scholarly journals On the construction of quasimodes associated with stable periodic orbits

1976 ◽  
Vol 51 (3) ◽  
pp. 219-242 ◽  
Author(s):  
J. V. Ralston
Keyword(s):  
Periodic Orbits ◽  
Stable Periodic

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

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.

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

scholarly journals Periodic orbits relevant for the formation of inner rings in barred galaxies

1985 ◽  
Vol 106 ◽  
pp. 543-544
Author(s):  
M. Michalodimitrakis ◽  
Ch. Terzides
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional ◽  
Particle Trapping ◽  
Barred Galaxies ◽  
Future Paper ◽  
Stability Of Periodic Orbits ◽  
Wide Range ◽  
The Galaxy ◽  
Stable Periodic ◽  
The Stability

The study of orbits of a test particle in the gravitational field of a model barred galaxy is a first step toward the understanding of the origin of the morphological characterstics observed in real barred galaxies. In this paper we confine our attention to the inner rings. Inner rings are a very common characteristic of barred galaxies. They are narrow, round or slightly elongated along the bar (with typical axial ratios from 0.7 to near 1.0), and of the same size as the bar. A first step to test the old hypothesis that inner rings consist of stars trapped near stable periodic orbits would be a study of particle trapping around periodic orbits encircling the bar. Such a study is contained in the work of several authors (Danby 1965, de Vaucouleurs and Freeman 1972, Michalodimitrakis 1975, Contopoulos and Papayannopoulos 1980, Athanassoula et al. 1983). In the above works the stability of periodic orbits was studied with respect to perturbations which lie on the plane of motion z = 0 (planar stability). To ensure the possibility of formation of rings, a study of stability with respect to perturbations perpendicular to the plane of motion (vertical stability) is necessary. In this paper we investigate the properties of periodic orbits which we believe to be relevant for the inner-ring problem using a sufficiently general model for the galaxy and sets of values for the parameters which cover a wide range of different possible cases. We also study the stability, planar and vertical, with respect to large perturbations in order to estimate the extent of particle trapping. A detailed numerical investigation of three-dimensional periodic orbits will be given in a future paper.

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

Diverting homoclinic chaos in a class of piecewise smooth oscillators to stable periodic orbits using small parametrical perturbations

2015 ◽  
Vol 149 ◽  
pp. 1587-1595 ◽  
Author(s):  
Huaqing Li ◽  
Xiaofeng Liao ◽  
Junjian Huang ◽  
Guo Chen ◽  
Zhaoyang Dong ◽  
...  
Keyword(s):  
Periodic Orbits ◽  
Piecewise Smooth ◽  
Homoclinic Chaos ◽  
Stable Periodic

scholarly journals Building CX peanut-shaped disk galaxy profiles

2018 ◽  
Vol 612 ◽  
pp. A114 ◽  
Author(s):  
P. A. Patsis ◽  
M. Harsoula
Keyword(s):  
Phase Space ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Resonance Region ◽  
Space Structure ◽  
Jacobi Constant ◽  
Dynamical Mechanism ◽  
Standard Case ◽  
New Model ◽  
Stable Periodic

Context. We present and discuss the orbital content of a rather unusual rotating barred galaxy model, in which the three-dimensional (3D) family, bifurcating from x1 at the 2:1 vertical resonance with the known “frown-smile” side-on morphology, is unstable. Aims. Our goal is to study the differences that occur in the phase space structure at the vertical 2:1 resonance region in this case, with respect to the known, well studied, standard case, in which the families with the frown-smile profiles are stable and support an X-shaped morphology. Methods. The potential used in the study originates in a frozen snapshot of an N-body simulation in which a fast bar has evolved. We follow the evolution of the vertical stability of the central family of periodic orbits as a function of the energy (Jacobi constant) and we investigate the phase space content by means of spaces of section. Results. The two bifurcating families at the vertical 2:1 resonance region of the new model change their stability with respect to that of most studied analytic potentials. The structure in the side-on view that is directly supported by the trapping of quasi-periodic orbits around 3D stable periodic orbits has now an infinity symbol (i.e. ∞-type) profile. However, the available sticky orbits can reinforce other types of side-on morphologies as well. Conclusions. In the new model, the dynamical mechanism of trapping quasi-periodic orbits around the 3D stable periodic orbits that build the peanut, supports the ∞-type profile. The same mechanism in the standard case supports the X shape with the frown-smile orbits. Nevertheless, in both cases (i.e. in the new and in the standard model) a combination of 3D quasi-periodic orbits around the stable x1 family with sticky orbits can support a profile reminiscent of the shape of the orbits of the 3D unstable family existing in each model.

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