Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
MetadataShow full item record
Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the Flow-Box Theorem we restrict ourselves to neighborhoods of singularities.
In this sense, we characterize Poisson structures around the typical zero-Hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non-vanishing first jet by connecting with the classical Poincar\'e center problem. From the global point of view, we connect the property of being strictly conservative (the invariant measure must be positive) with the existence of a Poisson structure depending on the phase space dimension. Finally, weak conservativeness in dimension two is introduced by the extension of inverse Jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed. Examples including Lotka-Volterra systems, quadratic isochronous centers, and non-smooth oscillators are provided.
Is part ofJournal of Physics A: Mathematical and Theoretical, 2017, vol. 50, p. 1-17
Showing items related by title, author, creator and subject.
García, I. A. (Isaac A.); Hernández Bermejo, Benito (Taylor & Francis, 2020-02-05)A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will ...
Buica, Adriana; García, I. A. (Isaac A.) (Springer Basel, 2015-05-01)In this paper we consider nonautonomous differential systems of arbitrary dimension and first find expressions for their inverse Jacobi multipliers and first integrals in some nonautonomous invariant set in terms of the ...
García, I. A. (Isaac A.); Llibre, Jaume (World Scientific Publishing, 2017-11-15)In this paper we show planar quadratic polynomial differentialsystems exhibiting as solutions some famous planar invariant algebraic curves. Also we put particular attention to the Darboux integrability of these differential ...