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October  2001, 7(4): 747-762. doi: 10.3934/dcds.2001.7.747

On subharmonics bifurcation in equations with homogeneous nonlinearities

Alexander Krasnosel'skii 1, and  Alexei Pokrovskii 2,

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetny Lane, Moscow 101447, Russian Federation
2. 

Institute for Nonlinear Science, Department of Physics, National University of Ireland, University College, Cork, Ireland

Received  January 2001 Published  July 2001

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.
Mathematics Subject Classification: 34C23, 34C35, 54H25, 34H0.
Citation: Alexander Krasnosel'skii, Alexei Pokrovskii. On subharmonics bifurcation in equations with homogeneous nonlinearities. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 747-762. doi: 10.3934/dcds.2001.7.747
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