scholarly journals On an integrability criterion for a family of cubic oscillators

2021 ◽  
Vol 6 (11) ◽  
pp. 12902-12910
Author(s):  
Dmitry Sinelshchikov ◽  
Keyword(s):  
Canonical Form ◽  
Irreducible Polynomial ◽  
First Integral ◽  
Nonlinear Oscillators ◽  
Invariant Curves ◽  
Polynomial Invariant ◽  
Equivalence Transformations ◽  
First Derivative ◽  
Equivalence Criterion ◽  
Integrability Criterion

<abstract><p>In this work we consider a family of cubic, with respect to the first derivative, nonlinear oscillators. We obtain the equivalence criterion for this family of equations and a non-canonical form of Ince Ⅶ equation, where as equivalence transformations we use generalized nonlocal transformations. As a result, we construct two integrable subfamilies of the considered family of equations. We also demonstrate that each member of these two subfamilies possesses an autonomous parametric first integral. Furthermore, we show that generalized nonlocal transformations preserve autonomous invariant curves for the equations from the studied family. As a consequence, we demonstrate that each member of these integrable subfamilies has two autonomous invariant curves, that correspond to irreducible polynomial invariant curves of the considered non-canonical form of Ince Ⅶ equation. We illustrate our results by two examples: An integrable cubic oscillator and a particular case of the Liénard (4, 9) equation.</p></abstract>

A general time-dependent invariant for and integrability of the quadratic system

1994 ◽  
Vol 444 (1922) ◽  
pp. 643-650 ◽  
Keyword(s):  
Direct Method ◽  
First Integral ◽  
Quadratic System ◽  
Time Dependent ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Linear Polynomial ◽  
A Direct Method ◽  
Special Case ◽  
General Time

We derive a general time-dependent invariant (first integral) for the quadratic system (QS) that requires only one condition on the coefficients of the QS. The general invariant could yield asymptotic behaviour of phase-space trajectories. With more conditions imposed on the coefficients, the general invariant reduces to polynomial form and is equivalent to polynomial invariants found using a direct method. For the special case of a linear polynomial invariant where one of the variables is analytically invertible, the solution of the QS is reduced to a quadrature.

Nonlinear Oscillations in Chemical Networks as Symmetries of Transformation Groups

1977 ◽  
Vol 32 (1) ◽  
pp. 40-46
Author(s):  
H. G. Busse ◽  
B. Havsteen
Keyword(s):  
Canonical Form ◽  
Limit Cycles ◽  
Nonlinear Oscillations ◽  
Nonlinear Oscillators ◽  
Chemical Model ◽  
Transformation Groups ◽  
Mathematical Approach ◽  
Form Theorem

Abstract The description of phenomena created by nonlinear oscillators with the aid of transformation groups has been attempted. Such a treatment of the chemical model of Dreitlein and Smoes was met with success. The mathematical approach is based on an application of the canonical form theorem for one-parameter groups. Conditions for the occurence of limit cycles have been derived.

scholarly journals PROPERTIES OF GROUPS G OF DOUBLE ERRORS AND ITS INVARIANTS IN BCH CODES

2018 ◽  
pp. 40-46
Author(s):  
V. A. Lipnitskij ◽  
A. V. Serada
Keyword(s):  
Irreducible Polynomial ◽  
Galois Field ◽  
Bch Codes ◽  
Bch Code ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
The Core ◽  
Complex Process ◽  
Scope Of Application ◽  
Complete Set

The goal of the work is the further extending the scope of application of code automorthism in methods and algorithms of error correction by these codes. The effectiveness of such approach was demonstrated by norm of syndrome theory that was developed by Belarusian school of noiseless coding at the turn of the XX and XXI century. The group Г of the cyclical shift of vector component lies at the core of the theory. Under its action The error vectors are divided into disjoint Г-orbits with definite spectrum of syndromes. This allowed to introduce norms of syndrome of a family of BCH codes that are invariant over action of group Г. Norms of syndrome are unique characteristic of error orbit Г of any decoding set, hence it is the basis of permutation norm methods of error decoding. Looking over the Г-orbits of errors not the errors these methods are faster than classic syndrome methods of error decoding, are avoided from the complex process of solving the algebraic equation in Galois field, are simply implemented.A detailed theory for automorphism group G of BCH codes obtained by adding cyclotomic substitution to the group Г develops in the article. The authors held a detailed study of structure of G-orbit of errors as union of orbits Г of error vectors; one-to-one mapping of this structure on the norm structure of group Г. These norms being interconnected by Frobenius automorphism in the Galois field – field of BCH code constitute the complete set of roots of the only irreducible polynomial. It is a polynomial invariant of its orbit G. The main focus of the work is on the description of properties and specific features of groups G of double errors and its polynomial invariants.

On certain properties of linear iterative equations

2014 ◽  
Vol 12 (4) ◽  
pp. 648-657 ◽  
Author(s):  
Jean-Claude Ndogmo ◽  
Fazal Mahomed
Keyword(s):  
Normal Form ◽  
Canonical Form ◽  
Standard Form ◽  
Equivalence Transformations ◽  
Iterative Equation ◽  
Symmetry Properties ◽  
General Order ◽  
Iterative Equations

Abstract An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator that generates an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. An expression for the parameters of the source equation of the transformed equation under equivalence transformations is obtained, and this gives rise to the derivation of important symmetry properties for iterative equations. The transformation mapping a given iterative equation to the canonical form is obtained in terms of the simplest determining equation, and several examples of application are discussed.

scholarly journals On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Taha Aziz
Keyword(s):  
Heat Equation ◽  
Linear Space ◽  
Canonical Form ◽  
Space Time ◽  
Time Dependent ◽  
Canonical Forms ◽  
Pricing Model ◽  
Partial Differential ◽  
Equivalence Transformations ◽  
Parabolic Pde

The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.

scholarly journals The role of algebraic solutions in planar polynomial differential systems

2007 ◽  
Vol 143 (2) ◽  
pp. 487-508 ◽  
Author(s):  
HÉCTOR GIACOMINI ◽  
JAUME GINÉ ◽  
MAITE GRAU
Keyword(s):  
Logarithmic Derivative ◽  
Common Factor ◽  
Irreducible Polynomial ◽  
First Integral ◽  
Exponential Factor ◽  
Algebraic Functions ◽  
Darboux Function ◽  
Polynomial Differential Systems ◽  
Particular Solution ◽  
Planar Polynomial Differential Systems

AbstractWe study a planar polynomial differential system, given by$\dot{x}=P(x,y)$, $\dot{y}=Q(x,y)$. We consider a function$I(x,y)=\exp\!\{h_2(x) A_1(x,y) \diagup A_0(x,y) \}$ $ h_1(x)\prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}$, wheregi(x) are algebraic functions of$x$, $A_1(x,y)=\prod_{k=1}^r (y-a_k(x))$, $A_0(x,y)=\prod_{j=1}^s (y-\tilde{g}_j(x))$withak(x) and$\tilde{g}_j(x)$algebraic functions,A0(x,y) andA1(x,y) do not share any common factor,h2(x) is a rational function,h(x) andh1(x) are functions ofxwith a rational logarithmic derivative and$\alpha_i \in \mathbb{C}$. We show that ifI(x,y) is a first integral or an integrating factor, thenI(x,y) is a Darboux function. A Darboux function is a function of the form$f_1^{\lambda_1} \cdots f_p^{\lambda_p} \exp\{h/f_0\}$, wherefiandhare polynomials in$\mathbb{C}[x,y]$and the λi's are complex numbers. In order to prove this result, we show that ifg(x) is an algebraic particular solution, that is, if there exists an irreducible polynomialf(x,y) such thatf(x,g(x)) ≡ 0, thenf(x,y) = 0 is an invariant algebraic curve of the system. In relation with this fact, we give some characteristics related to particular solutions and functions of the formI(x,y) such as the structure of their cofactor.Moreover, we considerA0(x,y),A1(x,y) andh2(x) as before and a function of the form$\Phi(x,y):= \exp \{h_2(x)\, A_1(x,y)/A_0 (x,y) \}$. We show that if the derivative of Φ(x,y) with respect to the flow is well defined over {(x,y):A0(x,y) = 0} then Φ(x,y) gives rise to an exponential factor. This exponential factor has the form exp {R(x,y)} where$R=h_2 A_1/A_0 + B_1/B_0$and withB1/B0a function of the same form ash2A1/A0. Hence, exp {R(x,y)} factorizes as the product Φ(x,y) Ψ(x,y), for Ψ(x,y): = exp {B1/B0.

Performance analysis of global local mean square error criterion of stochastic linearization for nonlinear oscillator

2019 ◽  
Author(s):  
Nguyen Cao Thang ◽  
Luu Xuan Hung
Keyword(s):  
Performance Analysis ◽  
Mean Square Error ◽  
Nonlinear Oscillators ◽  
Degree Of Freedom ◽  
Equivalent Linearization ◽  
Mean Square ◽  
Error Criterion ◽  
Stochastic Linearization ◽  
Local Mean ◽  
Mean Square Error Criterion

The paper presents a performance analysis of global-local mean square error criterion of stochastic linearization for some nonlinear oscillators. This criterion of stochastic linearization for nonlinear oscillators bases on dual conception to the local mean square error criterion (LOMSEC). The algorithm is generally built to multi degree of freedom (MDOF) nonlinear oscillators. Then, the performance analysis is carried out for two applications which comprise a rolling ship oscillation and two degree of freedom one. The improvement on accuracy of the proposed criterion has been shown in comparison with the conventional Gaussian equivalent linearization (GEL).

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