scholarly journals Integrating Factors and First Integrals for Liénard Type and Frequency-Damped Oscillators

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Emrullah Yaşar
Keyword(s):  
First Integral ◽  
First Integrals ◽  
The Other ◽  
First Order ◽  
Integrating Factor ◽  
Damped Oscillators ◽  
Damped Oscillator ◽  
Integrating Factors ◽  
Oscillator Equations

We consider Liénard type and frequency-damped oscillator equations. Integrating factors and the associated first integrals are derived from the method to compute -symmetries and the associated reduction algorithm. The knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. We will compare our results with those gained by the other methods.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

scholarly journals Polynomial Inverse Integrating Factors, First Integral and Non-Existence of Limit Cycles in the Plane for Quadratic Systems

2017 ◽  
Vol 5 (2) ◽  
pp. 232
Author(s):  
Ahmed M. Hussien
Keyword(s):  
Limit Cycles ◽  
First Integral ◽  
Quadratic Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor ◽  
Integrating Factors

The main purpose of this paper is to study the existence of polynomial inverse integrating factor and first integral, and non-existence of limit cycles for all systems. Furthermore, we consider some applications.

scholarly journals The orbital evolution of resonant chains of exoplanets incorporating circularisation produced by tidal interaction with the central star with application to the HD 158259 and EPIC 245950175 systems

2021 ◽  
Vol 133 (6) ◽  
Author(s):  
J. C. B. Papaloizou
Keyword(s):  
Central Star ◽  
Orbital Evolution ◽  
The Other ◽  
Mass Ratio ◽  
Tidal Effects ◽  
First Order ◽  
Tidal Interaction ◽  
Nearest Neighbours ◽  
Three Body

AbstractWe study orbital evolution of multi-planet systems that form a resonant chain, with nearest neighbours close to first order commensurabilities, incorporating orbital circularisation produced by tidal interaction with the central star. We develop a semi-analytic model applicable when the relative proximities to commensurability, though small, are large compared to $$\epsilon ^{2/3},$$ ϵ 2 / 3 , with $$\epsilon $$ ϵ being a measure of the characteristic planet to central star mass ratio. This enables determination of forced eccentricities as well as which resonant angles enter libration. When there are no active linked three body Laplace resonances, the rate of evolution of the semi-major axes may also be determined. We perform numerical simulations of the HD 158259 and EPIC 245950175 systems finding that the semi-analytic approach works well in the former case but not so well in the latter case on account of the effects of three active three body Laplace resonances which persist during the evolution. For both systems we estimate that if the tidal parameter, $$Q',$$ Q ′ , significantly exceeds 1000,  tidal effects are unlikely to have influenced period ratios significantly since formation. On the other hand if $$Q' < \sim 100$$ Q ′ < ∼ 100 tidal effects may have produced significant changes including the formation of three body Laplace resonances in the case of the EPIC 245950175 system.

First Integrals and Integrating Factors of Second-Order Autonomous Systems

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Tamás Kalmár-Nagy ◽  
Balázs Sándor
Keyword(s):  
Harmonic Oscillator ◽  
Autonomous Systems ◽  
Dynamical Symmetry ◽  
Nonlinear Oscillators ◽  
First Integrals ◽  
Second Order ◽  
Point Symmetry ◽  
New Approach ◽  
First Order ◽  
Integrating Factors

We present a new approach to the construction of first integrals for second-order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first-order equations and their Lie point symmetry and integrating factors of second-order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first-order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.

Determination of D[C6F5CH2—Br] by the toluene carrier method

1980 ◽  
Vol 58 (18) ◽  
pp. 1906-1908 ◽  
Author(s):  
R. John Kominar ◽  
Michael J. Krech ◽  
Stanley James W. Price
Keyword(s):  
Arrhenius Equation ◽  
Parent Compound ◽  
Material Balance ◽  
Internal Standard ◽  
Standard Technique ◽  
The Other ◽  
Reasonable Estimate ◽  
First Order ◽  
A Value

The pyrolysis of C6F5CH2Br has been studied by the toluene carrier technique over the temperature range 727–800 °C. Rate constants are based on analysis for residual parent compound by gas chromatography using an internal standard technique. In selected runs a material balance of 100 ± 2% was obtained for bromine based on C6F5CH2Br plus HBr. Within the limits of the experimental technique the process appears to be first order and homogeneous. In addition to HBr the other major products of the thermal decomposition are C6F5CH3, (C6F5CH2)2, C6F5CH2CH2C6H5, and (C6H5CH2)2. The Arrhenius equation obtained is[Formula: see text]The log A value is very close to the value of 14.6 recommended by Benson and O'Neal for the decomposition of C6H5CH2Br. The activation energy, 225 ± 6 kJ mol−1, should be a reasonable estimate of D[C6F5CH2—Br].

New Conservation Laws, Lagrangian Forms, and Exact Solutions of Modified Emden Equation

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Gülden Gün Polat ◽  
Teoman Özer
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
First Integrals ◽  
Lie Symmetries ◽  
Jacobi Method ◽  
Emden Equation ◽  
Mathematical Connections ◽  
Integrating Factors

This study deals with the determination of Lagrangians, first integrals, and integrating factors of the modified Emden equation by using Jacobi and Prelle–Singer methods based on the Lie symmetries and λ-symmetries. It is shown that the Jacobi method enables us to obtain Jacobi last multipliers by means of the Lie symmetries of the equation. Additionally, via the Lie symmetries of modified Emden equation, we analyze some mathematical connections between λ-symmetries and Prelle–Singer method. New and nontrivial Lagrangian forms, conservation laws, and exact solutions of the equation are presented and discussed.

scholarly journals Characterization of Symmetry Properties of First Integrals for Submaximal Linearizable Third-Order ODEs

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Differential Equations ◽  
Lie Algebras ◽  
First Integral ◽  
First Integrals ◽  
The Other ◽  
Symmetry Class ◽  
Third Order ◽  
Symmetry Properties ◽  
The Relationship

The relationship between first integrals of submaximal linearizable third-order ordinary differential equations (ODEs) and their symmetries is investigated. We obtain the classifying relations between the symmetries and the first integral for submaximal cases of linear third-order ODEs. It is known that the maximum Lie algebra of the first integral is achieved for the simplest equation and is four-dimensional. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the first integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2, and 3 symmetries, and for the 4 symmetry class of linear third-order ODEs, they are 0, 1, and 2 symmetries, respectively. In the case of submaximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals.

scholarly journals SYMMETRIES AND FIRST INTEGRALS FOR NON-VARIATIONAL EQUATIONS

2007 ◽  
Vol 04 (07) ◽  
pp. 1217-1230
Author(s):  
DIEGO CATALANO FERRAIOLI ◽  
PAOLA MORANDO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
First Integrals ◽  
Partial Differential ◽  
Variational Equations ◽  
First Order ◽  
The Ideal

For a class of exterior ideals, we present a method associating first integrals of the characteristic distributions to symmetries of the ideal. The method is applied, under some assumptions, to the study of first integrals of ordinary differential equations and first order partial differential equations as well as to the determination of first integrals for integrable distributions of vector fields.

scholarly journals Generalized symmetries, first integrals, and exact solutions of chains of differential equations

2021 ◽  
Vol Volume 1 ◽  
Author(s):  
C. Muriel ◽  
M. C. Nucci
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
First Integral ◽  
First Integrals ◽  
Order Equation ◽  
Generalized Symmetries ◽  
Complete Set

New integrability properties of a family of sequences of ordinary differential equations, which contains the Riccati and Abel chains as the most simple sequences, are studied. The determination of n generalized symmetries of the nth-order equation in each chain provides, without any kind of integration, n-1 functionally independent first integrals of the equation. A remaining first integral arises by a quadrature by using a Jacobi last multiplier that is expressed in terms of the preceding equation in the corresponding sequence. The complete set of n first integrals is used to obtain the exact general solution of the nth-order equation of each sequence. The results are applied to derive directly the exact general solution of any equation in the Riccati and Abel chains. Comment: 16 pages

Determination of the lattice translation across {110} APBs in GaAs

1988 ◽  
Vol 46 ◽  
pp. 600-601
Author(s):  
D.R. Rasmussen ◽  
N.-H. Cho ◽  
C.B. Carter
Keyword(s):  
Grain Boundaries ◽  
Lower Energy ◽  
Antiphase Boundary ◽  
The Other ◽  
Boundary Structure ◽  
Interface Plane ◽  
Inversion Symmetry ◽  
High Angle ◽  
Equilibrium Distance

Domains in GaAs can exist which are related to one another by the inversion symmetry, i.e., the sites of gallium and arsenic in one domain are interchanged in the other domain. The boundary between these two different domains is known as an antiphase boundary [1], In the terminology used to describe grain boundaries, the grains on either side of this boundary can be regarded as being Σ=1-related. For the {110} interface plane, in particular, there are equal numbers of GaGa and As-As anti-site bonds across the interface. The equilibrium distance between two atoms of the same kind crossing the boundary is expected to be different from the length of normal GaAs bonds in the bulk. Therefore, the relative position of each grain on either side of an APB may be translated such that the boundary can have a lower energy situation. This translation does not affect the perfect Σ=1 coincidence site relationship. Such a lattice translation is expected for all high-angle grain boundaries as a way of relaxation of the boundary structure.

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