Emergence and Approximation of Tori

Taoufik Bakri; Ferdinand Verhulst

doi:10.1007/978-981-99-7958-5_1

Emergence and Approximation of Tori

Taoufik Bakri^*
, Ferdinand Verhulst

^*Corresponding author for this work

TNO Sustainable Urban Mobility and Safety

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

Abstract

We will consider isolated quasi-periodic solutions and families of quasi-periodic solutions. A remarkable aspect of the system Sprott A and a few related systems is the observed presence of families of quasi-periodic solutions organised on invariant tori, a phenomenon known earlier in the context of conservative systems and the so-called KAM-theorem. We describe how the tori bifurcation phenomenon in dissipative systems has been linked to time-reversal and canards. This will be illustrated for the systems NE8 and NE9. Numerical bifurcation techniques in combination with averaging are used to study the presence of periodic and quasi-periodic solutions and possibly invariant manifolds. An interesting mechanical application will be the interaction of a parametrically excited oscillator and an oscillator with self-excitation where Hopf and Neimark–Sacker bifurcations occur naturally. This illustrates the strength of combining analytical and numerical techniques.

Original language	English
Title of host publication	Advances in Nonlinear Dynamics and Control of Mechanical and Physical Systems
Subtitle of host publication	Selected Articles from CSNDD 2023; 15–17 May; Marrakesh, Morocco
Editors	Mohamed Belhaq
Place of Publication	Singapore
Publisher	Springer
Pages	1-22
Number of pages	22
Edition	1
ISBN (Electronic)	978-981-99-7958-5
ISBN (Print)	978-981-99-7957-8
DOIs	https://doi.org/10.1007/978-981-99-7958-5_1
Publication status	Published - 27 Feb 2024
Event	5th Conference on Structural Nonlinear Dynamics and Diagnosis, CSNDD 2023 - Marrakesh, Morocco Duration: 15 May 2023 → 17 May 2023

Publication series

Name	Springer Proceedings in Physics
Publisher	Springer
Volume	301
ISSN (Print)	0930-8989
ISSN (Electronic)	1867-4941

Conference

Conference	5th Conference on Structural Nonlinear Dynamics and Diagnosis, CSNDD 2023
Country/Territory	Morocco
City	Marrakesh
Period	15/05/23 → 17/05/23

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024.

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10.1007/978-981-99-7958-5_1

Cite this

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title = "Emergence and Approximation of Tori",

abstract = "We will consider isolated quasi-periodic solutions and families of quasi-periodic solutions. A remarkable aspect of the system Sprott A and a few related systems is the observed presence of families of quasi-periodic solutions organised on invariant tori, a phenomenon known earlier in the context of conservative systems and the so-called KAM-theorem. We describe how the tori bifurcation phenomenon in dissipative systems has been linked to time-reversal and canards. This will be illustrated for the systems NE8 and NE9. Numerical bifurcation techniques in combination with averaging are used to study the presence of periodic and quasi-periodic solutions and possibly invariant manifolds. An interesting mechanical application will be the interaction of a parametrically excited oscillator and an oscillator with self-excitation where Hopf and Neimark–Sacker bifurcations occur naturally. This illustrates the strength of combining analytical and numerical techniques.",

author = "Taoufik Bakri and Ferdinand Verhulst",

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Bakri, T & Verhulst, F 2024, Emergence and Approximation of Tori. in M Belhaq (ed.), Advances in Nonlinear Dynamics and Control of Mechanical and Physical Systems: Selected Articles from CSNDD 2023; 15–17 May; Marrakesh, Morocco. 1 edn, Springer Proceedings in Physics, vol. 301, Springer, Singapore, pp. 1-22, 5th Conference on Structural Nonlinear Dynamics and Diagnosis, CSNDD 2023, Marrakesh, Morocco, 15/05/23. https://doi.org/10.1007/978-981-99-7958-5_1

Emergence and Approximation of Tori. / Bakri, Taoufik; Verhulst, Ferdinand.
Advances in Nonlinear Dynamics and Control of Mechanical and Physical Systems: Selected Articles from CSNDD 2023; 15–17 May; Marrakesh, Morocco. ed. / Mohamed Belhaq. 1. ed. Singapore: Springer, 2024. p. 1-22 (Springer Proceedings in Physics; Vol. 301).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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AB - We will consider isolated quasi-periodic solutions and families of quasi-periodic solutions. A remarkable aspect of the system Sprott A and a few related systems is the observed presence of families of quasi-periodic solutions organised on invariant tori, a phenomenon known earlier in the context of conservative systems and the so-called KAM-theorem. We describe how the tori bifurcation phenomenon in dissipative systems has been linked to time-reversal and canards. This will be illustrated for the systems NE8 and NE9. Numerical bifurcation techniques in combination with averaging are used to study the presence of periodic and quasi-periodic solutions and possibly invariant manifolds. An interesting mechanical application will be the interaction of a parametrically excited oscillator and an oscillator with self-excitation where Hopf and Neimark–Sacker bifurcations occur naturally. This illustrates the strength of combining analytical and numerical techniques.

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Emergence and Approximation of Tori

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