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Razvan M. Tudoran

Publications and source records attributed to Razvan M. Tudoran.

15 recordsLinked to original sources
arXiv2024

Natural geometric Fourier transforms and the associated fractional Laplacian

To each arbitrary given general geometric structure on $\mathbb{R}^{n}$, we associate a pair of compatible Fourier transforms, that prove to appear naturally in the framework of Poisson's summation formula for full lattices. We study their properties and the compatibility with the classical $n-$dimensional Fourier transform. In the case of a positive definite geometric structure, we show that these geometric Fourier transforms induce a geometric fractional Laplacian, with properties similar to those of the classical fractional Laplacian.

math.CA↗
arXiv2019

On the coercivity of continuously differentiable vector fields

Given an arbitrary fixed continuously differentiable vector field on $\mathbb{R}^n$, we prove that this vector field is coercive if and only if its conservative part is coercive. We apply this result in order to provide sufficient conditions to guarantee the co-existence of equilibrium states of a continuously differentiable vector field and its conservative part.

math.CA↗
arXiv2019

On the rattleback dynamics

In this paper we present some relevant dynamical properties of an idealized conservative model of the rattleback, from the Poisson dynamics point of view. In the first half of the article, along with a dynamical study of the orbits, using a Hamilton-Poisson realization of the dynamical system, we provide a geometric characterization of the space of orbits in terms of Whitney stratifications associated to the image of the energy-Casimir mapping. In the second half of the article we provide an explicit method to stabilize asymptotically any arbitrary fixed orbit/cycle of the rattleback system and to keep unchanged the geometry of the model space.

math.DS↗
arXiv2019

A note on the structure of prescribed gradient--like domains of non--integrable vector fields

Given a geometric structure on $\mathbb{R}^{n}$ with $n$ even (e.g. Euclidean, symplectic, Minkowski, pseudo-Euclidean), we analyze the set of points inside the domain of definition of an arbitrary given $\mathcal{C}^1$ vector field, where the value of the vector field equals the value of the left/right gradient--like vector field of some fixed $\mathcal{C}^2$ potential function, although a non-integrability condition holds at each such a point. Particular examples of gradient--like vector fields include the class of gradient vector fields with respect to Euclidean or pseudo-Euclidean inner products, and the class of Hamiltonian vector fields associated to symplectic structures on $\mathbb{R}^{n}$ (with $n$ even). The main result of this article provides a geometric version of the main result of [1].

math.CA↗
arXiv2018

Asymptotic bp-stabilization of a given closed invariant set

Given a closed invariant set $\mathcal{C}$ of a dynamical system generated by a smooth vector field, $X$, for each $λ> 0$, we construct a control vector field, $X_{0}^λ$, such that the perturbed dynamics generated by the vector field $X+X_{0}^λ$, globally asymptotically bp-stabilizes the invariant set $\mathcal{C}$, that is, $\mathcal{C}$ attracts every bounded positive orbit of the perturbed dynamical system.

math.CA↗
arXiv2017

A global geometric decomposition of vector fields and applications to topological conjugacy

We give a global geometric decomposition of continuously differentiable vector fields on $\mathbb{R}^n$. More precisely, given a vector field of class $\mathcal{C}^{1}$ on $\mathbb{R}^{n}$, and a geometric structure on $\mathbb{R}^n$, we provide a unique global decomposition of the vector field as the sum of a left (right) gradient--like vector field (naturally associated to the geometric structure) with potential function vanishing at the origin, and a vector field which is left (right) orthogonal to the identity, with respect to the geometric structure. As application, we provide a criterion to decide topological conjugacy of complete vector fields of class $\mathcal{C}^1$ on $\mathbb{R}^{n}$ based on topological conjugacy of the corresponding parts given by the associated geometric decompositions.

math.DS↗
arXiv2017

Asymptotic stabilization with phase of periodic orbits of three-dimensional Hamiltonian systems

We provide a geometric method to stabilize asymptotically with phase an arbitrary fixed periodic orbit of a locally generic three-dimensional Hamiltonian dynamical system. The main advantage of this method is that one needs not know a parameterization of the orbit to be stabilized, but only the values of the Hamiltonian and a fixed Casimir (of the Poisson configuration manifold) at that orbit. The stabilization procedure is illustrated in the case of the Rikitake model of geomagnetic reversal.

math.DS↗
arXiv2016

A stability criterion for non-degenerate equilibrium states of completely integrable systems

We provide a criterion in order to decide the stability of non-degenerate equilibrium states of completely integrable systems. More precisely, given a Hamilton-Poisson realization of a completely integrable system generated by a smooth $n-$ dimensional vector field, $X$, and a non-degenerate regular (in the Poisson sense) equilibrium state, $\overline{x}_e$, we define a scalar quantity, $\mathcal{I}_{X}(\overline{x}_e)$, whose sign determines the stability of the equilibrium. Moreover, if $\mathcal{I}_{X}(\overline{x}_e)>0$, then around $\overline{x}_e$ there exist one-parameter families of periodic orbits shrinking to $\{\overline{x}_e \}$, whose periods approach $2π/\sqrt{\mathcal{I}_{X}(\overline{x}_e)}$ as the parameter goes to zero. The theoretical results are illustrated in the case of the Rikitake dynamical system.

math.DS↗
arXiv2014

Controlling the stability of periodic orbits of completely integrable systems

We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.

math.DS↗
arXiv2014

A method to generate first integrals from infinitesimal symmetries

We propose a method to construct first integrals of a dynamical system, starting with a given set of independent infinitesimal symmetries. In the case of two infinitesimal symmetries, a rank two Poisson structure on the ambient space it is found, such that the vector field that generates the dynamical system, becomes a Poisson vector field. Moreover, the symplectic leaves and the Casimir functions of the associated Poisson manifold are characterized. Explicit conditions that guarantee Hamilton-Poisson realizations of the dynamical system are also given.

math-ph↗
arXiv2014

On local generators of affine distributions on Riemannian manifolds

Using a coordinate free characterization of hyperplanes intersection, we provide explicitly a set of local generators for a smooth affine distribution given by those smooth vector fields $X\in\mathfrak{X}(U)$ defined eventually on an open subset $U\subseteq M$ of a smooth Riemannian manifold $(M,g)$, that verifies the relations $g(X,X_1)=\dots =g(X,X_k)=0$, $g(X,Y_1)=h_1, \dots, g(X,Y_p)=h_p$, where $X_1,\dots, X_k, Y_1, \dots, Y_p \in\mathfrak{X}(U)$, and respectively $h_1,\dots, h_p \in \mathcal{C}^{\infty}(U,\mathbb{R})$, are a-priori given quantities. In the case when $X_1,\dots, X_k, Y_1, \dots, Y_p$ are gradient vector fields associated with some smooth functions $I_1,\dots, I_k, D_1, \dots, D_p \in \mathcal{C}^{\infty}(U,\mathbb{R})$, i.e., $X_1 =\nabla_{g}I_1$, $\dots$, $X_k = \nabla_{g}I_k$, $Y_1 = \nabla_{g}D_1$, $\dots, Y_p = \nabla_{g}D_p$, then we obtain a set of local generators for the smooth affine distribution of smooth vector fields which conserve the quantities $I_1, \dots, I_k$ and dissipate the scalar quantities $D_1, \dots, D_p$ with prescribed rates $h_1, \dots, h_p$.

math-ph↗
arXiv2013

The free rigid body dynamics: generalized versus classic

In this paper we analyze the normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ - skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric matrix. A consequence of the main results is that any first-order autonomous three-dimensional differential equation possessing two independent quadratic constants of motion which admits a positive/negative definite linear combination, is affinely equivalent to the classical "relaxed" free rigid body dynamics with linear controls.

math-ph↗