Resonant forced oscillations in systems with periodic nonlinearities
Alexander Krasnosel'skii
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.
keywords: higher order ODE resonance unbounded set of solutions method of stationary phase. Forced periodic oscillations periodic nonlinearities
On subharmonics bifurcation in equations with homogeneous nonlinearities
Alexander Krasnosel'skii Alexei Pokrovskii
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.
keywords: Bifurcation rotation of vector fields periodic solution bifurcation at infinity. periodic orbits Liénard equation subfurcation subharmonics
The index at infinity for some vector fields with oscillating nonlinearities
Alexander Krasnosel'skii Jean Mawhin
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.
keywords: oscillating nonlinearities. Vector fields
Peter E. Kloeden Alexander M. Krasnosel'skii Pavel Krejčí Dmitrii I. Rachinskii
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.

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Periodic canard trajectories with multiple segments following the unstable part of critical manifold
Alexander M. Krasnosel'skii Edward O'Grady Alexei Pokrovskii Dmitrii I. Rachinskii
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.
keywords: stability. canard trajectory degree theory Periodic solution

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