Persistent homoclinic tangencies in the Hénon family

Ittai Kan; Hüseyin Koçak; James A. Yorke

doi:10.1016/0167-2789(94)00226-G

Persistent homoclinic tangencies in the Hénon family

Ittai Kan, Hüseyin Koçak, James A. Yorke

Computer Science

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Abstract

Homoclinic tangencies in the Hénon family f_θ{symbol} (x, y) = (λ - x² + by, x) for the parameter values b = 0.3 and λ ε{lunate} [1.270, 1.420] are investigated. Our main observation is that there exist three intervals comprising 93 percent of the values of the parameter λ such that for a dense set of parameter values in these intervals the Hénon family possesses a homoclinic tangency. Therefore, one should expect long parameter intervals where the Hénon family is not structurally stable. Strong numerical support for this observation is provided.

Original language	English (US)
Pages (from-to)	313-325
Number of pages	13
Journal	Physica D: Nonlinear Phenomena
Volume	83
Issue number	4
DOIs	https://doi.org/10.1016/0167-2789(94)00226-G
State	Published - Jun 1 1995

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/0167-2789(94)00226-G

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title = "Persistent homoclinic tangencies in the H{\'e}non family",

abstract = "Homoclinic tangencies in the H{\'e}non family fθ{symbol} (x, y) = (λ - x2 + by, x) for the parameter values b = 0.3 and λ ε{lunate} [1.270, 1.420] are investigated. Our main observation is that there exist three intervals comprising 93 percent of the values of the parameter λ such that for a dense set of parameter values in these intervals the H{\'e}non family possesses a homoclinic tangency. Therefore, one should expect long parameter intervals where the H{\'e}non family is not structurally stable. Strong numerical support for this observation is provided.",

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T1 - Persistent homoclinic tangencies in the Hénon family

AU - Kan, Ittai

AU - Koçak, Hüseyin

AU - Yorke, James A.

PY - 1995/6/1

Y1 - 1995/6/1

N2 - Homoclinic tangencies in the Hénon family fθ{symbol} (x, y) = (λ - x2 + by, x) for the parameter values b = 0.3 and λ ε{lunate} [1.270, 1.420] are investigated. Our main observation is that there exist three intervals comprising 93 percent of the values of the parameter λ such that for a dense set of parameter values in these intervals the Hénon family possesses a homoclinic tangency. Therefore, one should expect long parameter intervals where the Hénon family is not structurally stable. Strong numerical support for this observation is provided.

AB - Homoclinic tangencies in the Hénon family fθ{symbol} (x, y) = (λ - x2 + by, x) for the parameter values b = 0.3 and λ ε{lunate} [1.270, 1.420] are investigated. Our main observation is that there exist three intervals comprising 93 percent of the values of the parameter λ such that for a dense set of parameter values in these intervals the Hénon family possesses a homoclinic tangency. Therefore, one should expect long parameter intervals where the Hénon family is not structurally stable. Strong numerical support for this observation is provided.

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Persistent homoclinic tangencies in the Hénon family

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