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On bifurcations and local stability in 1-D nonlinear discrete dynamical systems

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Abstract

In this paper, a theory of bifurcations and local stability of fixed-points (or period-1 solutions) in one-dimensional nonlinear discrete dynamical systems is presented. The linearized discrete dynamical systems are discussed first, and the higher-order singularity and monotonic and oscillatory stability of fixed-points for one-dimensional nonlinear discrete dynamical systems are presented. The monotonic and oscillatory bifurcations of fixed-points (period-1 solutions) are presented. A few special examples in 1-dimensional maps are presented for a better understanding of the general theory for the stability and bifurcation of nonlinear discrete dynamical systems. Global analysis of period-2 motions for the sampled nonlinear discrete dynamical systems are carried out, and global illustrations of period-1 to period-2 solutions in the sampled nonlinear discrete dynamical systems are given.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1805, USA

    Albert C. J. Luo

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  1. Albert C. J. Luo
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Correspondence to Albert C. J. Luo.

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Luo, A.C.J. On bifurcations and local stability in 1-D nonlinear discrete dynamical systems. Int. J. Dynam. Control 9, 1–29 (2021). https://doi.org/10.1007/s40435-020-00632-z

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  • DOI: https://doi.org/10.1007/s40435-020-00632-z

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