scholarly journals Periodic Solutions for Circular Restricted -Body Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xiaoxiao Zhao ◽  
Wei Li ◽  
Peng Wang
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Regular Polygon ◽  
Small Mass ◽  
Mass Point ◽  
Zero Mass

For circular restricted -body problems, we study the motion of a sufficiently small mass point (called the zero mass point) in the plane of equal masses located at the vertices of a regular polygon. By using variational minimizing methods, for some , we prove the existence of the noncollision periodic solution for the zero mass point with some fixed wingding number.

scholarly journals The Characterization of the Variational Minimizers for Spatial RestrictedN+1-Body Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Fengying Li ◽  
Shiqing Zhang ◽  
Xiaoxiao Zhao
Keyword(s):  
Periodic Solution ◽  
Loop Space ◽  
Vertical Axis ◽  
Small Mass ◽  
The Other ◽  
Necessary Condition ◽  
Zero Mass ◽  
Lagrangian Action ◽  
Variational Minimizers

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restrictedN+1-body problems with a zero mass on the vertical axis of the plane forNequal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any2≤N≤472; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.

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.

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.

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.

scholarly journals Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Giovanni Colombo ◽  
Paolo Gidoni ◽  
Emilio Vilches
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Finite Time ◽  
Dry Friction ◽  
Average Velocity ◽  
Well Posedness ◽  
Asymptotic Velocity ◽  
Sweeping Processes

<p style='text-indent:20px;'>We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger <inline-formula><tex-math id="M1">\begin{document}$ W^{1,2} $\end{document}</tex-math></inline-formula> convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.</p>

On the existence of periodic solutions for autonomous retarded functional differential equations on R2

1986 ◽  
Vol 102 (3-4) ◽  
pp. 259-262 ◽  
Author(s):  
J. G. Dos Reis ◽  
R. L. S. Baroni
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Continuous Functions ◽  
Retarded Functional Differential Equations ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

SynopsisLet Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

scholarly journals Variational Approach to Quasi-Periodic Solution of Nonautonomous Second-Order Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Juhong Kuang
Keyword(s):  
Periodic Solution ◽  
Variational Methods ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Variational Approach ◽  
Second Order ◽  
New Approach ◽  
Second Order Hamiltonian Systems

We deal with the quasi-periodic solutions of the following second-order Hamiltonian systemsx¨(t)=∇F(t,x(t)), wherex(t)=(x1(t),…,xN(t)), and we present a new approach via variational methods and Minmax method to obtain the existence of quasi-periodic solutions to the above equation.

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