Orbital Reversibility of Planar Vector Fields

Abstract

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible.

Orbitally symmetric systems with applications to planar centers

“…See these properties together with the phase portrait of 6.2 in Figure 8. We notice that the implicit relation 3.3 with a = 1 for the equilibrium points (2,8) and (−8/13, 0) gives δ = 0. Consequently, the involution ϕ is not well defined on them because their images by ϕ would go to the origin.…”

“…In [1] is proved that there exists a normal form change of variables, ψ, such that the Hamiltonian H(x, y) = −(2x 6 + 12ax 3 y 2 + 3y 4 + 12ax 5 y + 4bx 2 y 3 )/12 is transformed in the time-reversible Hamiltonian H(ψ(x, y)) = −(2x 6 + 12ax 3 y 2 + 3y 4 + 12cx 4 y 2 )/12. This property is used in [2] to prove its orbital reversibility. The associated vector field satisfies 1.3 with the transformed involution, using Lemma 2.6, given by the change ψ and the classical involution ϕ 1 (x, y) = (x, −y).…”

“…The notion of orbital reversibility for centers was presented previously by Giné & Maza in [19] and recently by Algaba, García & Giné in [2], but only for the classical time-reversibility. They use the Montgomery-Bochner theorem, see [29].…”

Orbital Hypernormal Forms

“…Moreover, the orbital hypernormal form is the starting point in the study of local bifurcations in linear degeneracies such as saddle-node-Hopf, Hopf-Hopf and triple-zero cases (see [28]), as well as their nonlinear degenerate cases (see [29][30][31]). It is of great interest in the study of bifurcations in control systems (see [26,32]), in the study of the orbital reversibility problem, because the orbital hypernormal form uses as evidence the invariants that prevent this kind of symmetry (see [33][34][35][36][37]), as well as in the study of the center and integrability problems (see [38][39][40][41]).…”

On Some Reversible Cubic Systems

“…Up to now two main known mechanisms yielding integrability in polynomial systems are the Darboux integrability [19][20][21][22][23][24] and time reversibility [25][26][27][28]. In 1994, in the renowned work [29], Żoła ¸dek presented a classification of reversible centers of a cubic system.…”