Canonical form under strong equivalence transformations and controllability indexes in singular systems

1988 ◽  
Vol 35 (11) ◽  
pp. 1438-1441
Author(s):  
S. Tan ◽  
J. Vandewalle
Keyword(s):  
Canonical Form ◽  
Singular Systems ◽  
Equivalence Transformations

Feedback canonical form for singular systems

1990 ◽  
Vol 52 (2) ◽  
pp. 347-376 ◽  
Author(s):  
HEIDE GLÜSING-LÜERBEN
Keyword(s):  
Canonical Form ◽  
Singular Systems

A Jordan control canonical form for singular systems

1988 ◽  
Vol 48 (5) ◽  
pp. 1769-1785 ◽  
Author(s):  
H. GLÜSING-LÜERßEN ◽  
D. HINRICHSEN
Keyword(s):  
Canonical Form ◽  
Singular Systems

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>

scholarly journals A canonical form for controllable singular systems

1989 ◽  
Vol 12 (2) ◽  
pp. 111-122 ◽  
Author(s):  
U. Helmke ◽  
M.A. Shayman
Keyword(s):  
Canonical Form ◽  
Singular Systems

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.

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