Existence Of Harmonic Solutions Research Articles

Harmonic solutions for a class of non-autonomous piecewise linear oscillators

The study deals with the existence and uniqueness conditions of harmonic solutions of a class of piecewise linear oscillators with a general periodic excitation. Based on the relationship between natural frequency and driving frequency, we divide the discussion of the harmonic solutions into resonant case and non-resonant case. For resonant case, the existence conditions dependent on periodic excitation and clearance of harmonic solutions are given based on the Poincaré-Bohl fixed point theorem. Moreover, we give a necessary and sufficient condition of bounded solutions, where harmonic solutions, n-subharmonic solutions, and quasi-periodic solutions can occur simultaneously. For non-resonant case, the existence of harmonic solutions are analyzed based on contract mapping. The uniqueness of harmonic solutions for resonant and non-resonant cases are proven by variational method and reduction to absurdity. Numerical examples are presented to illustrate the theoretical results.

Harmonic and subharmonic solutions of quadratic Liénard type systems with sublinearity

<abstract><p>In this paper, we prove the existence of harmonic solutions and infinitely many subharmonic solutions of dissipative second order sublinear differential equations named quadratic Liénard type systems. The method of the proof is based on the Poincaré-Birkhoff twist theorem.</p></abstract>

On the existence of harmonic solutions for higher-order Lienard systems

In this paper, by using of the theory of coincidence degree, we obtain the new conditions which guarantee the existence of harmonic solutions for Lienard Systems $$\ddot x = \frac{d}{{dt}}gradF(x) + gradG(x) = p(t)$$ our results do not require that the damping must be positive.

On the existence of harmonic solutions of Liénard systems

We discuss the existence of harmonic solutions of Lienard systems of the form x+d/dtgradF(x)+gradG(x)=p(t), where F∈C 2 (R n ,R), G∈C 1 (R n ,R), p∈C 0 (R,R n ) and p(t+T)=p(t) for a constant T>0

Fundamental solutions for the anisotropic neutron transport equation

SynopsisWe construct a fundamental solution for the n dimensional time independent anisotropic neutron transport equation. This is an operator valued distribution G(x) with a singularity at the origin. By estimating G(x) we are able to construct smooth solutions to the transport equation. We are also able to derive in a straightforward fashion results of Birkhoff and Abu-Shumays on the existence of harmonic solutions to the isotropic transport equation. When n = 1, G(x) is a function which is continuous except at x = 0. We show that the classical formula for the jump of G(x) at the origin is equivalent to the completeness of Case's full range eigenfunction expansion.