Periodic solutions and associated limit cycle for the generalised Chazy equation

2008 ◽  
pp. 327-335 ◽  
Author(s):  
Claude Géronimi ◽  
Marc R. Feix ◽  
Peter G. L. Leach
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Chazy Equation

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

Periodic Solutions and Their Regions of Attraction for Flexible Structures Under Relay Feedback Control With Nonlinear Control Law

2011 ◽  
Author(s):  
Michael Borre ◽  
Henryk Flashner
Keyword(s):  
Feedback Control ◽  
Nonlinear Control ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Dead Zone ◽  
Flexible Structures ◽  
Convergence Criterion ◽  
Control Law ◽  
Cell Mapping ◽  
Nonlinear Control Law

A method for calculating all periodic solutions and their domains of attraction for flexible systems under nonlinear feedback control is presented. The systems considered consist of mechanical systems with many flexible modes and a relay type controller coupled with a nonlinear control law operating in a feedback configuration. The proposed approach includes three steps. First, limit cycle frequencies and periodic fixed points are computed exactly, using a block diagonal state-space modal representation of the plant dynamics. Then the relay switching surface is chosen as the Poincare mapping surface and is discretized using the cell mapping method. Finally, the region of attraction for each limit cycle is computed using the cell mapping algorithm and employing an error based convergence criterion. An example consisting of a system with two modes, a relay with dead-zone and hysteresis, and a nonlinear control law with a signed velocity squared term is used to demonstrate the proposed approach.

LIMIT CYCLES IN 3D LOTKA–VOLTERRA SYSTEMS APPEARING AFTER PERTURBATION OF HOPF CENTER

2008 ◽  
Vol 18 (12) ◽  
pp. 3647-3656 ◽  
Author(s):  
Ł. J. GOŁASZEWSKI ◽  
P. SŁAWIŃSKI ◽  
H. ŻOŁADEK
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Invariant Torus ◽  
Small Amplitude ◽  
Parameter Family ◽  
Volterra Systems ◽  
Parameter Perturbations ◽  
Two Parameter

We study the system ẋ = x(y+2z+(15/2η2)u), ẏ = y(x-2z-(7/2η2)u), ż = -z(x+y+(4/η2)u), u = x+y+z-1, and its two-parameter perturbations. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. Moreover, we found that a secondary Hopf bifurcation to an invariant torus occurs after the perturbation.

scholarly journals Local Hopf Birurcation and Stability of Limit Cycle in a Delayed Kaldor-Kalecki Model

2009 ◽  
Vol 14 (3) ◽  
pp. 333-343 ◽  
Author(s):  
A. Kaddar ◽  
H. Talibi Alaoui
Keyword(s):  
Hopf Bifurcation ◽  
Business Cycle ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Cycle Model ◽  
Explicit Algorithm ◽  
Local Hopf Bifurcation ◽  
Business Cycle Model ◽  
The Stability

We consider a delayed Kaldor-Kalecki business cycle model. We first consider the existence of local Hopf bifurcation, and we establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions using the methods presented by O. Diekmann et al. in [1]. In the end, we conclude with an application.

scholarly journals On the existence of periodic and eventually periodic solutions of a fluid dynamic forced harmonic oscillator

1986 ◽  
Vol 9 (2) ◽  
pp. 381-385
Author(s):  
Charlie H. Cooke
Keyword(s):  
Differential Equation ◽  
Harmonic Oscillator ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Nonlinear Differential Equation ◽  
Fluid Dynamic ◽  
Department Store ◽  
Symmetry Properties ◽  
All Solutions ◽  
Vertical Jet

For certain flow regimes, the nonlinear differential equationY¨=F(Y)−G,Y≥0,G>0and constant, models qualitatively the behaviour of a forced, fluid dynamic, harmonic oscillator which has been a popular department store attraction. The device consists of a ball oscillating suspended in the vertical jet from a household fan. From the postulated form of the model, we determine sets of attraction and exploit symmetry properties of the system to show that all solutions are either initially periodic, with the ball never striking the fan, or else eventually approach a periodic limit cycle, after a sufficient number of bounces away from the fan.

Numerical Analysis of a Novel 3D Chaotic System with Period-Subtracting Structures

2021 ◽  
Vol 31 (11) ◽  
pp. 2150169
Author(s):  
Maryam Zolfaghari-Nejad ◽  
Hossein Hassanpoor ◽  
Mostafa Charmi
Keyword(s):  
Numerical Analysis ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Chaotic System ◽  
Strange Attractor ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Point Attractor

In this work, we present a novel three-dimensional chaotic system with only two cubic nonlinear terms. Dynamical behavior of the system reveals a period-subtracting bifurcation structure containing all [Formula: see text]th-order ([Formula: see text]) periods that are found in the dynamical evolution of the novel system concerning different values of parameters. The new system could be evolved into different states such as point attractor, limit cycle, strange attractor and butterfly strange attractor by changing the parameters. Also, the system is multistable, which implies another feature of a chaotic system known as the coexistence of numerous spiral attractors with one limit cycle under different initial values. Furthermore, bifurcation analysis reveals interesting phenomena such as period-doubling route to chaos, antimonotonicity, periodic solutions, and quasi-periodic motion. In the meantime, the existence of periodic solutions is confirmed via constructed Poincaré return maps. In addition, by studying the influence of system parameters on complexity, it is confirmed that the chaotic system has high spectral entropy. Numerical analysis indicates that the system has a wide variety of strong dynamics. Finally, a message coding application of the proposed system is developed based on periodic solutions, which indicates the importance of studying periodic solutions in dynamical systems.

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.

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