Lienard-type Equations Research Articles

Bell solitons in ultra-cold atomic Fermi gas

We demonstrate the existence of supersonic bell solitons in the Bardeen–Cooper–Schrieffer–Bose–Einstein condensate crossover regime. Starting from the extended Thomas–Fermi density functional theory of superfluid order parameter, a density transformation is used to map the hydrodynamic mean field equation to a Lienard-type equation. As a result, bell solitons are obtained as exact solutions, which is further verified by the numerical solution of the dynamical equation. The stability of the soliton is established and its behaviour in the entire crossover domain is obtained. It is found that, akin to the case of vortices, bell solitons yield highest contrast in the BEC regime.

Global existence and boundedness for a class of second-order nonlinear differential equations

In this paper we obtain new conditions for the global existence and boundedness of solutions for nonlinear second-order equations of the form ( r ( t ) | u ′ | p − 2 u ′ ) ′ + g ( t , u , u ′ ) u ′ + a ( t ) f ( u ) = e ( t ) , where p > 1 is a real constant. The results are applicable to well-known Emden–Fowler and Lienard type equations. An illustrative example is also provided.

On the approximate solution of an autonomous boundary-value problem by the Newton–Kantorovich method

We establish necessary and sufficient conditions for the existence of solutions of an autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. For the construction of solutions of a nonlinear Noetherian boundary-value problem in the critical case, we propose an iterative scheme that combines the Newton–Kantorovich method and the leastsquares technique. The efficiency of the proposed method is demonstrated in the analysis of a periodic problem for a Lienard-type equation.

Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

The problem of local linearizability of the planar linear center perturbed by cubic non- linearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., and Kurths, J., 2003, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, UK, and Blekhman, I. I., 1988, Synchronisation in Science and Technology, ASME Press Translations, New York) consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a five parameters non isochronous Chouikha cubic system (Chouikha, A. R., 2007, “Isochronous Centers of Lienard Type Equations and Applications,” J. Math. Anal. Appl., 331, pp. 358–376) we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system. The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb (J. C. Faugère, “FGb Salsa Software,” http://fgbrs.lip6.fr) for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts are different since our object is the synchronisation rather than the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers, (in particular Bardet, M., and Boussaada, I., 2011, “Compexity Reduction of C-algorithm,” App. Math. Comp., in press; Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(1), pp. 89–112; Bardet, M., Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(2), pp. 230–249; and furthermore, it can be considered as an adaptation of a qualitative theory method to a synchronization problem.

The role of the Jacobi last multiplier and isochronous systems

We employ Jacobi’s last multiplier (JLM) to study planar differential systems. In particular, we examine its role in the transformation of the temporal variable for a system of ODEs originally analysed by Calogero–Leyvraz in course of their identification of isochronous systems. We also show that JLM simplifies to a great extent the proofs of isochronicity for the Lienard-type equations.

On the uniform boundedness of solutions of Lienard type equations with multiple deviating arguments

This paper considers a Lienard type equation with multiple variable deviating arguments. We find some sufficient conditions for the solution of this equation to be uniform bounded by utilizing Lyapunov functional approach. An example is also given to show the effectiveness of our result.

New stability and boundedness results of Liénard type equations with multiple deviating arguments

The paper considers a Lienard type equation withmultiple variable deviating arguments. Some sufficient conditions, under which the solution of this equation is asymptotically stable and bounded by means of the Lyapunov functional approach, are found. An example showing the effectiveness of the result is given.

Factorization and Lie point symmetries of general Lienard-type equation in the complex plane

We present a variational approach to a general Lienard-type equation in order to linearize it and, as an example, the Van der Pol oscillator is discussed. The new equation which is almost linear is factorized. The point symmetries of the deformed equation are also discussed and the two-dimensional Lie algebraic generators are obtained.

New Results on Existence of Asymptotically Stable Periodic Solutions of a Forced Liénard Type Equation

In this work, we will be concerned with the following forced Lienard type equation: $$x^{\prime \prime} (t) + f(x(t))x^{\prime} (t) + g(t, x(t)) = e(t)$$ . Under certain assumptions, some new criteria for guaranteeing the existence of asymptotically stable (in the sense of Liapunov) periodic solutions of this equation are presented by applying Manasevich–Mawhin continuation theorem, Floquet theory, Liapunov stability theory and some analysis techniques. Moreover, an example is provided to demonstrate the applications of our results.

Existence and uniqueness of periodic solutions for forced Liénard-type equations

By using topological degree theory and some analysis skill, some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced Lienard-type equations are obtained.

Isochronous centers of Lienard type equations and applications

In this work we study Eq. ( E ) x ¨ + f ( x ) x ˙ 2 + g ( x ) = 0 with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of ( E ) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems ( L D , F ) . We also classify a 5-parameters family of reversible cubic systems with isochronous centers.

$p$-Laplacian and Lienard-Type Equation

$p$-Laplacian and Lienard-Type Equation

Existence and nonexistence of periodic solutions for the Li�nard and related equations

The existence of periodic solutions of Lienard type equations is converted into the existence of special fixed point problems with auxiliary conditions. A general method for exact calculation of the limit cycles is given, and the corresponding numerical iterative procedure is carried out in significant cases, with comparison to standard Runge-Kutta numerical integration. On the basis of the general theory, criteria for the existence of limit cycles are given and tested in particular cases.

The stability domain of the Lienard type equation

The author proves that the Lienard equation, ẍ + ƒ( x) ẍ + g( x) = 0, with appropritae conditions on ƒ( x) and g( x) has a minimum domain of stability. It is also inferred that if ƒ( x) = μ( x 2 −1) and g( x) = x, the upper limit on x is in the neighborhood of 2. This does not constitute a proof for the upper limit, but the computations of others gives the upper limit as 2.0235.