New global solutions of the Jacobi partial differential equations

Abstract

A new family of solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is investigated. Such family is mathematically remarkable, as far as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves. This kind of Poisson structures (termed distinguished solutions or D-solutions) are defined for every nontrivial combination of values of the dimension and the rank, and are also determined in terms of functions of arbitrary nonlinearity, properties usually not present simultaneously in the already known solution families. In addition, D-solutions display several properties allowing the generation of an infinity of D-solutions from a given one, which is an uncommon feature in the framework of the Jacobi equations. Furthermore, a special family of D-solutions complying to the previous requirements is constructively characterized and analyzed. Examples are discussed focusing on physical implications and including an application for the global construction of the Darboux canonical form. : 02.30.Jr,45.20.Jj. PACSMSC: 35Q72, 37J15, 37N05.

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

“…Unfortunately the massive computations to obtain f 7 (z; λ), hencef 7 (z; λ), in the proof of Theorem 4 do not seem to be possible in our computer. In other words, for family (15) we are unable to get the ideal stabilization explained in Remark 2. The reason is that we can check that I 6 = I because there are parameters in I 6 for which the origin is not a center of (15) as can be easily seen by using Remark 1.…”

“…; z, λ, ε) is defined on the interval [0, 2π] provided that ε is close enough to 0, hence we can define the displacement map d : R + × R 12 × I → R + with I some real interval containing the origin as d(z, λ, ε) = r(2π; z, λ, ε)−z. From this definition we see that the isolated positive zeros z 0 ∈ R + of d(., λ, ε) are just the initial conditions for the 2π-periodic solutions of (16), which clearly are in one-to-one correspondence with the limit cycles of system (15) bifurcating from the circle y 2 1 + y 2 2 = z 2 0 included in the period annulus P of the unperturbed harmonic oscillator.…”

“…Introducing polar coordinates y 1 = r cos θ, y 2 = r sin θ, and for |ε| sufficiently small, any systemẏ 1 = y 2 + εP (y 1 , y 2 ; ε),ẏ 2 = −y 1 + εQ(y 1 , y 2 ; ε) and in particular system (15) is transformed into the analytic differential equation…”

“…Theorem 4. Let us consider the perturbed harmonic oscillator given by family (15) and the following set of polynomials in their parameters λ ∈ R 10 : ξ 20 (λ) = a 50 (a 30 − a 60 ), ξ 30 (λ) = a 31 a 50 + a 30 a 51 − a 51 a 60 − a 50 a 61 , ξ 40 (λ) = a 51 (a 31 − a 61 ), ξ 42 (λ) = −a 20 a 40 (5a 30 + a 40 − 5a 60 )(a 30 − a 60 ), ξ 51 (λ) = 5a 21 Let N(λ) be the number of limit cycles that bifurcate from its period annulus P = R 2 \{(0, 0)} as the perturbation parameter ε slightly varies from zero. Then the following holds:…”

“…(v) If ξ 20 =ξ 30 =ξ 40 =ξ 42 = 0 andξ 51 = 0 then N = 0;(vi) If ξ 20 =ξ 30 =ξ 40 =ξ 42 =ξ 51 = 0 butξ 63 = then, defining s 2 =ξ 61 /ξ 63 , we have that N = 1 or N = 0 according to whether s 2 < 0 or s 2 ≥ 0, respectively.Proof. Straightforward computations produce the following averaged functions for system(15):f 1 (z; λ) ≡ 0,…”