Computation of Periodic Solutions of Conservative Systems with Application to the 3-Body Problem

2003 ◽  
Vol 13 (06) ◽  
pp. 1353-1381 ◽  
Author(s):  
E. J. Doedel ◽  
R. C. Paffenroth ◽  
H. B. Keller ◽  
D. J. Dichmann ◽  
J. Galán-Vioque ◽  
...  
Keyword(s):  
Periodic Solution ◽  
Phase Shift ◽  
Periodic Solutions ◽  
Body Problem ◽  
Scaling Law ◽  
Main Idea ◽  
Continuous Path ◽  
Point Boundary ◽  
Conservative Systems ◽  
Branch Switching

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simó, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

LONG-RANGE CORRELATIONS IN THE PHASE-SHIFTS OF NUMERICAL SIMULATIONS OF BIOCHEMICAL OSCILLATIONS AND IN EXPERIMENTAL CARDIAC RHYTHMS

1999 ◽  
Vol 07 (02) ◽  
pp. 113-130 ◽  
Author(s):  
I. M. DE LA FUENTE ◽  
L. MARTINEZ ◽  
J. M. AGUIRREGABIRIA ◽  
J. VEGUILLAS ◽  
M. IRIARTE
Keyword(s):  
Phase Shift ◽  
Periodic Solutions ◽  
Long Range ◽  
Phase Shifts ◽  
Long Range Correlations ◽  
Rescaled Range ◽  
Biochemical Oscillations ◽  
Phase Shift Data ◽  
Relationship Of ◽  
The Relationship

In biochemical dynamical systems during each transition between periodical behaviors, all metabolic intermediaries of the system oscillate with the same frequency but with different phase-shifts. We have studied the behavior of phase-shift records obtained from random transitions between periodic solutions of a biochemical dynamical system. The phase-shift data were analyzed by means of Hurst's rescaled range method (introduced by Mandelbrot and Wallis). The results show the existence of persistent behavior: each value of the phase-shift depends not only on the recent transitions, but also on previous ones. In this paper, the different kind of periodic solutions were determined by different small values of the control parameter. It was assessed the significance of this results through extensive Monte Carlo simulations as well as quantifying the long-range correlations. We have also applied this type of analysis on cardiac rhythms, showing a clear persistent behavior. The relationship of the results with the cellular persistence phenomena conditioned by the past, widely evidenced in experimental observations, is discussed.

Network periodic solutions: patterns of phase-shift synchrony

Nonlinearity ◽  
2012 ◽  
Vol 25 (4) ◽  
pp. 1045-1074 ◽  
Author(s):  
Martin Golubitsky ◽  
David Romano ◽  
Yunjiao Wang
Keyword(s):  
Phase Shift ◽  
Periodic Solutions

scholarly journals Multiplicity of positive periodic solutions to second order differential equations

2006 ◽  
Vol 73 (2) ◽  
pp. 175-182 ◽  
Author(s):  
Jifeng Chu ◽  
Xiaoning Lin ◽  
Daqing Jiang ◽  
Donal O'Regan ◽  
R. P. Agarwal
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Positive Periodic Solution ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Second Order Differential Equations

In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

2021 ◽  
Vol 31 (10) ◽  
pp. 2150147
Author(s):  
Yo Horikawa
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Output Function ◽  
Piecewise Constant ◽  
Border Collision Bifurcations ◽  
Saddle Node ◽  
Periodic Input ◽  
Symmetric Periodic Solution ◽  
Period Doubling Bifurcations

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

New styles of periodic solutions of the classical six-body problem

2019 ◽  
Vol 159 ◽  
pp. 183-196 ◽  
Author(s):  
Ahmed Alsaedi ◽  
Feras Yousef ◽  
Samia Bushnaq ◽  
Shaher Momani
Keyword(s):  
Periodic Solutions ◽  
Body Problem

scholarly journals Periodic solution for ϕ-Laplacian neutral differential equation

2019 ◽  
Vol 17 (1) ◽  
pp. 172-190 ◽  
Author(s):  
Shaowen Yao ◽  
Zhibo Cheng
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
A Priori ◽  
Neutral Differential Equation ◽  
A Priori Bounds ◽  
T Tau

Abstract This paper is devoted to the existence of a periodic solution for ϕ-Laplacian neutral differential equation as follows $$\begin{array}{} (\phi(x(t)-cx(t-\tau))')'=f(t,x(t),x'(t)). \end{array}$$ By applications of an extension of Mawhin’s continuous theorem due to Ge and Ren, we obtain that given equation has at least one periodic solution. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature.

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