A Computer Method for Verification of Asymptotically Stable Periodic Orbits

1979 ◽  
Vol 10 (3) ◽  
pp. 614-628 ◽  
Author(s):  
John E. Franke ◽  
James F. Selgrade
Keyword(s):  
Periodic Orbits ◽  
Computer Method ◽  
Asymptotically Stable ◽  
Stable Periodic

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

Asymptotically stable periodic orbits of a coupled electromechanical system

2014 ◽  
Vol 78 (1) ◽  
pp. 29-35 ◽  
Author(s):  
M. J. H. Dantas ◽  
R. Sampaio ◽  
R. Lima
Keyword(s):  
Periodic Orbits ◽  
Electromechanical System ◽  
Asymptotically Stable ◽  
Stable Periodic

On a bifurcation structure mimicking period adding

2011 ◽  
Vol 467 (2129) ◽  
pp. 1503-1518 ◽  
Author(s):  
Björn Schenke ◽  
Viktor Avrutin ◽  
Michael Schanz
Keyword(s):  
Periodic Orbits ◽  
Parameter Space ◽  
Piecewise Linear ◽  
The Other ◽  
Asymptotically Stable ◽  
Periodic Domain ◽  
Bifurcation Structure ◽  
Symbolic Sequences ◽  
Stable Periodic ◽  
The One

In this work, we investigate a piecewise-linear discontinuous scalar map defined on three partitions. This map is specifically constructed in such a way that it shows a recently discovered bifurcation scenario in its pure form. Owing to its structure on the one hand and the similarities to the nested period-adding scenario on the other hand, we denoted the new bifurcation scenario as nested period-incrementing bifurcation scenario. The new bifurcation scenario occurs in several physical and electronical systems but usually not isolated, which makes the description complicated. By isolating the scenario and using a suitable symbolic description for the asymptotically stable periodic orbits, we derive a set of rules in the space of symbolic sequences that explain the structure of the stable periodic domain in the parameter space entirely. Hence, the presented work is a necessary step for the understanding of the more complicated bifurcation scenarios mentioned above.

scholarly journals Stable periodic orbits about the sun perturbed earth-moon triangularpoints.

AIAA Journal ◽  
1968 ◽  
Vol 6 (7) ◽  
pp. 1301-1304 ◽  
Author(s):  
RONALD KOLENKIEWICZ ◽  
LLOYD CARPENTER
Keyword(s):  
Periodic Orbits ◽  
The Sun ◽  
Stable Periodic

STABILIZATION OF CHAOTIC DYNAMICS OF ONE-DIMENSIONAL MAPS BY A CYCLIC PARAMETRIC TRANSFORMATION

1996 ◽  
Vol 06 (04) ◽  
pp. 725-735 ◽  
Author(s):  
ALEXANDER Yu. LOSKUTOV ◽  
VALERY M. TERESHKO ◽  
KONSTANTIN A. VASILIEV
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Exponential Map ◽  
One Dimensional ◽  
Stable Periodic ◽  
One Dimensional Maps ◽  
Parameter Values ◽  
Parametric Transformation

We consider one-dimensional maps, the logistic map and an exponential map, in those sets of parameter values which correspond to their chaotic dynamics. It is proven that such dynamics may be stabilized by a certain cyclic parametric transformation operating strictly within the chaotic set. The stabilization is a result of the creation of stable periodic orbits in the initially chaotic maps. The period of these stable orbits is a multiple of the period of the cyclic transformation. It is shown that stabilized behavior cannot be destroyed by a weak noise smearing of the required parameter values. The regions where the behavior stabilization takes place are numerically estimated. Periods of the created stabile periodic orbits are calculated.

scholarly journals Stable Periodic Orbits for a Class of Three Dimensional Competitive Systems

1994 ◽  
Vol 110 (1) ◽  
pp. 143-156 ◽  
Author(s):  
H.R. Zhu ◽  
H.L. Smith
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional ◽  
Competitive Systems ◽  
Stable Periodic
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