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### Open Access Journals

DCDS

We present an approach to study degenerate ODE with periodic nonlinearities;
for resonant higher order nonlinear equations $L(p)x=f(x)+b(t),\;p=d/dt$
with $2\pi$-periodic forcing $b$ and periodic $f$ we give multiplicity results,
in particular, conditions of existence of infinite and unbounded
sets of $2\pi$-periodic solutions.

DCDS

The bifurcation of subharmonics for resonant nonautonomous equations
of the second order is studied. The set of subharmonics is defined by principal
homogeneous parts of the nonlinearities

*provided these parts are not polynomials*. Analogous statements are proved for bifurcations of $p$-periodic orbits of a planar dynamical system. The analysis is based on topological methods and harmonic linearization.
DCDS

This paper is devoted to the computation of the index at
infinity for some asymptotically linear completely
continuous vector fields $x-T(x)$, when the principal
linear part $x-Ax$ is degenerate ($1$ is an eigenvalue
of $A$), and the sublinear part is not asymptotically
homogeneous (in particular do not satisfy Landesman-Lazer
conditions).
In this work we consider only the case of a one-dimensional
degeneration of the linear part, i.e.s $1$ is a simple
eigenvalue of $A$.
For this case we formulate an
abstract theorem and give some general examples for vector
fields of Hammerstein
type and for a two point boundary value problem.

DCDS-B

Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and
a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62.
For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

For more information please click the “Full Text” above

For more information please click the “Full Text” above

keywords:

DCDS-B

Periodic
canard trajectories with multiple segments following the unstable
part of critical manifold

We consider a scalar fast differential equation which is periodically driven by a slowly varying input.
Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions
for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions
between the stable branch and the unstable branch of the folded critical curve.
The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$
portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the
length of the time intervals of the slow motion separated by the short time intervals of fast transitions
between the branches are controlled by the parameters.

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