The transversal homoclinic point of a saddle-node implies horseshoe

1999 ◽  
Vol 42 (7) ◽  
pp. 699-703
Author(s):  
Weigu Li
Keyword(s):  
Homoclinic Point ◽  
Transversal Homoclinic Point ◽  
Saddle Node

scholarly journals A generalization of a theorem of marotto

1981 ◽  
Vol 82 ◽  
pp. 83-97 ◽  
Author(s):  
Kenichi Shiraiwa ◽  
Masahiro Kurata
Keyword(s):  
Euclidean Space ◽  
Periodic Point ◽  
Real Line ◽  
Chaotic Behavior ◽  
Dimensional Euclidean Space ◽  
Homoclinic Point ◽  
Minimal Period ◽  
The Real ◽  
Continuous Map ◽  
Transversal Homoclinic Point

In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.

Attractor-preserving control to avoid saddle-node bifurcation

2014 ◽  
Vol 2 ◽  
pp. 150-153
Author(s):  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Shigeki Tsuji ◽  
Kazuyuki Aihara
Keyword(s):  
Saddle Node Bifurcation ◽  
Saddle Node

BASIN ORGANIZATION PRIOR TO A TANGLED SADDLE-NODE BIFURCATION

1991 ◽  
Vol 01 (01) ◽  
pp. 107-118 ◽  
Author(s):  
MOHAMED S. SOLIMAN ◽  
J. M. T. THOMPSON
Keyword(s):  
Fractal Structure ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Homoclinic Connections

Heteroclinic and homoclinic connections of saddle cycles play an important role in basin organization. In this study, we outline how these events can lead to an indeterminate jump to resonance from a saddle-node bifurcation. Here, due to the fractal structure of the basins in the vicinity of the saddle-node, we cannot predict to which available attractor the system will jump in the presence of even infinitesimal noise.

scholarly journals Global structural stability of a saddle node bifurcation

1978 ◽  
Vol 236 ◽  
pp. 155-155 ◽  
Author(s):  
Clark Robinson
Keyword(s):  
Structural Stability ◽  
Saddle Node Bifurcation ◽  
Saddle Node

scholarly journals Highly Excited Motion in Molecules:  Saddle-Node Bifurcations and Their Fingerprints in Vibrational Spectra

2002 ◽  
Vol 106 (22) ◽  
pp. 5407-5421 ◽  
Author(s):  
M. Joyeux ◽  
S. C. Farantos ◽  
R. Schinke
Keyword(s):  
Vibrational Spectra ◽  
Saddle Node

Existence of a homoclinic point for the H�non map

1980 ◽  
Vol 75 (3) ◽  
pp. 285-291 ◽  
Author(s):  
Micha? Misiurewicz ◽  
Boles?aw Szewc
Keyword(s):  
Homoclinic Point

Bifurcation analysis of steady natural convection in a tilted cubical cavity with adiabatic sidewalls

2014 ◽  
Vol 756 ◽  
pp. 650-688 ◽  
Author(s):  
J. F. Torres ◽  
D. Henry ◽  
A. Komiya ◽  
S. Maruyama
Keyword(s):  
Natural Convection ◽  
Rotation Axis ◽  
Numerical Technique ◽  
Continuation Method ◽  
Rotation Vector ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Stable Solutions ◽  
The Stability ◽  
Cubical Cavity

AbstractNatural convection in an inclined cubical cavity heated from two opposite walls maintained at different temperatures and with adiabatic sidewalls is investigated numerically. The cavity is inclined by an angle $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\theta $ around a lower horizontal edge and the isothermal wall set at the higher temperature is the lower wall in the horizontal situation ($\theta = 0^\circ $). A continuation method developed from a three-dimensional spectral finite-element code is applied to determine the bifurcation diagrams for steady flow solutions. The numerical technique is used to study the influence of ${\theta }$ on the stability of the flow for moderate Rayleigh numbers in the range $\mathit{Ra} \leq 150\, 000$, focusing on the Prandtl number $\mathit{Pr} = 5.9$. The results show that the inclination breaks the degeneracy of the stable solutions obtained at the first bifurcation point in the horizontal cubic cavity: (i) the transverse stable rolls, whose rotation vector is in the same direction as the inclination vector ${\boldsymbol{\Theta}}$, start from $\mathit{Ra} \to 0$ forming a leading branch and becoming more predominant with increasing $\theta $; (ii) a disconnected branch consisting of transverse rolls, whose rotation vector is opposite to ${\boldsymbol{\Theta}}$, develops from a saddle-node bifurcation, is stabilized at a pitchfork bifurcation, but globally disappears at a critical inclination angle; (iii) the semi-transverse stable rolls, whose rotation axis is perpendicular to ${\boldsymbol{\Theta}}$ at $\theta \to 0^\circ $, develop from another saddle-node bifurcation, but the branch also disappears at a critical angle. We also found the stability thresholds for the stable diagonal-roll and four-roll solutions, which increase with $\theta $ until they disappear at other critical angles. Finally, the families of stable solutions are presented in the $\mathit{Ra}-\theta $ parameter space by determining the locus of the primary, secondary, saddle-node, and Hopf bifurcation points as a function of $\mathit{Ra}$ and $\theta $.

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

2021 ◽  
Vol 31 (10) ◽  
pp. 2150147
Author(s):  
Yo Horikawa
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Output Function ◽  
Piecewise Constant ◽  
Border Collision Bifurcations ◽  
Saddle Node ◽  
Periodic Input ◽  
Symmetric Periodic Solution ◽  
Period Doubling Bifurcations

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

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