Linear water waves with vorticity: rotational features and particle paths

Steady linear gravity waves of small amplitude travelling on a current of constant vorticity are found. For negative vorticity we show the appearance of internal waves and vortices, wherein the particle trajectories are not any more closed ellipses. For positive vorticity the situation resembles that of Stokes waves, but for large vorticity the trajectories are affected

An improvement of Massera’s theorem for the existence and uniqueness of a periodic solution for the Li´enard equation

In this paper we prove the existence and uniqueness of a periodic solution for the Liénard equation x¨ + f (x) x˙ + x = 0. The classical Massera’s monotonicity assumptions, which are required in the whole line, are relaxed to the interval (\alpha,\delta ), where \alpha and \delta can be easily determined. In the final part of the paper a simple perturbation criterion of uniqueness is presented

Existence of limit cycles for some generalisation of the Liénard equations: the relativistic and the prescribed curvature cases.

We study the problem of existence of periodic solutions for some generalisations of the relativistic Liénard equation \begin{equation*} \frac{d}{dt}\frac{\dot{x}}{\sqrt{1-\dot{x}^{2}}}+\hat{f}(x,\dot{x})\dot{x}+g(x)=0 , \end{equation*} and the prescribed curvature Liénard equation \begin{equation*} \frac{d}{dt}\frac{\dot{x}}{\sqrt{1+\dot{x}^{2}}}+\hat{f}(x,\dot{x})\dot{x}+g(x)=0 , \end{equation*} where the damping function depends both on the position and the velocity. In the associated phase-plane this corresponds to a term of the form $f(x; y)$ instead of the standard dependence on x alone. By controlling the continuability of the solutions, we are able to prove the existence of at least a limit cycle in the associated phase-plane for both cases, moreover we provide results with a prefixed arbitrary number of limit cycles. Some examples are given to show the applicability of these results

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

n/

On the uniqueness of the limit cycle for the Li\ue9nard equation with f(x) not sign-definite

The problem of uniqueness of limit cycles for the Li\ue9nard equation \u1e8d+f(x)\u1e8b+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. \ua9 2017 Elsevier Lt