November  2021, 26(11): 5849-5871. doi: 10.3934/dcdsb.2021232

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

1. 

University of Agder, School of Business and Law, Department of Economics and Finance, Servicebox 422, N-4604 Kristiansand S, Norway

2. 

Ural Federal University, Institute of Natural Science and Mathematics, 51 Lenin Avenue, Ekaterinburg 620000, Russian Federation

3. 

Ural Federal University, Institute of Natural Science and Mathematics, Ural Mathematical Center, 51 Lenin Avenue, Ekaterinburg 620000, Russian Federation

* Corresponding author: Jochen Jungeilges

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

Received  November 2020 Revised  July 2021 Published  November 2021 Early access  September 2021

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.

Citation: Jochen Jungeilges, Trygve Kastberg Nilssen, Tatyana Perevalova, Alexander Satov. Transitions between metastable long-run consumption behaviors in a stochastic peer-driven consumer network. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5849-5871. doi: 10.3934/dcdsb.2021232
References:
[1]

I. Bashkirtseva and L. Ryashko, Stochastic sensitivity of the closed invariant curves for discrete-time systems, Phys. A, 410 (2014), 236-243.  doi: 10.1016/j.physa.2014.05.037.  Google Scholar

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I. BashkirtsevaL. Ryashko and A. Sysolyatina, Analysis of stochastic effects in Kaldor-type business cycle discrete model, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 446-456.  doi: 10.1016/j.cnsns.2015.12.020.  Google Scholar

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H. W. Broer, M. Golubitsky and G. Vegter, Geometry of resonance tongues, Singularity Theory, 327–356, World Sci. Publ., Hackensack, NJ, (2007). https://www.researchgate.net/publication/252963138_Geometry_of_resonance_tongues doi: 10.1142/9789812707499_0012.  Google Scholar

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E. EkaterinchukJ. JungeilgesT. Ryazanova and I. Sushko, Dynamics of a minimal consumer network with bi-directional influence, Commun. Nonlinear Sci. Numer. Simul., 58 (2018), 107-118.  doi: 10.1016/j.cnsns.2017.04.007.  Google Scholar

[6]

E. EkaterinchukJ. JungeilgesT. Ryazanova and I. Sushko, Dynamics of a minimal consumer network with uni-directional influence, Journal of Evolutionary Economics, 27 (2017), 831-857.  doi: 10.1007/s00191-017-0517-5.  Google Scholar

[7]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 3rd edition, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.  Google Scholar

[8]

W. Gaertner and J. Jungeilges, A non-linear model of interdependent consumer behaviour, Economics Letters, 27 (1988), 145-150.  doi: 10.1016/0165-1765(88)90087-0.  Google Scholar

[9]

W. Gaertner and J. Jungeilges, "Spindles" and coexisting attractors in a dynamic model of interdependent consumer behavior: A note, Journal of Economic Behavior & Organization, 21 (1993), 223-231.  doi: 10.1016/0167-2681(93)90049-U.  Google Scholar

[10]

J. Jungeilges, E. Maklakova and T. Perevalova, Stochastic sensitivity of bull and bear states, Journal of Economic Interaction and Cooperation, (2021). doi: 10.1007/s11403-020-00313-2.  Google Scholar

[11]

J. Jungeilges and T. Ryazanova, Transitions in consumption behaviors in a peer-driven stochastic consumer network, Chaos Solitons Fractals, 128 (2019), 144-154.  doi: 10.1016/j.chaos.2019.07.042.  Google Scholar

[12]

J. Jungeilges, T. Ryazanova, A. Mitrofanova and I. Popova, Sensitivity analysis of consumption cycles, Chaos, 28 (2018), 055905, 12 pp. doi: 10.1063/1.5024033.  Google Scholar

[13]

Z. Li, K. Guo, J. Jiang and L. Hong, Study on critical conditions and transient behavior in noise-induced bifurcations, Control of Self-Organizing Nonlinear Systems, 169–187, Underst. Complex Syst., Springer, [Cham], (2016). doi: 10.1007/978-3-319-28028-8_9.  Google Scholar

[14]

G. Mil'shtein and L. Ryashko, The first approximation in the quasipotential problem of stability of non-degenerate systems with random perturbations, Journal of Applied Mathematics and Mechanics, 59 (1995), 47-56.   Google Scholar

[15]

A. Panchuk, CompDTIMe: Computing one-dimensional invariant manifolds for saddle points of discrete time dynamical systems, Gecomplexity Discussion Paper Series 11, Action IS1104 "The EU in the new complex geography of economic systems: Models, tools and policy evaluation", 2015, https://EconPapers.repec.org/RePEc:cst:wpaper:11. Google Scholar

[16]

L. Ryashko, Noise-induced transformations in corporate dynamics of coupled chaotic oscillators, Mathematical Methods in the Applied Sciences. doi: 10.1002/mma.6578.  Google Scholar

[17]

A. N. SilchenkoS. BeriD. G. Luchinsky and P. V. E. McClintock, Fluctuational transitions through a fractal basin boundary, Phys. Rev. Lett., 91 (2003), 174104.  doi: 10.1103/PhysRevLett.91.174104.  Google Scholar

[18]

E. Slepukhina, L. Ryashko and P. Kügler, Noise-induced early afterdepolarizations in a three-dimensional cardiac action potential model, Chaos, Solitons & Fractals, 131 (2020), 109515. doi: 10.1016/j.chaos.2019.109515.  Google Scholar

[19]

Y. TadokoroH. Tanaka and M. I. Dykman, Noise-induced switching from a symmetry-protected shallow metastable state, Scientific Reports, 10 (2020), 1-10.   Google Scholar

[20]

J. Xu, T. Zhang and K. Song, A stochastic model of bacterial infection associated with neutrophils, Appl. Math. Comput., 373 (2020), 125025, 12 pp. doi: 10.1016/j.amc.2019.125025.  Google Scholar

[21]

Z. T. ZhusubaliyevE. Soukhoterin and E. Mosekilde, Quasiperiodicity and torus breakdown in a power electronic dc/dc converter, Math. Comput. Simulation, 73 (2007), 364-377.  doi: 10.1016/j.matcom.2006.06.021.  Google Scholar

show all references

References:
[1]

I. Bashkirtseva and L. Ryashko, Stochastic sensitivity of the closed invariant curves for discrete-time systems, Phys. A, 410 (2014), 236-243.  doi: 10.1016/j.physa.2014.05.037.  Google Scholar

[2]

I. BashkirtsevaL. Ryashko and A. Sysolyatina, Analysis of stochastic effects in Kaldor-type business cycle discrete model, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 446-456.  doi: 10.1016/j.cnsns.2015.12.020.  Google Scholar

[3]

J. Benhabib and R. H. Day, Rational choice and erratic behaviour, Rev. Econom. Stud., 48 (1981), 459-471.  doi: 10.2307/2297158.  Google Scholar

[4]

H. W. Broer, M. Golubitsky and G. Vegter, Geometry of resonance tongues, Singularity Theory, 327–356, World Sci. Publ., Hackensack, NJ, (2007). https://www.researchgate.net/publication/252963138_Geometry_of_resonance_tongues doi: 10.1142/9789812707499_0012.  Google Scholar

[5]

E. EkaterinchukJ. JungeilgesT. Ryazanova and I. Sushko, Dynamics of a minimal consumer network with bi-directional influence, Commun. Nonlinear Sci. Numer. Simul., 58 (2018), 107-118.  doi: 10.1016/j.cnsns.2017.04.007.  Google Scholar

[6]

E. EkaterinchukJ. JungeilgesT. Ryazanova and I. Sushko, Dynamics of a minimal consumer network with uni-directional influence, Journal of Evolutionary Economics, 27 (2017), 831-857.  doi: 10.1007/s00191-017-0517-5.  Google Scholar

[7]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 3rd edition, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-25847-3.  Google Scholar

[8]

W. Gaertner and J. Jungeilges, A non-linear model of interdependent consumer behaviour, Economics Letters, 27 (1988), 145-150.  doi: 10.1016/0165-1765(88)90087-0.  Google Scholar

[9]

W. Gaertner and J. Jungeilges, "Spindles" and coexisting attractors in a dynamic model of interdependent consumer behavior: A note, Journal of Economic Behavior & Organization, 21 (1993), 223-231.  doi: 10.1016/0167-2681(93)90049-U.  Google Scholar

[10]

J. Jungeilges, E. Maklakova and T. Perevalova, Stochastic sensitivity of bull and bear states, Journal of Economic Interaction and Cooperation, (2021). doi: 10.1007/s11403-020-00313-2.  Google Scholar

[11]

J. Jungeilges and T. Ryazanova, Transitions in consumption behaviors in a peer-driven stochastic consumer network, Chaos Solitons Fractals, 128 (2019), 144-154.  doi: 10.1016/j.chaos.2019.07.042.  Google Scholar

[12]

J. Jungeilges, T. Ryazanova, A. Mitrofanova and I. Popova, Sensitivity analysis of consumption cycles, Chaos, 28 (2018), 055905, 12 pp. doi: 10.1063/1.5024033.  Google Scholar

[13]

Z. Li, K. Guo, J. Jiang and L. Hong, Study on critical conditions and transient behavior in noise-induced bifurcations, Control of Self-Organizing Nonlinear Systems, 169–187, Underst. Complex Syst., Springer, [Cham], (2016). doi: 10.1007/978-3-319-28028-8_9.  Google Scholar

[14]

G. Mil'shtein and L. Ryashko, The first approximation in the quasipotential problem of stability of non-degenerate systems with random perturbations, Journal of Applied Mathematics and Mechanics, 59 (1995), 47-56.   Google Scholar

[15]

A. Panchuk, CompDTIMe: Computing one-dimensional invariant manifolds for saddle points of discrete time dynamical systems, Gecomplexity Discussion Paper Series 11, Action IS1104 "The EU in the new complex geography of economic systems: Models, tools and policy evaluation", 2015, https://EconPapers.repec.org/RePEc:cst:wpaper:11. Google Scholar

[16]

L. Ryashko, Noise-induced transformations in corporate dynamics of coupled chaotic oscillators, Mathematical Methods in the Applied Sciences. doi: 10.1002/mma.6578.  Google Scholar

[17]

A. N. SilchenkoS. BeriD. G. Luchinsky and P. V. E. McClintock, Fluctuational transitions through a fractal basin boundary, Phys. Rev. Lett., 91 (2003), 174104.  doi: 10.1103/PhysRevLett.91.174104.  Google Scholar

[18]

E. Slepukhina, L. Ryashko and P. Kügler, Noise-induced early afterdepolarizations in a three-dimensional cardiac action potential model, Chaos, Solitons & Fractals, 131 (2020), 109515. doi: 10.1016/j.chaos.2019.109515.  Google Scholar

[19]

Y. TadokoroH. Tanaka and M. I. Dykman, Noise-induced switching from a symmetry-protected shallow metastable state, Scientific Reports, 10 (2020), 1-10.   Google Scholar

[20]

J. Xu, T. Zhang and K. Song, A stochastic model of bacterial infection associated with neutrophils, Appl. Math. Comput., 373 (2020), 125025, 12 pp. doi: 10.1016/j.amc.2019.125025.  Google Scholar

[21]

Z. T. ZhusubaliyevE. Soukhoterin and E. Mosekilde, Quasiperiodicity and torus breakdown in a power electronic dc/dc converter, Math. Comput. Simulation, 73 (2007), 364-377.  doi: 10.1016/j.matcom.2006.06.021.  Google Scholar

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 3">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 2(a) and an enlargement (b) focussing on the interval $ 0.00145 \leq D_{12} \leq 0.001975 $ over which two attractors coexists">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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