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Errata

On Shil’nikov Analysis of Homoclinic and Heteroclinic Orbits of the T System

[+] Author and Article Information
Antonio Algaba

Professor
Department of Mathematics,
Facultad de Ciencias Experimentales,
University of Huelva,
Avda. Tres de Marzo s/n,
21071 Huelva, Spain
e-mail: algaba@uhu.es

Fernando Fernández-Sánchez

Professor
Department of Applied Mathematics II,
E.S. Ingenieros, University of Sevilla,
Camino de los Descubrimientos s/n,
41092 Sevilla, Spain
e-mail: fefesan@us.es

Manuel Merino

Professor
Department of Mathematics,
Facultad de Ciencias Experimentales,
University of Huelva,
Avda. Tres de Marzo s/n,
21071 Huelva, Spain
e-mail: merino@uhu.es

Alejandro J. Rodríguez-Luis

Professor
Department of Applied Mathematics II,
E.S. Ingenieros, University of Sevilla,
Camino de los Descubrimientos s/n,
41092 Sevilla, Spain
e-mail: ajrluis@us.es

J. Comput. Nonlinear Dynam 8(2), 027001 (Jul 23, 2012) (4 pages) Paper No: CND-11-1170; doi: 10.1115/1.4006788 History: Received October 08, 2011; Revised March 27, 2012

In the referenced paper, the authors use the undetermined coefficient method to analytically construct homoclinic and heteroclinic orbits in the T system. Unfortunately their method is not valid because they assume odd functions for the first component of the homoclinic and the heteroclinic orbit whereas these Shil'nikov global connections do not exhibit symmetry.

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Figures

Grahic Jump Location
Fig. 1

(a) Projection onto the x - z plane of the pair of heteroclinic orbits (which are mapped onto each other by the symmetry) joining E- and E+. (b) First component x(t) of both heteroclinic orbits. (c) Qualitative plot of the odd solution x(t) proposed in Eq. (3.18) of Ref. [1], for the heteroclinic orbit: it tends to x0 when t + (without oscillatory behavior), it satisfies x(0)=0 and it is an odd function.

Grahic Jump Location
Fig. 2

For c=30 and b=0.1, projection onto the x – z plane of the Shil’nikov heteroclinic orbit that exists when: (a) a≈1.7269; (b) a≈1.7272

Grahic Jump Location
Fig. 3

(a) Shil’nikov homoclinic orbit to the origin. (b) First component x(t) of the Shil’nikov homoclinic orbit. (c) Qualitative plot of the odd solution x(t) proposed in Eq. (3.27) of Ref. [1], for the homoclinic orbit: it tends to 0 when t + (without oscillatory behavior), it satisfies x(0)=0 and it is an odd function.

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