac{1}{4}, 1\right), \ (b_1, b_2) = (10,20), \ \alpha_1 = 0.0002, \ \alpha_2 = 0.00052 $. $ NS $ indicates the Neimark-Sacker bifurcation curve related to the fixed point. $ SN_3 $ curve gives the loci at which a saddle 3-cycle is born together with the attracting 3-cycle ($ C_3 $) via a saddle-node bifurcation. $ NS_3 $ designates the Neimark-Sacker bifurcation curve of the 3-cycle. The horizontal line through $ D_{21} = 0.0075 $ indicates the interval of parameter values for which our study of transitions between coexisting attractors focuses on. The $ NS $ and $ NS_3 $ curves are crossed twice at $ \star $ (red star) and $ \star $ (green star). Also the saddle node bifurcation curve $ SN_3 $ is intersected twice. The intersection points are indicated by $ \bullet $ (blue circles). Related details are revealed in Figure 3

    Figure 3.  For $ D_{21} = 0.0075 $, we give bifurcation diagrams for $ 0 \leq D_{12} \leq 0.00245 $ linked to the horizontal black line in Figure 2(a) and an enlargement (b) focussing on the interval $ 0.00145 \leq D_{12} \leq 0.001975 $ over which two attractors coexists

    Figure 4.  Bifurcation diagram for the case of additive noise with $ \varepsilon = 0.1 $ ($ D_{21} = 0.0075 $). If the initial value $ (x_{1,0},x_{2,0}) $ lies on the deterministic blue (red) attractor, then elements of the trajectory are colored light blue (red)

    Figure 5.  Bifurcation diagram for the case of parametric noise with $ \varepsilon = 0.1 $ ($ D_{21} = 0.0075 $). If $ (x_{1,0},x_{2,0}) $ lies on the deterministic blue (red) attractor, then elements of the trajectory are colored light blue (red)

    Figure 6.  Confidence sets for fixed point $ E $ ($ \bullet $) and 3-cycle $ C_3 $ ($ \bullet $) at $ D_{12} = 0.00195 $, $ D_{21} = 0.0075 $ with trajectories superimposed ($ \varepsilon = 0.1 $ (white), $ \varepsilon = 0.05 $ (grey))

    Figure 7.  The top panel shows the graph of the sensitivity function for $ \Gamma $, i.e. a plot of the maximum eigenvalue ($ \lambda $) of the sensitivity matrix at a point on $ \Gamma $ versus the angle $ \phi $ identifying the point on the attractor. The subfigures on the bottom give the confidence sets $ \mathcal{C}(\Gamma, \varepsilon = 0.1) $ at $ D_{12} = 0.00157 $ for additive (a) and parametric noise (b)

    Figure 8.  The figure shows the attractor $ \Gamma_3 $ at $ D_{12} = 0.0017 $ (a), the sensitivity functions for $ \Gamma_3 $ (b) as well as the related confidence sets $ \mathcal{C}(\Gamma_3, \varepsilon = 0.1) $ for additive (c) and parametric noise (d)

    Figure 9.  1D bifurcation diagrams ((a),(b)) and critical intensities for coexisting attractors (c) with additive (solid lines) and parametric (dashed lines) noise for $ D_{12} \in D^{ms} $

    Figure 10.  For $ (D_{12}, D_{21}) = (0.001706,0.0075) $ we show the state space representation of the coexisting attractors $ \Gamma_3 $ (dark red curves) and $ \Gamma $ (blue curve) together with their immediate basins $ \mathcal{B}(\Gamma_3) $ (light red) and $ \mathcal{B}(\Gamma) $ (light blue). The confidence sets are superimposed ($ \varepsilon \in \{ 0.1, 0.2, 0.3\} $). Periodic points (red triangles) of the 3-saddle cycle are exhibited together with its stable (black lines) and unstable (red lines) manifolds. In addition, the unstable fixed point $ E $ (blue circle) and the unstable 3-cylce (periodic point given by red circles) are given

    Figure 11.  ( $ \Gamma $, $ \Gamma_3 $) at $ (D_{12}, D_{21}) = (0.001706,0.0075) $ with sample trajectory (single simulation run with $ \varepsilon = 0.2 $) superimposed

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