Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies

Xiujing Han*, Qinsheng Bi, Peng Ji, Juergen Kurths

*Corresponding author for this work

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

We present a general method for analyzing mixed-mode oscillations (MMOs) in parametrically and externally excited systems with two low excitation frequencies (PEESTLEFs) for the case of arbitrary m:n relation between the slow frequencies of excitations. The validity of the approach has been demonstrated using the equations of Duffing and van der Pol, separately. Our study shows that, by introducing a slow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a single slow variable and therefore MMOs observed in PEESTLEFs can be understood by the classical machinery of fast subsystem analysis of the transformed fast-slow system.

Original languageEnglish
Article number012911
Number of pages12
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume92
Issue number1
DOIs
Publication statusPublished - 15 Jul 2015

Keywords

  • mixed-mode oscillations
  • HOPF-bifurcation
  • catalyst deactivation
  • passage
  • modulation
  • dynamics
  • patterns
  • period
  • chaos
  • weak

Cite this

Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies. / Han, Xiujing; Bi, Qinsheng; Ji, Peng; Kurths, Juergen.

In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, Vol. 92, No. 1, 012911, 15.07.2015.

Research output: Contribution to journalArticle

@article{a3a49b94cc034221bc8c46ccee625a17,
title = "Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies",
abstract = "We present a general method for analyzing mixed-mode oscillations (MMOs) in parametrically and externally excited systems with two low excitation frequencies (PEESTLEFs) for the case of arbitrary m:n relation between the slow frequencies of excitations. The validity of the approach has been demonstrated using the equations of Duffing and van der Pol, separately. Our study shows that, by introducing a slow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a single slow variable and therefore MMOs observed in PEESTLEFs can be understood by the classical machinery of fast subsystem analysis of the transformed fast-slow system.",
keywords = "mixed-mode oscillations, HOPF-bifurcation, catalyst deactivation, passage, modulation, dynamics, patterns, period, chaos, weak",
author = "Xiujing Han and Qinsheng Bi and Peng Ji and Juergen Kurths",
note = "ACKNOWLEDGMENTS The authors express their gratitude to the anonymous reviewers for their valuable comments and suggestions that help to improve the paper. This work was supported by the National Natural Science Foundation of China (Grants No. 11202085, No. 21276115, No. 11302087, No. 11302086, and No. 11402226), the Natural Science Foundation of Jiangsu Province (Grant No. BK20130479), and the Research Foundation for Advanced Talents of Jiangsu University (Grant No. 11JDG075 ).",
year = "2015",
month = "7",
day = "15",
doi = "10.1103/PhysRevE.92.012911",
language = "English",
volume = "92",
journal = "Physical Review. E, Statistical, Nonlinear and Soft Matter Physics",
issn = "1539-3755",
publisher = "AMER PHYSICAL SOC",
number = "1",

}

TY - JOUR

T1 - Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies

AU - Han, Xiujing

AU - Bi, Qinsheng

AU - Ji, Peng

AU - Kurths, Juergen

N1 - ACKNOWLEDGMENTS The authors express their gratitude to the anonymous reviewers for their valuable comments and suggestions that help to improve the paper. This work was supported by the National Natural Science Foundation of China (Grants No. 11202085, No. 21276115, No. 11302087, No. 11302086, and No. 11402226), the Natural Science Foundation of Jiangsu Province (Grant No. BK20130479), and the Research Foundation for Advanced Talents of Jiangsu University (Grant No. 11JDG075 ).

PY - 2015/7/15

Y1 - 2015/7/15

N2 - We present a general method for analyzing mixed-mode oscillations (MMOs) in parametrically and externally excited systems with two low excitation frequencies (PEESTLEFs) for the case of arbitrary m:n relation between the slow frequencies of excitations. The validity of the approach has been demonstrated using the equations of Duffing and van der Pol, separately. Our study shows that, by introducing a slow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a single slow variable and therefore MMOs observed in PEESTLEFs can be understood by the classical machinery of fast subsystem analysis of the transformed fast-slow system.

AB - We present a general method for analyzing mixed-mode oscillations (MMOs) in parametrically and externally excited systems with two low excitation frequencies (PEESTLEFs) for the case of arbitrary m:n relation between the slow frequencies of excitations. The validity of the approach has been demonstrated using the equations of Duffing and van der Pol, separately. Our study shows that, by introducing a slow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a single slow variable and therefore MMOs observed in PEESTLEFs can be understood by the classical machinery of fast subsystem analysis of the transformed fast-slow system.

KW - mixed-mode oscillations

KW - HOPF-bifurcation

KW - catalyst deactivation

KW - passage

KW - modulation

KW - dynamics

KW - patterns

KW - period

KW - chaos

KW - weak

U2 - 10.1103/PhysRevE.92.012911

DO - 10.1103/PhysRevE.92.012911

M3 - Article

VL - 92

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 1

M1 - 012911

ER -