Gabriele Villari

Periodic solutions of the Liénard equation with one-sided growth restrictions

In this article we are concerned with the problem of the existence of periodic solutions to the periodically forced scalar Lienard equation x” + f(x) x’ + g(x) = e( t ) (’ = d/dt), (1) where ,f, g, e: R + R are continuous functions and e(.) is periodic. We are looking for solutions of (1) having the same period of the forcing term e(.). Throughout the paper, we assume, without loss of generality (see e.g. [22; 36, p. 693]), that

Steady Water Waves with Multiple Critical Layers: Interior Dynamics

We study small-amplitude steady water waves with multiple critical layers. Those are rotational two-dimensional gravity-waves propagating over a perfect fluid of finite depth. It is found that arbitrarily many critical layers with cat’s-eye vortices are possible, with different structure at different levels within the fluid. The corresponding vorticity depends linearly on the stream function.

Limit cycle uniqueness for a class of planar dynamical systems

Abstract A uniqueness theorem for limit cycles of a class of plane differential systems is proved. The main result is applicable to second order O.D.E.’s with a dissipative term depending both on the position and on the velocity.

On Massera’s theorem concerning the uniqueness of a periodic solution for the Liénard equation. When does such a periodic solution actually exist?

AbstractIn this note we consider the classical Massera theorem, which proves the uniqueness of a periodic solution for the Liénard equation x¨+f(x)x˙+x=0, and investigate the problem of the existence of such a periodic solution when f is monotone increasing for x>0 and monotone decreasing for x<0 but with a single zero, because in this case the existence is not granted. Sufficient conditions for the existence of a periodic solution and also a necessary condition, which proves that with this assumptions actually it is possible to have no periodic solutions, are presented.MSC:34C05, 34C25.

Some remarks on non-conservative oscillatory systems with periodic solutions

Abstract We prove the existence or a family of periodic orbits for a wide class of two-dimensional differential systems verifying suitable symmetry conditions. Elementary phase plane analysis is used.

Periodic solutions of a switching dynamical system in the plane

The non linear differential equation is considered. We investigate the number of nontrivial periodic solutions of this equation depending on k ≥ 0 for µ < 0 fixed. The results obtained can be used in a problem of time optimal control for the Van der Pol equation in order to describe the process of control.

On the uniqueness of the limit cycle for the Liénard equation with f(x) not sign-definite

Abstract The problem of uniqueness of limit cycles for the Lienard equation x + f ( x ) x + g ( x ) = 0 is investigated. The classical assumption of sign-definiteness of f ( x ) is relaxed. The effectiveness of our result as a perturbation technique is illustrated by some constructive examples of small amplitude limit cycles, coming from bifurcation theory.

A Survival Kit in Phase Plane Analysis: Some Basic Models and Problems

The aim of this note is to show how phase plane analysis is a strong tool for the study of a mathematical model, in view of its application in water waves theory. This because only in recent work such method was actually used in water waves theory and people working in this field area might be interested in a discussion of the basic ideas of phase plane analysis, which we call “a survival kit”. In this light, at first we review some classical results in Dynamics of Population and Epidemiology, and then we investigate more carefully the phase portrait of the classical Lienard equation. In particular, starting from the Van Der Pol equation, the problem of existence and uniqueness of limit cycles will be treated and the methods used to attack this problem will be presented. Finally we come back to water waves theory and present in details the results of a joint paper with Constantin (Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, Wiley, New York, 1977) in which, as far as we know, for the first time phase plane analysis was used in this kind of problems.