A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

Chandrasekar, V. K.; Senthilvelan, M.; Kundu, Anjan; Lakshmanan, M.

doi:10.1088/0305-4470/39/31/006

Cited by 42 publications

(65 citation statements)

References 28 publications

(57 reference statements)

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“…Their approach was confined to the determination of these nonlocal symmetries that reduce to point symmetries under reduction of order by ∂ ∂t . Later several authors have studied nonlocal symmetries of nonlinear ODEs [25,26,27,28,29,30]. Nucci and Leach have introduced a way to find the nonlocal symmetries [29].…”

Section: Nonlocal Symmetriesmentioning

confidence: 99%

“…To understand the integrability of these nonlinear ODEs, through Lie symmetry analysis, attempts have been made to extend Lie's theory of continuous group of point transformations in several directions. A few notable extensions which have been developed for this purpose are (i) contact symmetries [23,21,22,24], (ii) hidden and nonlocal symmetries [36,37,31,25,26,27,28,29,30,33,34,35], (iii) λ-symmetries [20,38,39,40], (iv) adjoint symmetries [10,9,41] and (v) telescopic vector fields [42,40].…”

Section: Introductionmentioning

confidence: 99%

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Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

2015

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Abstract. Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, λ-symmetries, adjoint symmetries and telescopic vector fields of a second-order ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between (i) symmetries and integrating factors and (ii) symmetries and integrals are also discussed and illustrated through the same example. The interconnections between some of the above symmetries, that is (i) Lie point symmetries and λ-symmetries and (ii) exponential nonlocal symmetries and λ-symmetries are also discussed. The order reduction procedure is invoked to derive the general solution of the second-order equation.

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Section: Nonlocal Symmetriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

2015

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“…Equation (70) is a generalized special case of the Chazy equation XII [1] and the subcases of (70) has been studied in detail in Refs. [23][24][25][26][27][28][29].…”

Section: Third Order Odesmentioning

confidence: 99%

A nonlocal connection between certain linear and nonlinear ordinary differential equations – Part II: Complex nonlinear oscillators

Mohanasubha

Sheeba

Chandrasekar

et al. 2013

Applied Mathematics and Computation

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Abstract. In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations (ODEs) with complex linear ODEs, thereby proving the integrability of the former. We also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples.1. Overview MotivationIn a previous paper [1] we have developed a new way of identifying integrable nonlinear ordinary differential equations (ODEs) by relating linear and nonlinear oscillator equations of any order through suitable nonlocal transformations. We have also devised a method to derive explicit general solution of the nonlinear ODEs. Interestingly the nonlocal oscillators identified through this procedure posses certain remarkable properties [2]. In this paper we extend the underlying features of this method to the case of complex ODEs and identify a class of integrable complex nonlinear oscillators including hierarchy of Stuart-Landau equations, complex modified Emden equation, complex Duffing-van der Pol oscillator equation and so on.The necessity of this analysis comes from the fact that complex nonlinear ODEs are being used to describe autoresonance/parametric resonance phenomenon in a number of problems in different branches of physics and are being widely used in the contemporary nonlinear dynamics literature [3,4]. Some of the interesting systems include, the Poincaré equation dz dt +αz+β|z|z = 0 and Stuart-Landau oscillator equation dz dt + αz + β|z| 2 z = 0 which play crucial roles in explaining complex network properties of systems in physics, chemistry, biology and sociology, where coupled models of these systems are of basic interest. Examples include Josephson junction arrays, lasers,

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“…To give support for this type of linearizing transformations, we note that the MEE (2.10) can also be linearized to the free particle equation (d 2 w/dz 2 = 0) and third-order linear equation (d 3 w/dz 3 = 0) through the nonlocal transformations (i) w = xe R xdt , z = t and (ii) w = e R xdt , z = t, respectively [6]. However, in this paper we restrict our attention only to the case in which the new dependent variable is not a nonlocal one.…”

Section: ẋ) and Z = G(t Xẋ)dt And (Ii) W = F (T Xẋ)dt And Z = G(mentioning

confidence: 99%

“…Recently, the present authors have proposed certain generalized linearizing transformations in which the new independent variable is allowed to have derivative terms also besides being nonlocal [5]. Apart from these, attempts have also been made to linearize certain second-order nonlinear ODEs by specific nonlocal transformations [6]. As far as the third-order nonlinear ODEs are concerned the study on linearization was started by Lie himself.…”

Section: Introductionmentioning

confidence: 99%

A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations I: Scalar Case

Chandrasekar¹,

Senthilvelan²,

Lakshmanan³

2012

JNMP

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In this paper we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focus our attention on a system of two coupled second-order ODEs. In the case of scalar ODEs, we consider second and third-order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third-order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second-order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.

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A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

Cited by 42 publications

References 28 publications

Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

A nonlocal connection between certain linear and nonlinear ordinary differential equations – Part II: Complex nonlinear oscillators

A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations I: Scalar Case

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