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Title: Biparametric investigation of the general standard map: multistability and global bifurcations

Abstract

We investigate multistability and global bifurcations in the general standard map, a biparametric two-dimensional map. Departing from the conservative case of the map, we describe the evolution of periodic solutions and their basins of attraction as dissipation builds up, paying special attention on how the biparametric variation affects multistability. We examine general and specific phenomena and behavior for three distinct dynamical regimes, namely small, moderate, and large damping and different forcing amplitudes. Also, we report numerically the mechanism of global bifurcations associated to small chaotic attractors in the multistable system. Several global bifurcations are investigated as dissipation increases. Specifically, through the characterization of an interior, a merging and a boundary crisis, we study the crucial role played by fundamental hyperbolic invariant structures, such as unstable periodic orbits and their stable and unstable invariant manifolds, in the mechanisms by which the phase space is globally transformed.

Authors:
 [1];  [2]
  1. UNESP, Universidade Estatual Paulista (Brazil)
  2. Instituto Tecnológico de Aeronáutica (Brazil)
Publication Date:
OSTI Identifier:
22769259
Resource Type:
Journal Article
Journal Name:
Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 37; Journal Issue: 3; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; BIFURCATION; CHAOS THEORY; MATHEMATICAL SOLUTIONS; PHASE SPACE

Citation Formats

Sousa-Silva, Priscilla A., E-mail: priscilla.silva@sjbv.unesp.br, and Terra, Maisa O., E-mail: maisa@ita.br. Biparametric investigation of the general standard map: multistability and global bifurcations. United States: N. p., 2018. Web. doi:10.1007/S40314-017-0544-8.
Sousa-Silva, Priscilla A., E-mail: priscilla.silva@sjbv.unesp.br, & Terra, Maisa O., E-mail: maisa@ita.br. Biparametric investigation of the general standard map: multistability and global bifurcations. United States. doi:10.1007/S40314-017-0544-8.
Sousa-Silva, Priscilla A., E-mail: priscilla.silva@sjbv.unesp.br, and Terra, Maisa O., E-mail: maisa@ita.br. Sun . "Biparametric investigation of the general standard map: multistability and global bifurcations". United States. doi:10.1007/S40314-017-0544-8.
@article{osti_22769259,
title = {Biparametric investigation of the general standard map: multistability and global bifurcations},
author = {Sousa-Silva, Priscilla A., E-mail: priscilla.silva@sjbv.unesp.br and Terra, Maisa O., E-mail: maisa@ita.br},
abstractNote = {We investigate multistability and global bifurcations in the general standard map, a biparametric two-dimensional map. Departing from the conservative case of the map, we describe the evolution of periodic solutions and their basins of attraction as dissipation builds up, paying special attention on how the biparametric variation affects multistability. We examine general and specific phenomena and behavior for three distinct dynamical regimes, namely small, moderate, and large damping and different forcing amplitudes. Also, we report numerically the mechanism of global bifurcations associated to small chaotic attractors in the multistable system. Several global bifurcations are investigated as dissipation increases. Specifically, through the characterization of an interior, a merging and a boundary crisis, we study the crucial role played by fundamental hyperbolic invariant structures, such as unstable periodic orbits and their stable and unstable invariant manifolds, in the mechanisms by which the phase space is globally transformed.},
doi = {10.1007/S40314-017-0544-8},
journal = {Computational and Applied Mathematics},
issn = {0101-8205},
number = 3,
volume = 37,
place = {United States},
year = {2018},
month = {7}
}