The Bifurcation Behaviour of Periodic Solutions of Impact Oscillators

1997 ◽  
pp. 101-108
Author(s):  
J. P. Meijaard
Keyword(s):  
Periodic Solutions ◽  
Impact Oscillators ◽  
Bifurcation Behaviour

scholarly journals Periodic solutions of n-dofs autonomous vibro-impact oscillators with one lasting contact phase

2017 ◽  
Vol 90 (3) ◽  
pp. 1771-1783 ◽  
Author(s):  
Anders Thorin ◽  
Pierre Delezoide ◽  
Mathias Legrand
Keyword(s):  
Periodic Solutions ◽  
Contact Phase ◽  
Impact Oscillators

Existence of quasi-periodic solutions and Littlewood’s boundedness problem of sub-linear impact oscillators

2011 ◽  
Vol 74 (16) ◽  
pp. 5606-5617 ◽  
Author(s):  
Zhiguo Wang ◽  
Qihuai Liu ◽  
Dingbian Qian
Keyword(s):  
Periodic Solutions ◽  
Impact Oscillators ◽  
Boundedness Problem

Periodic Solutions and Chaotic Dynamics in Forced Impact Oscillators

2013 ◽  
Vol 12 (1) ◽  
pp. 383-414 ◽  
Author(s):  
Alfonso Ruiz-Herrera ◽  
Pedro J. Torres
Keyword(s):  
Periodic Solutions ◽  
Chaotic Dynamics ◽  
Impact Oscillators

Hopf bifurcation for equivariant conservative and time-reversible systems

1990 ◽  
Vol 116 (1-2) ◽  
pp. 103-128 ◽  
Author(s):  
A. Vanderbauwhede
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Reversible Systems ◽  
Schmidt Method ◽  
Bifurcation Equation ◽  
Hamiltonian Hopf Bifurcation ◽  
Bifurcation Behaviour ◽  
Small Periodic

SynopsisWe study the bifurcation of small periodic solutions at a non-semi-simple 1:1-resonance in equivariant conservative or equivariant time-reversible systems. By using an equivariant Liapunov-Schmidt method and restricting to solutions with an appropriate isotropy, we reduce the problem to a scalar bifurcation equation. The analysis of this equation shows a bifurcation behaviour similar to that found for the Hamiltonian Hopf bifurcation.

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.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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