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Dissecting shrimps: results for some one-dimensional physical models

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  • Gallas, Jason A.C.

Abstract

This paper describes how certain shrimp-like clusters of stability organize themselves in the parameter space of dynamical systems. Clusters are composed of an infinite affine-similar repetition of a basic elementary cell containing two primay noble points, a head and a tail, defining an axis of approximate symmetry. Knowledge of the axis and the skewness of the k-periodic main cell of k×2n cluster is enough to define the orientation of the whole cluster in space. Peculiarly simple directions along which shrimp-like clusters align are formed by the locus of doubly degenerate saddle zero multipliers corresponding to the main shrimp head. In addition, we report a family of models having the boundaries of all isoperiodic domains of stability totally degenerate and describe different aspects of their mathematical arrangement and some of their consequences for example, that shrimps are diffeomorphic copies of shrimps.

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  • Gallas, Jason A.C., 1994. "Dissecting shrimps: results for some one-dimensional physical models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 202(1), pages 196-223.
    • Handle: RePEc:eee:phsmap:v:202:y:1994:i:1:p:196-223
    DOI: 10.1016/0378-4371(94)90174-0
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    Cited by:

    1. de Souza, S.L.T. & Batista, A.M. & Baptista, M.S. & Caldas, I.L. & Balthazar, J.M., 2017. "Characterization in bi-parameter space of a non-ideal oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 224-231.
    2. Medeiros, E.S. & de Souza, S.L.T. & Medrano-T, R.O. & Caldas, I.L., 2011. "Replicate periodic windows in the parameter space of driven oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 982-989.
    3. Oliveira, Diego F.M. & Leonel, Edson D., 2014. "Statistical and dynamical properties of a dissipative kicked rotator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 498-514.
    4. Gallas, Jason A.C., 1995. "Units: Remarkable points in dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 222(1), pages 125-151.
    5. dos Santos, Vagner & Szezech Jr., José D. & Baptista, Murilo S. & Batista, Antonio M. & Caldas, Iberê L., 2016. "Unstable dimension variability structure in the parameter space of coupled Hénon maps," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 23-28.
    6. Rao, Xiao-Bo & Zhao, Xu-Ping & Chu, Yan-Dong & Zhang, Jian-Gang & Gao, Jian-She, 2020. "The analysis of mode-locking topology in an SIR epidemic dynamics model with impulsive vaccination control: Infinite cascade of Stern-Brocot sum trees," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).

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