Periodic orbits and bifurcation diagrams of acetylene/vinylidene revisited

2003 ◽  
Vol 118 (18) ◽  
pp. 8275-8280 ◽  
Author(s):  
Rita Prosmiti ◽  
Stavros C. Farantos
Keyword(s):  
Periodic Orbits ◽  
Bifurcation Diagrams

CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS

2008 ◽  
Vol 18 (12) ◽  
pp. 3689-3701 ◽  
Author(s):  
YANCONG XU ◽  
DEMING ZHU ◽  
FENGJIE GENG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Bifurcation ◽  
Reversible Systems ◽  
Reversible System ◽  
Bifurcation Diagrams ◽  
Symmetric Periodic Orbits ◽  
The Continuum

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

Linear Scaling Laws in Bifurcations of Scalar Maps

1994 ◽  
Vol 49 (12) ◽  
pp. 1207-1211 ◽  
Author(s):  
Celso Grebogi
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Scaling Laws ◽  
Linear Scaling ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Canonical Map ◽  
Global Scaling

Abstract A global scaling property for bifurcation diagrams of periodic orbits of smooth scalar maps with both one and two dimensional parameter spaces is examined. It is argued that for both parameter spaces bifurcations within a periodic window of a given scalar map are well approximated by a linear transformation of the bifurcation diagram of a canonical map.

Bifurcation diagrams of periodic orbits for unbound molecular systems: FH2

1997 ◽  
Vol 277 (5-6) ◽  
pp. 456-464 ◽  
Author(s):  
M. Founargiotakis ◽  
S.C. Farantos ◽  
Ch. Skokos ◽  
G. Contopoulos
Keyword(s):  
Periodic Orbits ◽  
Bifurcation Diagrams ◽  
Molecular Systems

Detecting Parameter Changes Using Experimental Nonlinear Dynamics and Chaos

1996 ◽  
Vol 118 (3) ◽  
pp. 375-383 ◽  
Author(s):  
R. S. Chancellor ◽  
R. M. Alexander ◽  
S. T. Noah
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Phase Portraits ◽  
Unstable Periodic Orbits ◽  
Bifurcation Diagrams ◽  
Frequency Spectra ◽  
Nonlinear Dynamics And Chaos ◽  
Chaotic Nature ◽  
Parameter Values

A method of detecting parameter changes using analytical and experimental nonlinear dynamics and chaos is applied to a piecewise-linear oscillator. Experimental data show the chaotic nature of the system through phase portraits, Poincare´ maps, frequency spectra and bifurcation diagrams. Unstable periodic orbits were extracted from each chaotic time series obtained from the system with six different parameter values. Movement of the unstable periodic orbits in phase space is used to detect parameter changes in the system.

scholarly journals CONNECTING PERIOD-DOUBLING CASCADES TO CHAOS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250022 ◽  
Author(s):  
EVELYN SANDER ◽  
JAMES A. YORKE
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Duffing Equation ◽  
Period Doubling Bifurcation ◽  
Bifurcation Diagrams ◽  
The Third ◽  
Infinite Collection

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value μ2 of the map at which there is chaos. We show that often virtually all (i.e. all but finitely many) "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired — connected to exactly one other cascade, or solitary — connected to exactly one regular periodic orbit at μ2. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ2, ⋅). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

Periodic orbits, bifurcation diagrams and the spectroscopy of C2H2 system

1995 ◽  
Vol 103 (9) ◽  
pp. 3299-3314 ◽  
Author(s):  
R. Prosmiti ◽  
S. C. Farantos
Keyword(s):  
Periodic Orbits ◽  
Bifurcation Diagrams

scholarly journals Bogdanov Map for Modelling a Phase-Conjugated Ring Resonator

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 384
Author(s):  
Vicente Aboites ◽  
David Liceaga ◽  
Rider Jaimes-Reátegui ◽  
Juan Hugo García-López
Keyword(s):  
Phase Space ◽  
Dynamic Behavior ◽  
Periodic Orbits ◽  
Ring Resonator ◽  
Matrix Elements ◽  
Bifurcation Diagrams ◽  
Direct Dependence ◽  
Computer Calculations ◽  
Generating Matrix ◽  
Explicit Expressions

In this paper, we propose using paraxial matrix optics to describe a ring-phase conjugated resonator that includes an intracavity chaos-generating element; this allows the system to behave in phase space as a Bogdanov Map. Explicit expressions for the intracavity chaos-generating matrix elements were obtained. Furthermore, computer calculations for several parameter configurations were made; rich dynamic behavior among periodic orbits high periodicity and chaos were observed through bifurcation diagrams. These results confirm the direct dependence between the parameters present in the intracavity chaos-generating element.

Bifurcation Diagram of a Map with Multiple Critical Points

2018 ◽  
Vol 28 (05) ◽  
pp. 1850065 ◽  
Author(s):  
M. Romera ◽  
G. Pastor ◽  
M.-F. Danca ◽  
A. Martin ◽  
A. B. Orue ◽  
...  
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Critical Points ◽  
Bifurcation Diagrams

In this work a conjecture to draw the bifurcation diagram of a map with multiple critical points is enunciated. The conjecture is checked by using two quartic maps in order to verify that the bifurcation diagrams obtained according to the conjecture contain all the periodic orbits previously counted by Xie and Hao for maps with four laps. We show that a map with split bifurcation contains more periodic orbits than those counted by these authors for a map with the same number of laps.

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