On subharmonics bifurcation in equations with homogeneous nonlinearities
Alexander Krasnosel'skii Alexei Pokrovskii
Discrete & Continuous Dynamical Systems - A 2001, 7(4): 747-762 doi: 10.3934/dcds.2001.7.747
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
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 165-174 doi: 10.3934/dcds.2000.6.165
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
Resonant forced oscillations in systems with periodic nonlinearities
Alexander Krasnosel'skii
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 239-254 doi: 10.3934/dcds.2013.33.239
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

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