Nonconstant Periodic Solutions Research Articles

Periodic Solutions to a Nonlinear Volterra Integro-Differential Equation

A nonlinear Volterra integro-differential equation arising from the theory of population dynamics is shown to have a nonconstant periodic solution whenever the (biologically important) steady state is unstable. Existence of the periodic solution is proved by a fixed point argument.

Periodic solutions for certain protein synthesis models

In this paper, we investigate the qualitative behavior of a class of deterministic models with delays that arise in the study of protein synthesis. We demonstrate the existence of nonconstant periodic solutions to our system of differential-delay equations under certain conditions on the parameters. We conclude with a brief discussion of the biological significance of our results.

Periodic solutions of a nonlinear second order differential equation with delay

It is proved that the autonomous difference-differential equation x ̈ (t) + (a + b) x ̇ (t) + ab x(t) = −f(x(t − 1)) (0) has, for a broad class of functions f, a nonconstant periodic solution whenever the associated characteristic equation has a root with positive real part.

Periodic solutions of special differential equations: an example in non-linear functional analysis

SynopsisWe consider differential-delay equations which can be written in the formThe functions fi and gk are all assumed odd. The equationis a special case of such equations with q = N + 1 (assuming f is an odd function). We obtain an essentially best possible theorem which ensures the existence of a non-constant periodic solution x(t) with the properties (1) x(t)≧0 for 0≦t≦q, (2) x(–t) = –x(t) for all t and (3) x(t + q) = –x(t) for all t. We also derive uniqueness and constructibility results for such special periodic solutions. Our theorems answer a conjecture raised in [8].

On the stability of the stationary state of a population growth equation with time-lag.

If in the Verhulst equation for population growth the reproduction factor depends on the history then the equilibrium may become unstable and oscillations and even non-constant periodic solutions may occur. It is shown that the equilibrium is unstable if the reproduction factor at time t is, up to a sufficiently large factor, an arbitrary average of the population densities in the interval (t-2, t-1).

Existence of a non-constant periodic solution of a non-linear autonomous functional differential equation representing the growth of a single species population.

We consider an integro-differential equation for the densityn of a single species population where the birth rate is constant and the death rate depends on the values ofn in an interval of length τ - 1 > 0. We prove the existence of a non-constant periodic solution under the conditions birth rate b > π/2 and τ- 1 small enough. The basic idea of proof (due to R. D. Nussbaum) is to employ a theorem about non-ejective fixed points for a translation operator associated with the solutions of the equation.

Period bounds for generalized Rayleigh equation

Simple upper and lower bounds are obtained for the least period T of any non-constant periodic solution x( t) of the differential equation x″ − F( x') + g( x) = 0.

Electrostatic oscillations in inhomogeneous cold plasmas: E. M. Barston, Microwave Laboratory, W. W. Hansen Laboratories of Physics, Stanford University, Stanford, California

We investigate the existence of collision-free nonconstant periodic solutions of the N-vortex problem in domains Ω⊂C. These are solutions z(t)=(z1(t),…,zN(t))∈ΩN of the first order Hamiltonian system where the Hamiltonian HΩ has the form The function F:ΩN→R depends on the regular part of the hydrodynamic Green's function and is unbounded from above: F(z)→∞ if zk→∂Ω for some k. The Hamiltonian is unbounded from above and below, it is singular, not integrable, energy surfaces are not compact and not known to be of contact type. Using singular perturbation techniques and Conley index theory we prove the existence of a family of periodic solutions zr(t), 0<r<r0, with arbitrarily small minimal period Tr→0 as r→0. The solutions are close to the singular set of HΩ. Our result applies in particular to generic bounded domains, which may be simply or multiply connected. It also applies to certain unbounded domains. Depending on the domain there are multiple such families.