Equilibrium points and periodic orbits of higher order autonomous generalized Birkhoff system

2013 ◽  
Vol 224 (8) ◽  
pp. 1593-1599 ◽  
Author(s):  
Xiangwei Chen ◽  
Yanmin Li
Keyword(s):  
Periodic Orbits ◽  
Equilibrium Points ◽  
Higher Order ◽  
Birkhoff System

A Higher-Order Period Function and Its Application

2015 ◽  
Vol 25 (10) ◽  
pp. 1550140 ◽  
Author(s):  
Linping Peng ◽  
Lianghaolong Lu ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Upper Bound ◽  
Critical Period ◽  
Higher Order ◽  
Quadratic System ◽  
Critical Periods ◽  
Explicit Formulas ◽  
Period Function ◽  
Isochronous System ◽  
Isochronous Centers

This paper derives explicit formulas of the q th period bifurcation function for any perturbed isochronous system with a center, which improve and generalize the corresponding results in the literature. Based on these formulas to the perturbed quadratic and quintic rigidly isochronous centers, we prove that under any small homogeneous perturbations, for ε in any order, at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system. For ε in order of 1, 2, 3, 4 and 5, at most three critical periods bifurcate from the periodic orbits of the unperturbed quintic system. Moreover, in each case, the upper bound is sharp. Finally, a family of perturbed quintic rigidly isochronous centers is shown, which has three, for ε in any order, as the exact upper bound of the number of critical periods.

scholarly journals Orbital Structure of the Two Fixed Centres Problem

1999 ◽  
Vol 172 ◽  
pp. 463-464
Author(s):  
A. Cordero ◽  
J. Martínez Alfaro ◽  
P. Vindel
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Artificial Satellite ◽  
Equilibrium Points ◽  
Integrable Hamiltonian System ◽  
Three Dimensional ◽  
First Integral ◽  
Material Point ◽  
Dimensional Sphere

The set of orbits of the Two Fixed Centres problem has been known for a long time (Chartier, 1902, 1907; Pars, 1965), since it is an integrable Hamiltonian system.We consider a plane that contains the fixed masses. Denote by φ the angle denned by this plane and the one that contains also the third body. The momentum pφ is a first integral of the system and when pφ is different from zero, the manifold generated by the generalized coordinates and momenta are two copies of the three-dimensional sphere S3. If pφ = 0, that is to say when the planet crosses the line joining both suns, the motion is restricted to a planar one. All the equilibrium points appears in this case and therefore the phase spaces are more complex. We restrict our attention to this case which has two degrees of freedom.It is again a Bott-integrable Hamiltonian system. The set of periodic orbits of this systems can be studied from a subset of them, the Non-Singular Morse-Smale type orbits (see Casasayas, 1992). It is proved in Campos (1997) that a small perturbation of a Bott-integrable Hamiltonian system transforms it into a Non-Singular Morse-Smale system. The NMS periodic orbits belong to both the NMS system and the Hamiltonian one. Moreover, The NMS p.o. can be continued to nearly Hamiltonian systems. For instance, in our case to the Restricted Three Body Problem and in the study of the motion of a material point moving inside the gravitational field generated by two stars. This approximation is also useful when the motion of an artificial satellite around a spheroidal body is considered.

Non-linear dynamics of the photodissociation of nitrous oxide: Equilibrium points, periodic orbits, and transition states

2011 ◽  
Vol 134 (24) ◽  
pp. 244302 ◽  
Author(s):  
Frederic Mauguiere ◽  
Stavros C. Farantos ◽  
Jaime Suarez ◽  
Reinhard Schinke
Keyword(s):  
Nitrous Oxide ◽  
Periodic Orbits ◽  
Equilibrium Points ◽  
Transition States ◽  
Linear Dynamics ◽  
Non Linear

scholarly journals Multistability and Multiperiodicity for a General Class of Delayed Cohen-Grossberg Neural Networks with Discontinuous Activation Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yanke Du ◽  
Yanlu Li ◽  
Rui Xu
Keyword(s):  
Neural Networks ◽  
Periodic Orbits ◽  
General Class ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Multiple Equilibrium ◽  
Activation Functions ◽  
Discontinuous Activation ◽  
Discontinuous Activation Functions ◽  
Multiple Equilibrium Points

A general class of Cohen-Grossberg neural networks with time-varying delays, distributed delays, and discontinuous activation functions is investigated. By partitioning the state space, employing analysis approach and Cauchy convergence principle, sufficient conditions are established for the existence and locally exponential stability of multiple equilibrium points and periodic orbits, which ensure thatn-dimensional Cohen-Grossberg neural networks withk-level discontinuous activation functions can haveknequilibrium points orknperiodic orbits. Finally, several examples are given to illustrate the feasibility of the obtained results.

Local Manifolds and Periodic Orbits Around Equilibrium Points of the Rotating Mass Dipole

2016 ◽  
Author(s):  
Xiangyuan Zeng ◽  
S. R. Vadali ◽  
Kyle T. Alfriend ◽  
Quan Hu ◽  
Hexi Baoyin
Keyword(s):  
Periodic Orbits ◽  
Equilibrium Points ◽  
Rotating Mass Dipole

scholarly journals Stability of Periodic Orbits in the Averaging Theory: Applications to Lorenz and Thomas Differential Systems

2018 ◽  
Vol 28 (03) ◽  
pp. 1830007 ◽  
Author(s):  
Murilo R. Cândido ◽  
Jaume Llibre
Keyword(s):  
Periodic Orbits ◽  
Higher Order ◽  
Averaging Theory ◽  
Differential Systems ◽  
Stability Of Periodic Orbits

We provide new results in studying a kind of stability of periodic orbits provided by the higher-order averaging theory. Then, we apply these results to determining the [Formula: see text]-hyperbolicity of some periodic orbits of the Lorenz and Thomas differential systems.

Sign in / Sign up

Export Citation Format

Share Document