scholarly journals Bifurcations of Cubic Homoclinic Tangencies in Two-dimensional Symplectic Maps

2017 ◽  
Vol 12 (1) ◽  
pp. 41-61 ◽  
Author(s):  
M. Gonchenko ◽  
S. Gonchenko ◽  
I. Ovsyannikov
Keyword(s):  
Two Dimensional ◽  
Symplectic Maps ◽  
Homoclinic Tangencies

ON GEOMETRICAL PROPERTIES OF TWO-DIMENSIONAL DIFFEOMORPHISMS WITH HOMOCLINIC TANGENCIES

1995 ◽  
Vol 05 (03) ◽  
pp. 819-829 ◽  
Author(s):  
S.V. GONCHENKO ◽  
L.P. SHIL’NIKOV
Keyword(s):  
Fixed Point ◽  
Invariant Manifolds ◽  
Two Dimensional ◽  
Geometrical Properties ◽  
Saddle Value ◽  
Almost All ◽  
Homoclinic Tangencies

Two-dimensional diffeomorphisms with a quadratic tangency of invariant manifolds of a saddle fixed point are considered in the cases where the saddle value σ is either less than 1 or equal to it. A description of the structure of hyperbolic subsets is given. In the case σ=1, it is shown that almost all such diffeomorphisms admit the complete description in distinction with the case σ<1.

scholarly journals Existence of generic cubic homoclinic tangencies for Hénon maps

2012 ◽  
Vol 33 (4) ◽  
pp. 1029-1051 ◽  
Author(s):  
SHIN KIRIKI ◽  
TERUHIKO SOMA
Keyword(s):  
Arbitrary Order ◽  
Homoclinic Orbits ◽  
Steklov Inst ◽  
Strange Attractors ◽  
Heteroclinic Cycle ◽  
Two Dimensional ◽  
Henon Map ◽  
Cubic Polynomial ◽  
New Phenomena ◽  
Homoclinic Tangencies

AbstractIn this paper, we show that the Hénon map $\varphi _{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko, Shilnikov and Turaev [On models with non-rough Poincaré homoclinic curves. Physica D 62(1–4) (1993), 1–14; Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos 6(1) (1996), 15–31; On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle. Proc. Steklov Inst. Math.216 (1997), 70–118; Homoclinic tangencies of an arbitrary order in Newhouse domains. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. 67 (1999), 69–128, translation in J. Math. Sci. 105 (2001), 1738–1778; Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20 (2007), 241–275]. Combining this fact with theorems in Kiriki and Soma [Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies. Nonlinearity 21(5) (2008), 1105–1140], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.

Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

Nonlinearity ◽  
2007 ◽  
Vol 20 (2) ◽  
pp. 241-275 ◽  
Author(s):  
Sergey Gonchenko ◽  
Dmitry Turaev ◽  
Leonid Shilnikov
Keyword(s):  
Two Dimensional ◽  
Homoclinic Tangencies

Homoclinic and Big Bang Bifurcations of an Embedding of 1D Allee’s Functions into a 2D Diffeomorphism

2017 ◽  
Vol 27 (09) ◽  
pp. 1730030 ◽  
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha ◽  
D. Fournier-Prunaret
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Big Bang ◽  
Complete Classification ◽  
Two Dimensional ◽  
Contour Lines ◽  
Dimension One ◽  
Necessary And Sufficient ◽  
Homoclinic Tangencies

In this work a thorough study is presented of the bifurcation structure of an embedding of one-dimensional Allee’s functions into a two-dimensional diffeomorphism. A complete classification of the nature and stability of the fixed points, on the contour lines of the two-dimensional diffeomorphism, is provided. A necessary and sufficient condition so that the Allee fixed point is a snapback repeller is established. Sufficient conditions for the occurrence of homoclinic tangencies of a saddle fixed point of the two-dimensional diffeomorphism are also established, associated to the snapback repeller bifurcation of the endomorphism defined by the Allee functions. The main results concern homoclinic and big bang bifurcations of the diffeomorphism as “germinal” bifurcations of the Allee functions. Our results confirm previous predictions of structures of homoclinic and big bang bifurcation curves in dimension one and extend these studies to “local” concepts of Allee effect and big bang bifurcations to this two-dimensional exponential diffeomorphism.

Homoclinic tangencies of arbitrarily high order in conservative two-dimensional maps

2006 ◽  
Vol 73 (2) ◽  
pp. 210-213
Author(s):  
S. V. Gonchenko ◽  
D. V. Turaev ◽  
L. P. Shil’nikov
Keyword(s):  
High Order ◽  
Two Dimensional ◽  
Homoclinic Tangencies

Homoclinic tangencies: moduli and topology of separatrices

1993 ◽  
Vol 13 (2) ◽  
pp. 369-385
Author(s):  
Rense A. Posthumus ◽  
Floris Takens
Keyword(s):  
Fixed Point ◽  
Two Dimensional ◽  
And Topology ◽  
Homoclinic Tangencies

We consider two-dimensional diffeomorphisms φ:M→M. For a fixed pointp, i.e.,p∈Mand φ(p) =p, we say thatqis homoclinic topifp≠qand if both limi→+∞φi(q) and limi→−∞φi(q) are equal top.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

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