Displaying similar documents to “Some examples of vector fields on the 3-sphere”

Some examples of nonsingular Morse-Smale vector fields on S 3

F. Wesley Wilson Jr (1977)

Annales de l'institut Fourier

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One wonders or not whether it is possible to determine the homotopy class of a vector field by examining some algebraic invariants associated with its qualitative behavior. In this paper, we investigate the algebraic invariants which are usually associated with the periodic solutions of non-singular Morse-Smale vector fields on the 3-sphere. We exhibit some examples for which there appears to be no correlation between the algebraic invariants of the periodic solutions and the homotopy...

On systems governed by two alternating vector fields

Alois Klíč, Jan Řeháček (1994)

Applications of Mathematics

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We investigate the nonautonomous periodic system of ODE’s of the form x ˙ = v → ( x ) + r p ( t ) ( w → ( x ) - v → ( x ) ) , where r p ( t ) is a 2 p -periodic function defined by r p ( t ) = 0 for t ∈ 〈 0 , p 〉 , r p ( t ) = 1 for t ∈ ( p , 2 p ) and the vector fields v → and w → are related by an involutive diffeomorphism.

Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers

Grzegorz Graff, Agnieszka Kaczkowska (2013)

Annales Polonici Mathematici

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Let f be a smooth self-map of an m-dimensional (m ≥ 4) closed connected and simply-connected manifold such that the sequence L ( f ⁿ ) n = 1 ∞ of the Lefschetz numbers of its iterations is periodic. For a fixed natural r we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to r. The resulting number is given by a topological invariant J[f] which is defined in combinatorial terms and is constant for all sufficiently large r. We compute J[f]...

Periodic solutions of the Rayleigh equation with damping of definite sign

Pierpaolo Omari, Gabriele Villari (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The existence of a non-trivial periodic solution for the autonomous Rayleigh equation x ¨ + F ⁢ x ˙ + g ⁢ x = 0 is proved, assuming conditions which do not imply that F ⁢ x ⁢ x has a definite sign for x large. A similar result is obtained for the periodically forced equation x ¨ + F ⁢ x ˙ + g ⁢ x = e ⁢ t .

Periodic segments and Nielsen numbers

Klaudiusz Wójcik (1999)

Banach Center Publications

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We prove that the Poincaré map φ ( 0 , T ) has at least N ( h ˜ , c l ( W 0 W 0 - ) ) fixed points (whose trajectories are contained inside the segment W) where the homeomorphism h ˜ is given by the segment W.

Invariant tori for periodically perturbed oscillators.

Carmen Chicone (1997)

Publicacions Matemàtiques

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The response of an oscillator to a small amplitude periodic excitation is discussed. In particular, sufficient conditions are formulated for the perturbed oscillator to have an invariant torus in the phase cylinder. When such an invariant torus exists, some perturbed solutions are in the basin of attraction of this torus and are thus entrained to the dynamical behavior of the perturbed system on the torus. In particular, the perturbed solutions in the basin of attraction of the invariant...