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Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which selfsimilarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N21 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features

漏 Physical Review E, 2000, vol. 62, n煤m. 1, p. 333-348

American Physical Society

Author: Rius, J.
Figueras, M.
Herrero, R.
Pi i Vila, Francesc
Farjas Silva, Jordi
Orriols Tubella, Gaspar
Date: 2000
Abstract: We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which selfsimilarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N21 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features
Format: application/pdf
ISSN: 1539-3755 (versi贸 paper)
1550-2376 (versi贸 electr貌nica)
Document access: http://hdl.handle.net/10256/7699
Language: eng
Publisher: American Physical Society
Collection: Reproducci贸 digital del document publicat a: http://link.aps.org/doi/10.1103/PhysRevE.62.333
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Is part of: 漏 Physical Review E, 2000, vol. 62, n煤m. 1, p. 333-348
Rights: Tots els drets reservats
Subject: Sistemes din脿mics diferenciables
Differentiable dynamical systems
H-espais
H-spaces
Oscil路lacions no lineals
Nonlinear oscillations
Title: Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field
Type: info:eu-repo/semantics/article
Repository: DUGiDocs

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