Search results for "Liénard equation"
showing 5 items of 5 documents
UNIQUENESS OF PERIODIC SOLUTIONS OF THE LIENARD EQUATION
1981
This chapter analyzes the uniqueness of periodic solutions of the Lienard equation. It considers the Lienard equation = y − F ( x ) and y = − x where F (0) = 0 , F ( x ) ∈ Lip( R ). The chapter discusses the existence of periodic solutions. It highlights that the origin is the only stationary point of the system = y − F ( x ) and y = − x , and therefore all nontrivial periodic solutions must circle around the origin. The existence of at least one periodic solution is proved by constructing a Poincare–Bendixson domain. The chapter also emphasizes that to prove the uniqueness of periodic solutions, additional assumptions are also needed. In the literature there are numerous uniqueness results…
Multiple period annuli in Liénard type equations
2010
Abstract We consider the equation x ″ x 1 − x 2 x ′ 2 + g ( x ) = 0 , where g ( x ) is a polynomial. We provide the conditions for existence of multiple period annuli enclosing several critical points.
Multi-layer canard cycles and translated power functions
2008
Abstract The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with n ⩾ 2 . The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n . We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions.
Modal expansions in lasers outside the uniform-field limit
2003
We show that, in lasers characterized by a slow population dynamics, the expansion of the electric field on longitudinal modes is useful even beyond the uniform-field limit. The dynamical behavior of the laser above the second threshold can be well reproduced by a set of ordinary differential equations, whose integration is much faster than that of the complete Maxwell–Bloch equations. The conditions for the uniform-field limit are also clarified.
More limit cycles than expected in Liénard equations
2007
The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x"+ f(x)x' + x = 0 where f is a polynomial. We prove that for a well-chosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n ≥ 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classic…