Center manifold reduction of periodic delay differential systems

Eric Butcher, Venkatesh Deshmukh, Ed Bueler

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method, Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.

Original languageEnglish (US)
Title of host publication2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007
Pages711-719
Number of pages9
Volume5 PART A
DOIs
StatePublished - 2008
Externally publishedYes
Event6th International Conference on Multibody Systems, Nonlinear Dynamics and Control, presented at - 2007 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE2007 - Las Vegas, NV, United States
Duration: Sep 4 2007Sep 7 2007

Other

Other6th International Conference on Multibody Systems, Nonlinear Dynamics and Control, presented at - 2007 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE2007
CountryUnited States
CityLas Vegas, NV
Period9/4/079/7/07

Fingerprint

Center Manifold Reduction
Delay-differential Systems
Periodic Systems
Ordinary differential equations
Periodic Coefficients
Delay Differential Equations
Ordinary differential equation
Differential equations
Perturbation Expansion
Inverted Pendulum
Linear Ordinary Differential Equations
Nonlinear Response
Parameter Perturbation
Direct numerical simulation
Pendulums
Flip
Spectral Methods
Collocation Method
Chebyshev
Limit Cycle

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Computer Science Applications
  • Mechanical Engineering
  • Modeling and Simulation

Cite this

Butcher, E., Deshmukh, V., & Bueler, E. (2008). Center manifold reduction of periodic delay differential systems. In 2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007 (Vol. 5 PART A, pp. 711-719) https://doi.org/10.1115/DETC2007-34583

Center manifold reduction of periodic delay differential systems. / Butcher, Eric; Deshmukh, Venkatesh; Bueler, Ed.

2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007. Vol. 5 PART A 2008. p. 711-719.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Butcher, E, Deshmukh, V & Bueler, E 2008, Center manifold reduction of periodic delay differential systems. in 2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007. vol. 5 PART A, pp. 711-719, 6th International Conference on Multibody Systems, Nonlinear Dynamics and Control, presented at - 2007 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE2007, Las Vegas, NV, United States, 9/4/07. https://doi.org/10.1115/DETC2007-34583
Butcher E, Deshmukh V, Bueler E. Center manifold reduction of periodic delay differential systems. In 2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007. Vol. 5 PART A. 2008. p. 711-719 https://doi.org/10.1115/DETC2007-34583
Butcher, Eric ; Deshmukh, Venkatesh ; Bueler, Ed. / Center manifold reduction of periodic delay differential systems. 2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007. Vol. 5 PART A 2008. pp. 711-719
@inproceedings{60787eeae7ec4717b8253c51abd03628,
title = "Center manifold reduction of periodic delay differential systems",
abstract = "A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method, Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.",
author = "Eric Butcher and Venkatesh Deshmukh and Ed Bueler",
year = "2008",
doi = "10.1115/DETC2007-34583",
language = "English (US)",
isbn = "0791848027",
volume = "5 PART A",
pages = "711--719",
booktitle = "2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007",

}

TY - GEN

T1 - Center manifold reduction of periodic delay differential systems

AU - Butcher, Eric

AU - Deshmukh, Venkatesh

AU - Bueler, Ed

PY - 2008

Y1 - 2008

N2 - A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method, Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.

AB - A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method, Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.

UR - http://www.scopus.com/inward/record.url?scp=44949261057&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44949261057&partnerID=8YFLogxK

U2 - 10.1115/DETC2007-34583

DO - 10.1115/DETC2007-34583

M3 - Conference contribution

AN - SCOPUS:44949261057

SN - 0791848027

SN - 9780791848029

SN - 079184806X

SN - 9780791848067

VL - 5 PART A

SP - 711

EP - 719

BT - 2007 Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007

ER -