scholarly journals A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

2020 ◽  
Vol 59 (2) ◽  
Author(s):  
Xijun Hu ◽  
Alessandro Portaluri ◽  
Ran Yang
Keyword(s):  
Periodic Orbits ◽  
Iteration Formula ◽  
Symmetric Periodic Orbits

scholarly journals Symmetric Periodic Orbits in the Anisotropic Kepler Problem

1983 ◽  
Vol 74 ◽  
pp. 263-270
Author(s):  
Josefina Casasaya ◽  
Jaume Llibre
Keyword(s):  
Energy Level ◽  
Periodic Orbits ◽  
Kepler Problem ◽  
Negative Energy ◽  
Velocity Curve ◽  
Zero Velocity ◽  
Anisotropic Kepler Problem ◽  
Symmetric Periodic Orbits ◽  
Group Of Symmetries

AbstractThe anisotropic Kepler problem has a group of symmetries with three generators; they are symmetries respect to zero velocity curve and the two axes of motion’s plane. For a fixed negative energy level it has four homothetic orbits. We describe the symmetric periodic orbits near these homothetic orbits. Full details and proofs will appear elsewhere (Casasayas-Llibre).

scholarly journals The Mechanism of Branching of Three-Dimensional Periodic Orbits from the Plane

1983 ◽  
Vol 74 ◽  
pp. 213-224
Author(s):  
I.A. Robin ◽  
V.V. Markellos
Keyword(s):  
Recent Work ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Three Dimensional ◽  
General Situation ◽  
Orbital Symmetry ◽  
Resonant Orbits ◽  
Symmetry Properties ◽  
Elliptic Restricted Problem ◽  
Symmetric Periodic Orbits

AbstractA linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

Bifurcations of Double Homoclinic Loops in Reversible Systems

2020 ◽  
Vol 30 (16) ◽  
pp. 2050246
Author(s):  
Yuzhen Bai ◽  
Xingbo Liu
Keyword(s):  
Coordinate System ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Reversible Systems ◽  
Bifurcation Equation ◽  
Bifurcation Phenomena ◽  
New Type ◽  
Symmetric Periodic Orbits

This paper is devoted to the study of bifurcation phenomena of double homoclinic loops in reversible systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equation, we perform a detailed study to obtain fruitful results, and demonstrate the existence of the R-symmetric large homoclinic orbit of new type near the primary double homoclinic loops, the existence of infinitely many R-symmetric periodic orbits accumulating onto the R-symmetric large homoclinic orbit, and the coexistence of R-symmetric large homoclinic orbit and the double homoclinic loops. The homoclinic bellow can also be found under suitable perturbation. The relevant bifurcation surfaces and the existence regions are located.

scholarly journals SYMMETRIC PERIODIC ORBITS NEAR HETEROCLINIC LOOPS AT INFINITY FOR A CLASS OF POLYNOMIAL VECTOR FIELDS

2006 ◽  
Vol 16 (11) ◽  
pp. 3401-3410 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Phase Portrait ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Phase Portraits ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Open Set ◽  
Symmetric Periodic Orbits

For polynomial vector fields in ℝ3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

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