Integrability and solvability of polynomial Liénard differential systems

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time-dependent exponential factor.     

On the Center of the Generalized Liénard System

In this paper, we discuss the conditions for a center for the generalized Liénard system (E)₁

On the criteria for integrability of the Liénard equation

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Liénard equation using an approach based on nonlocal transformations. We also obtain some of the previously known criteria for integrability of the Liénard equation as a straightforward consequence of our approach’s application. We illustrate our results by several new examples of integrable Liénard equations.     

On the number of limit cycles of polynomial Liénard systems

Liénard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Liénard systems of arbitrary degree on the plane, and develop a new method to obtain a lower bound of the maximal number of limit cycles. Using the method and basing on some known results for lower degree we obtain new estimations of the number of limit cycles in the systems which greatly improve existing results.     

On the Liénard system with two isoclines

In this paper, the non-existence of limit cycles of a Liénard system ẋ = y−F(x), ẏ = −g(x) is discussed. By using the transformation y = z+ϕ(x) to the system, the new system has two special isoclines. We call the curves Vertical isocline or Horizontal isocline, respectively. It shall be shown that the existence of these isoclines play an important role in the non-existence of limit cycles of the system. The results are applied to many examples, and the known results are improved in certain cases. The results were announced at Annual Meeting of Mathematical Society of Japan on September 19 of 2005. Also they were published at the Poster Competition of ICM(Madrid) on August 22–30 of 2006.     

On the Non-Existence of the Closed Orbit for a Liénard System

A sufficient condition for a Liénard system to have no non-trivial closed orbits is given by transforming the system into another system called the Bogdanov—Takens system. The result here (Theorem 2) is a partial improvement of that of Wang and Yu [2].AMS Subject Classification (2000), 34C07, 34C25, 34C26, 34D20     

On one conjecture in the theory of isochronous Liénard systems

New normal forms are obtained for the center as well as isochronous center of holomorphic Liénard systems.     

Generalized theorems of Liénard and Shepherd

The paper considers a real polynomial p(x)=₀+₁x+...+_nxⁿ(₀ > 0) for which there hold inequalities ₁>0, ₃>0, ... or ₂>0, ₄>0, ..., where ₁, ₂, ..., _jn are the Hurwitz determinants for polynomial p(x). It is proven that polynomial p(x) can have, in the right half-plane, only real roots, where the quantity of positive roots of polynomial p(x) equals the quantity of changes of sign in the system of coefficients a₀, a₂, ..., a_n–2, a_n, when n is even, and ₀, a₂,..., a_n–1, a_n, when n is odd. From the proven theorem, in particular, there follows the Liénard and Shepherd criterion of stability.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 13–21, July, 1977.     

Local Bifurcation of Critical Periods for a Class of Liénard Equations

In this paper,we study the local bifurcation of critical periods near the nondegenerate center(the origin) of a class of Liénard equations with degree 2n,and prove that at most 2n-2 critical periods(taken into account multiplicity) can be produced from a weak center of finite order.We also prove that it can have exactly2n-2 critical periods near the origin.     

The localization of attractors of the Liénard equation

A method for localizing the attractors of the Liénard equation is proposed, based on the construction of special piecewise-linear discontinuous comparison systems.     

Stability of Liénard Equation

We consider a balance stability problem for the second order nonlinear differential equations of the Liénard type. Investigations are carried out by means of constant-sign Lyapunov functions for problems of stability, asymptotical stability (local and global), and instability. We implicitly formulate a method of construction of constant-sign functions suitable for solving problems of motion stability. Special attention is paid to a problem of non-asymptotical stability, where we demonstrate possibilities of new assertions that rely upon a usage of constant-sign Lyapunov functions.     

Periodic solutions of the retarded Liénard equation

Consider the equation

On the Strong Solutions and the Structural Stability of the g-Bénard Problem

In this paper, we study a type of modified Boussinesq equations which is called g-Bénard problem. We show the existence and uniqueness of strong solutions of the problem in two dimensions, and then, we investigate the continuous dependence of the solutions on the viscosity parameter.