A neutral homoclinic bifurcation in a 3D map

Hil Meijer; Willy Govaerts; Yu.A. Kuznetsov; Niels Neirycnk

A neutral homoclinic bifurcation in a 3D map

Hil Meijer
, Willy Govaerts
, Yu.A. Kuznetsov
, Niels Neirycnk

Ghent University
University of Twente

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

Abstract

We investigate the structure of Arnol’d tongues passing through a quasi-periodic saddle-node bifurcation in a 3D map. Due to resonances, the bifurcation set has a complicated structure. Using numerical continuation in MatcontM and Lyapunov exponents we explore the tongues and the quasi-periodic saddle-node bifurcation set.
The bifurcation set emerges from two Chenciner bifurcations. Both sets terminate in a homoclinic bifurcation of a neutral saddle cycle of period 3. It is similar to a case for vector fields, but the first report of such a codim-2 homoclinic tangency bifurcation in a map. We also show how these manifolds oscillate around the saddle near the tangency, for real and complex multipliers.

Original language	English
Title of host publication	Proceedings of the 9th European Nonlinear Dynamics Conference
Editors	Gábor Stépán, Gábor Csernák
Place of Publication	Budapest
Publisher	CongressLIne Ltd.
Number of pages	2
ISBN (Electronic)	978-963-12-9168-1
Publication status	Published - 2017
Event	9th European Nonlinear Dynamics Conference - Budapest University of Technology end Economics, Budapest, Hungary Duration: 25 Jun 2017 → 30 Jun 2017 Conference number: 9

Conference

Conference	9th European Nonlinear Dynamics Conference
Abbreviated title	ENOC 2017
Country/Territory	Hungary
City	Budapest
Period	25/06/17 → 30/06/17

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http://congressline.hu/enoc2017/pdf/ENOC2017_Front_matter.pdf

Cite this

@inproceedings{13cc63a5894e457db43a50a47b79f2a5,

title = "A neutral homoclinic bifurcation in a 3D map",

abstract = "We investigate the structure of Arnol{\textquoteright}d tongues passing through a quasi-periodic saddle-node bifurcation in a 3D map. Due to resonances, the bifurcation set has a complicated structure. Using numerical continuation in MatcontM and Lyapunov exponents we explore the tongues and the quasi-periodic saddle-node bifurcation set. The bifurcation set emerges from two Chenciner bifurcations. Both sets terminate in a homoclinic bifurcation of a neutral saddle cycle of period 3. It is similar to a case for vector fields, but the first report of such a codim-2 homoclinic tangency bifurcation in a map. We also show how these manifolds oscillate around the saddle near the tangency, for real and complex multipliers.",

author = "Hil Meijer and Willy Govaerts and Yu.A. Kuznetsov and Niels Neirycnk",

year = "2017",

language = "English",

editor = "G{\'a}bor St{\'e}p{\'a}n and G{\'a}bor Csern{\'a}k",

booktitle = "Proceedings of the 9th European Nonlinear Dynamics Conference",

publisher = "CongressLIne Ltd.",

note = "9th European Nonlinear Dynamics Conference, ENOC 2017 ; Conference date: 25-06-2017 Through 30-06-2017",

}

Meijer, H, Govaerts, W, Kuznetsov, YA & Neirycnk, N 2017, A neutral homoclinic bifurcation in a 3D map. in G Stépán & G Csernák (eds), Proceedings of the 9th European Nonlinear Dynamics Conference., ID 290, CongressLIne Ltd., Budapest, 9th European Nonlinear Dynamics Conference, Budapest, Hungary, 25/06/17. <http://congressline.hu/enoc2017/pdf/ENOC2017_Front_matter.pdf>

TY - GEN

T1 - A neutral homoclinic bifurcation in a 3D map

AU - Meijer, Hil

AU - Govaerts, Willy

AU - Kuznetsov, Yu.A.

AU - Neirycnk, Niels

N1 - Conference code: 9

PY - 2017

Y1 - 2017

N2 - We investigate the structure of Arnol’d tongues passing through a quasi-periodic saddle-node bifurcation in a 3D map. Due to resonances, the bifurcation set has a complicated structure. Using numerical continuation in MatcontM and Lyapunov exponents we explore the tongues and the quasi-periodic saddle-node bifurcation set. The bifurcation set emerges from two Chenciner bifurcations. Both sets terminate in a homoclinic bifurcation of a neutral saddle cycle of period 3. It is similar to a case for vector fields, but the first report of such a codim-2 homoclinic tangency bifurcation in a map. We also show how these manifolds oscillate around the saddle near the tangency, for real and complex multipliers.

AB - We investigate the structure of Arnol’d tongues passing through a quasi-periodic saddle-node bifurcation in a 3D map. Due to resonances, the bifurcation set has a complicated structure. Using numerical continuation in MatcontM and Lyapunov exponents we explore the tongues and the quasi-periodic saddle-node bifurcation set. The bifurcation set emerges from two Chenciner bifurcations. Both sets terminate in a homoclinic bifurcation of a neutral saddle cycle of period 3. It is similar to a case for vector fields, but the first report of such a codim-2 homoclinic tangency bifurcation in a map. We also show how these manifolds oscillate around the saddle near the tangency, for real and complex multipliers.

M3 - Conference contribution

BT - Proceedings of the 9th European Nonlinear Dynamics Conference

A2 - Stépán, Gábor

A2 - Csernák, Gábor

PB - CongressLIne Ltd.

CY - Budapest

T2 - 9th European Nonlinear Dynamics Conference

Y2 - 25 June 2017 through 30 June 2017

ER -

A neutral homoclinic bifurcation in a 3D map

Abstract

Conference

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