Abstract
The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on 3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities 1 > 0, 2 > 0, 3 > 0 and |2|>|1|>|3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors. A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 793-821 |
| Number of pages | 29 |
| Journal | Ergodic Theory and Dynamical Systems |
| Volume | 10 |
| Issue number | 4 |
| DOIs | |
| State | Published - 1990 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
Lorenz attractors through Šill'nikov-type bifurcation. Part I. / Rychlik, Marek R.
In: Ergodic Theory and Dynamical Systems, Vol. 10, No. 4, 1990, p. 793-821.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Lorenz attractors through Šill'nikov-type bifurcation. Part I
AU - Rychlik, Marek R
PY - 1990
Y1 - 1990
N2 - The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on 3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities 1 > 0, 2 > 0, 3 > 0 and |2|>|1|>|3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors. A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.
AB - The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on 3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities 1 > 0, 2 > 0, 3 > 0 and |2|>|1|>|3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors. A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.
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UR - http://www.scopus.com/inward/citedby.url?scp=84971972587&partnerID=8YFLogxK
U2 - 10.1017/S0143385700005915
DO - 10.1017/S0143385700005915
M3 - Article
AN - SCOPUS:84971972587
VL - 10
SP - 793
EP - 821
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
IS - 4
ER -