Effective computation of periodic orbits and bifurcation diagrams in delay equations

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.

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Periodic orbits and bifurcation diagrams of acetylene/vinylidene revisited

The Journal of Chemical Physics ◽

10.1063/1.1565991 ◽

2003 ◽

Vol 118 (18) ◽

pp. 8275-8280 ◽

Cited By ~ 11

Author(s):

Rita Prosmiti ◽

Stavros C. Farantos

Keyword(s):

Periodic Orbits ◽

Bifurcation Diagrams

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On the stability of periodic orbits in delay equations with large delay

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2013.33.3109 ◽

2013 ◽

Vol 33 (7) ◽

pp. 3109-3134 ◽

Cited By ~ 17

Author(s):

Jan Sieber ◽

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Matthias Wolfrum ◽

Mark Lichtner ◽

Serhiy Yanchuk ◽

...

Keyword(s):

Periodic Orbits ◽

Delay Equations ◽

Large Delay ◽

Stability Of Periodic Orbits ◽

The Stability

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Linear Scaling Laws in Bifurcations of Scalar Maps

Zeitschrift für Naturforschung A ◽

10.1515/zna-1994-1216 ◽

1994 ◽

Vol 49 (12) ◽

pp. 1207-1211 ◽

Cited By ~ 4

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.

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Bifurcation diagrams of periodic orbits for unbound molecular systems: FH2

Chemical Physics Letters ◽

10.1016/s0009-2614(97)00931-7 ◽

1997 ◽

Vol 277 (5-6) ◽

pp. 456-464 ◽

Cited By ~ 7

Author(s):

M. Founargiotakis ◽

S.C. Farantos ◽

Ch. Skokos ◽

G. Contopoulos

Keyword(s):

Periodic Orbits ◽

Bifurcation Diagrams ◽

Molecular Systems

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Detecting Parameter Changes Using Experimental Nonlinear Dynamics and Chaos

Journal of Vibration and Acoustics ◽

10.1115/1.2888193 ◽

1996 ◽

Vol 118 (3) ◽

pp. 375-383 ◽

Cited By ~ 13

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.

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CONNECTING PERIOD-DOUBLING CASCADES TO CHAOS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412500228 ◽

2012 ◽

Vol 22 (02) ◽

pp. 1250022 ◽

Cited By ~ 25

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.

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