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G. Cicogna

Publications and source records attributed to G. Cicogna.

17 recordsLinked to original sources
arXiv2015

MHD equilibria with incompressible flows: symmetry approach

We identify and discuss a family of azimuthally symmetric, incompressible, magnetohydrodynamic plasma equilibria with poloidal and toroidal flows in terms of solutions of the Generalized Grad Shafranov (GGS) equation. These solutions are derived by exploiting the incompressibility assumption, in order to rewrite the GGS equation in terms of a different dependent variable, and the continuous Lie symmetry properties of the resulting equation and in particular a special type of "weak" symmetries.

physics.plasm-ph↗
arXiv2014

Side conditions for ordinary differential equations

We specialize Olver's and Rosenau's side condition heuristics for the determination of particular invariant sets of ordinary differential equations. It turns out that side conditions of so-called LaSalle type are of special interest. Moreover we put side condition properties of symmetric and partially symmetric equations in a wider context. In the final section we present an application to parameter-dependent systems, in particular to quasi-steady state for chemical reactions.

math.DS↗
arXiv2013

Convergence of normal form transformations: The role of symmetries

We discuss the convergence problem for coordinate transformations which take a given vector field into Poincaré-Dulac normal form. We show that the presence of linear or nonlinear Lie point symmetries can guaranteee convergence of these normalizing transformations, in a number of scenarios. As an application, we consider a class of bifurcation problems.

math-ph↗
arXiv2013

Normal Forms, symmetry, and linearization of dynamical systems

We discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system - or a parameter dependent family - via the Poincaré Normal Form approach. We discuss this at first formally, and later pay attention to the convergence of the linearizing procedure. We also discuss some generalizations of our main result

math-ph↗
arXiv2013

Partial Lie-point symmetries of differential equations

When we consider a differential equation $Δ=0$ whose set of solutions is ${\cal S}_Δ$, a Lie-point exact symmetry of this is a Lie-point invertible transformation $T$ such that $T({\cal S}_Δ)={\cal S}_Δ$, i.e. such that any solution to $Δ=0$ is tranformed into a (generally, different) solution to the same equation; here we define {\it partial} symmetries of $Δ=0$ as Lie-point invertible transformations $T$ such that there is a nonempty subset ${\cal P} \subset {\cal S}_Δ$ such that $T({\cal P}) = {\cal P}$, i.e. such that there is a subset of solutions to $Δ=0$ which are transformed one into the other. We discuss how to determine both partial symmetries and the invariant set ${\cal P} \subset {\cal S}_Δ$, and show that our procedure is effective by means of concrete examples. We also discuss relations with conditional symmetries, and how our discussion applies to the special case of dynamical systems. Our discussion will focus on continuous Lie-point partial symmetries, but our approach would also be suitable for more general classes of transformations; the discussion is indeed extended to partial generalized (or Lie-Bäcklund) symmetries along the same lines, and in the appendix we will discuss the case of discrete partial symmetries.

math-ph↗
arXiv2013

Symmetries of dynamical systems and convergent normal forms

It is shown that, under suitable conditions, involving in particular the existence of analytic constants of motion, the presence of Lie point symmetries can ensure the convergence of the transformation taking a vector field (or dynamical system) into normal form

math-ph↗
arXiv2010

Twisted symmetries and integrable systems

Symmetry properties are at the basis of integrability. In recent years, it appeared that so called "twisted symmetries" are as effective as standard symmetries in many respects (integrating ODEs, finding special solutions to PDEs). Here we discuss how twisted symmetries can be used to detect integrability of Lagrangian systems which are not integrable via standard symmetries.

math-ph↗
arXiv2008

Reduction of systems of first-order differential equations via Lambda-symmetries

The notion of lambda-symmetries, originally introduced by C. Muriel and J.L. Romero, is extended to the case of systems of first-order ODE's (and of dynamical systems in particular). It is shown that the existence of a symmetry of this type produces a reduction of the differential equations, restricting the presence of the variables involved in the problem. The results are compared with the case of standard (i.e. exact) Lie-point symmetries and are also illustrated by some examples.

nlin.SI↗
arXiv2007

Noether theorem for mu-symmetries

We give a version of Noether theorem adapted to the framework of mu-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of lambda-symmetries, and connects mu-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this "mu-conservation law'' actually reduces to a standard one; we also note a relation between mu-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under mu-prolonged variation fields, leading to modified Euler-Lagrange equations. In this setting mu-symmetries of the Lagrangian correspond to standard conservation laws as in the standard Noether theorem. We finally propose some applications and examples.

math-ph↗
arXiv2006

On the notion of conditional symmetry of differential equations

Symmetry properties of PDE's are considered within a systematic and unifying scheme: particular attention is devoted to the notion of conditional symmetry, leading to the distinction and a precise characterization of the notions of ``true'' and ``weak'' conditional symmetry. Their relationship with exact and partial symmetries is also discussed. An extensive use of ``symmetry-adapted'' variables is made; several clarifying examples, including the case of Boussinesq equation, are also provided.

math-ph↗
arXiv2004

Symmetry properties of some Euler-type equations in plasma physics

We consider a system of partial differential equations, of interest to plasma physics, and provide all its Lie point symmetries, with their respective invariant solutions. We also discuss some of its conditional and partial symmetries. We finally show that, although the system can be cast in divergence form and admits conserved currents, it does not admit potential symmetries.

math-ph↗
arXiv2002

Mel'nikov method revisited

We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be also applied to some ``degenerate'' situations, as to the case of nonhyperbolic unstable points, or of critical points located at the infinity (thus giving rise to unbounded orbits, e.g. the Keplerian parabolic orbits), and it is naturally ``compatible'' with the presence of general symmetry properties of the problem.

nlin.CD↗
arXiv1999

An approach to Mel'nikov theory in celestial mechanics

Using a completely analytic procedure - based on a suitable extension of a classical method - we discuss an approach to the Poincaré-Mel'nikov theory, which can be conveniently applied also to the case of non-hyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case, and precisely on problems described by Kepler-like potentials in one or two degrees of freedom, in the presence of general time-dependent perturbations. We show that the appearance of chaos (possibly including Arnol'd diffusion) can be proved quite easily and in a direct way, without resorting to singular coordinate transformations, such as the McGehee or blowing-up transformations. Natural examples are provided by the classical Gyldén problem, originally proposed in celestial mechanics, but also of interest in different fields, and by the general 3-body problem in classical mechanics.

chao-dyn↗
arXiv1999

Nonhyperbolic homoclinic chaos

Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. We consider here, following a (suitably adjusted) classical analytic method, the case of non-hyperbolic points and show that, under a Melnikov-type condition plus an additional assumption, the negatively and positively asymptotic sets persist under periodic perturbations, together with their infinitely many intersections on the Poincaré section. We also examine, by means of essentially the same procedure, the case of (heteroclinic) orbits tending to the infinity; this case includes in particular the classical Sitnikov 3--body problem.

chao-dyn↗
arXiv1997

Convergent Normal Forms of Symmetric Dynamical Systems

It is shown that the presence of Lie-point-symmetries of (non-Hamiltonian) dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical sytem (or vector field) into Poincaré-Dulac normal form.

solv-int↗
arXiv1993

Modifying the onset of homoclinic chaos. Application to a bistable potential

We analyze, by means of Melnikov method, the possibility of modifying the threshold of homoclinic chaos in general 1-dimensional problems, by introducing small periodic resonant modulations. We indicate in particular a prescription in order to increase the threshold (i.e. to prevent chaos), and consider then its application to the bistable Duffing-Holmes potential. All results are confirmed both by numerical and by analog simulations, showing that small modulations can in fact sensibly influence the onset of chaos.

chao-dyn↗