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Global dynamic scenarios in a discrete-time model of renewable resource exploitation: a mathematical study

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Abstract

We consider the two-dimensional map introduced in Bischi et al. (J Differ Equ Appl 21(10):954–973, 2015) formulated as a model for a renewable resource exploitation process in an evolutionary setting. The global dynamic scenarios displayed by the model are not so often encountered in smooth two-dimensional dynamical systems. We explain the occurrence of such scenarios at the light of the theory of noninvertible maps. Moreover, complex structures of basins of attraction of coexisting invariant sets are observed. We analyze such structures by examining stability properties of chaotic sets, in the case in which a non-topological Milnor attractor is present. Stability changes of a chaotic set occur through global bifurcations (such as riddling and blowout) and are detected by means of the study of the spectrum of Lyapunov exponents associated with the set.

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Notes

  1. Note that in general \(LC_{-1}\) is only included in the set of points at which the determinant of the Jacobi matrix vanishes.

  2. For a mixed chaotic area, the stable multipliers of the related saddle cycles must be positive, so that to prevent the points jumping outside this area (see [32]).

  3. An attractor \(A_i\) of the restriction \(f_{r=i}\) is stable with respect to perturbations along \(\mathcal {M}_i\).

  4. We recall that \(a_0^2q_0-a_1^2q_1<0\).

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Authors and Affiliations

  1. Department of Economics, Statistics and Finance, University of Calabria, Rende, Italy

    Lorenzo Cerboni Baiardi

  2. Institute of Mathematics, NAS of Ukraine, Kiev, Ukraine

    Anastasiia Panchuk

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  1. Lorenzo Cerboni Baiardi
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  2. Anastasiia Panchuk
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Correspondence to Anastasiia Panchuk.

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Cerboni Baiardi, L., Panchuk, A. Global dynamic scenarios in a discrete-time model of renewable resource exploitation: a mathematical study. Nonlinear Dyn 102, 1111–1127 (2020). https://doi.org/10.1007/s11071-020-05898-8

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