Clodoaldo Grotta Ragazzo

University of São Paulo | USP · Departamento de Matemática Aplicada (IME) (São Paulo)

The goal of this work is to investigate under which circumstances the tidal response of a stratified body can be approximated by that of a homogeneous body. We show that any multilayered planet model can be approximated by a homogeneous body, with the same dissipation of tidal energy as a function of the excitation frequency, as long as the rheolog...

The main propose of this work is to investigate under which circumstances the tidal response of a stratified body can be approximated by that of a homogeneous body. We show that any multilayered body can be approximated by a homogeneous body, with the same dissipation of tidal energy as a function of the excitation frequency, as long as the rheolog...

We present fully three-dimensional equations to describe the rotations of a body made of a deformable mantle and a fluid core. The model in its essence is similar to that used by INPOP19a (Integration Planétaire de l’Observatoire de Paris) Fienga et al. (INPOP19a planetary ephemerides. Notes Scientifiques et Techniques de l’Institut de Mécanique Cé...

We present fully three-dimensional equations to describe the rotations of a body made of a deformable mantle and a fluid core. The model in its essence is similar to that used by INPOP (Integration Plan\'{e}taire de l'Observatoire de Paris), e.g. Viswanathan et al. (2019), and by JPL (Jet Propulsion Laboratory), e.g. Folkner et al. (2014), to repre...

We study Birkhoff sums as distributions. We obtain regularity results on such distributions for various dynamical systems with hyperbolicity, as hyperbolic linear maps on the torus and piecewise expanding maps on the interval. We also give some applications, as the study of advection in discrete dynamical systems.

We study Birkhoff sums as distributions. We obtain regularity results on such distributions for various dynamical systems with hyperbolicity, as hyperbolic linear maps on the torus and piecewise expanding maps on the interval. We also give some applications, as the study of advection in discrete dynamical systems.

Often topological classes of one-dimensional dynamical systems are finite codimension smooth manifolds. We describe a method to prove this sort of statement that we believe can be applied in many settings. In this work we will implement it for piecewise expanding maps. The most important step will be the identification of infinitesimal deformations...

This paper contains a review of Clairaut’s theory with focus on the determination of a gravitational modulus \(\gamma\) defined as \(\left( \frac{C-{{\mathrm{I}}}_{\mathrm{o}}}{{\mathrm{I}}_{\mathrm{o}}}\right) \gamma =\frac{2}{3}\Omega ^2\), where C and \({{\mathrm{I}}}_{\mathrm{o}}\) are the polar and mean moment of inertia of the body and \(\Ome...

The main purpose of this work is to present a time-domain implementation of the Andrade rheology, instead of the traditional expansion in terms of a Fourier series of the tidal potential. This approach can be used in any fully three dimensional numerical simulation of the dynamics of a system of many deformable bodies. In particular, it allows larg...

The main purpose of this work is to present a time-domain implementation of the Andrade rheology, instead of the traditional expansion in terms of a Fourier series of the tidal potential. This approach can be used in any fully three dimensional numerical simulation of the dynamics of a system of many deformable bodies. In particular, it allows larg...

This paper contains a review of Clairaut's theory with focus on the determination of a gravitational rigidity modulus $\gamma$ defined as $\left(\frac{C-I_o}{I_o}\right)\gamma=\frac{2}{3}\Omega^2$, where $C$ and $I_o$ are the polar and mean moment of inertia of the body and $\Omega$ is the body spin.The constant $\gamma$ is related to the static fl...

The purpose of this work is to evaluate the effect of deformation inertia on tide dynamics, particularly within the context of the tide response equations proposed independently by Boué et al. (Celest Mech Dyn Astron 126:31–60, 2016) and Ragazzo and Ruiz (Celest Mech Dyn Astron 128(1):19–59, 2017). The singular limit as the inertia tends to zero is...

The equations of motion for a system of point vortices on an oriented Riemannian surface of finite topological type are presented. The equations are obtained from a Green’s function on the surface. The uniqueness of the Green’s function is established under hydrodynamic conditions at the surface’s boundaries and ends. The hydrodynamic force on a po...

The purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one. The algorithm is based on a relation between the Laplace-Beltrami Green function and the heat kernel. The algorithm is used to compute the motion of a vortex on the Bolz...

This paper contains equations for the motion of linear viscoelastic bodies interacting under gravity. The equations are fully three dimensional and allow for the integration of the spin, the orbit, and the deformation of each body. The goal is to present good models for the tidal forces that take into account the possibly different rheology of each...

This paper is devoted to an alternative model for a rotating, isolated, self-gravitating, viscoelastic body. The initial approach is quite similar to the classical one, present in the works of Dirichlet, Riemann, Chandrasekhar, among others. Our main contribution is to present a simplified model for the motion of an almost spherical body. The Lagra...

An explicit Lyapunov function is constructed for scalar parabolic reaction-advection-diffusion equations under periodic boundary conditions. The non-linearity is assumed to be even with respect to the advection term. The method followed was originally suggested by H. Matano for, and limited to, separated boundary conditions.

The problem of ascertaining that floating ocean structures exhibit adequate dynamic behavior under the action of frequently taxing environmental conditions is of major importance to safe offshore operations. Many times, two ships remain close to each other at sea; one such situation occurs during the widely employed offloading operations, in which...

This paper is about equations of the form ${\dot u=v,\dot v = F(u, v) \, \, {\rm where} \, \, (u,v) \in \mathbb {R}^{2}}$ and F is an infinitely differentiable function. Its main theorem states that if F(u, −v) = F(u, v) then, under some additional conditions, there exists an infinitely differentiable change of variables (u, v) → (x, y) onto ${...

We consider the scalar delayed differential equation $\ep\dot x(t)=-x(t)+f(x(t-r))$, where $\ep>0$, $r=r(x,\ep)$ and $f$ represents either a positive feedback $df/dx>0$ or a negative feedback $df/dx<0$. When the delay is a constant, i.e. $r(x,\ep)=1$, this equation admits metastable rapidly oscillating solutions that are transients whose duration i...

We consider the scalar delayed differential equation e[(x)\dot](t)=-x(t)+f(x(t-1)){\epsilon\dot x(t)=-x(t)\,+f(x(t-1))}, where ${\epsilon\,{>}\,0}${\epsilon\,{>}\,0} and f verifies either df/dx>0 or df/dx<0 and some other conditions. We present theorems indicating that a generic initial condition with sign changes generates a solution with a transi...

M. S. Howe [J. Fluid Mech. 206, 131 (1989)] presented integral formulas for the force and torque on a rigid body that permit the identification of the separate influences of added mass, normal stresses induced by free vorticity, and viscous skin friction. Here a simple extension of Howe's formulas is done for systems of several rigid bodies. The go...

In this paper we prove the existence of global sections of disk-type in non-regular and strictly convex energy levels of integrable and near-integrable Hamiltonian systems with two degrees of freedom. This extends a result of (Hofer etal. in Ann. Math.(2) 148(1):197–289, 1998) where the same statement is true provided the energy level is regular.

The dynamics of a charged particle under axisymmetric gravitational and electromagnetic forces is studied. The symmetry allows a phase space reduction from three to two degrees of freedom. The reduced system is a function of two parameters (essentially: the ratio “charge to mass” and the z-component of the angular momentum) and it has an invariant...

There are various situations in the operation of ships that re- quire them to remain positioned close to another vessel at sea under the action of environmental agents. This scenario is com- mon in the offshore oil exploitation industry, where shuttle ves- sels attach themselves to FPSOs for offloading operations that may last for several hours. Th...

We consider Hamiltonian systems with two degrees of freedom. We suppose the existence of a saddle-center equilibrium in a strictly convex component S of its energy level. Moser's normal form for such equilibriums and a theorem of Hofer, Wysocki and Zehnder are used to establish the existence of a periodic orbit in S with several topological propert...

We consider two-degree-of-freedom Hamiltonian systems with a saddle-center loop, namely an orbit homoclinic to a saddle-center equilibrium (related to pairs of pure real, ±ν, and pure imaginary, ±ωi, eigenvalues). We study the topology of the sets of orbits that have the saddle-center loop as their α and ω limit set. A saddle-center loop, as a peri...

The problem of motion of many solids through an unbounded ideal liquid (inviscid and irrotational) is considered. A Lagrangian formulation of the equations of motion leads to a set of ordinary differential equations (ODEs) coupled to an elliptic partial differential equation (PDE) [H. Lamb, Hydrodynamics, 6th ed., Dover, New York (1932; JFM 58.1298...

In the present paper the problem of slow dynamics (surge, sway, and yaw) of a tanker equipped with a turret system in deep water is considered. In this situation the mooring line damping dominates the overall damping related to surge and sway and attenuate these motions. The environmental input acts primarily on the yaw motion and the surge and swa...

This paper is a small collection of results about the topology of non-singular plane fields which are either transverse or tangent to nonsingular volume preserving vector fields on 3-manifolds. Emphasis is given to contact plane distributions and to restrictions of Hamiltonian vector fields to hypersurfaces in symplectic 4-manifolds.

We study a class of Hamiltonian systems on a 4 dimensional symplectic manifold which have a saddle-center fixed point and satisfy the following property: All the periodic orbits in the center manifold of the fixed point have an orbit homoclinic to them, although the fixed point itself does not. In addition, we prove that these systems have a chaoti...

We study a two-parameter family of twist maps defined on the torus. This family essentially determines the dynamics near saddle-centre loops of four-dimensional real analytic Hamiltonian systems. A saddle-centre loop is an orbit homoclinic to a saddle-centre equilibrium (related to pairs of pure real, ±ν, and pure imaginary, ±ωi, eigenvalues). We p...

An explicit set of ordinary differential equations that approximately describe the dynamics of many rigid bodies (without any hypothesis on their geometry) inside an ideal fluid is presented. Two applications of these equations are made. At first, it is shown how to use these equations to compute the added-mass tensor of a single body made up of si...

Scattering and escaping problems for Hamiltonian systems with two degrees of freedom of the type kinetic plus potential energy arise in many applications. Under some discrete symmetry assumptions, it is shown that important quantities in these problems are determined by a relation between two canonical invariant numbers that can be explicitly compu...

We consider the equation Delta u = -alpha u - u(3) in Omega = {(x, y)/lx is an element of R, y is an element of (-pi, pi)}, alpha is an element of R, with Dirichlet or periodic boundary conditions. We prove the existence of an infinite number of nontrivial classical solutions satisfying lim u(x, y) -> 0 as vertical bar x vertical bar -> +/-infinity...

In systems at phase transitions, two phases of the same substance may coexist for a long time before one of them dominates. We show that a similar phenomenon occurs in systems with delayed feedback, where short-term stable oscillatory patterns can also have very long lifetimes before vanishing into constant or periodic steady states.

In irreducible excitatory networks of analog graded-response neurons, the trajectories of most solutions tend to the equilibria. We derive sufficient conditions for such networks to be globally asymptotically stable. When the network possesses several locally stable equilibria, their location in the phase space is discussed and a description of the...

The equations describing the mean flow and small-scale interaction of a barotropic flow via topographic stress with layered topography are studied here through the interplay of theory and numerical experiments. Both a viewpoint toward atmosphere—ocean science and one toward chaotic nonlinear dynamics are emphasized. As regards atmosphere—ocean scie...

Consider the three degrees of freedom Hamiltonian system related to the following analytic Hamiltonian function $${2}\,\sum\limits_{k = 1}^3 {\omega _k (p_k^2 \, + \,q_k^2 )\, + \,higher\,order\,terms.} $$ The origin is an elliptic equilibrium which is characterized by the frequency vector ω = (ω1, ω2, ω3). The equilibrium is said to have a resonan...

Finite transmission times between neurons, referred to as delays, appear in hardware implementation of neural networks and may interfere with information processing by inducing oscillations. In some networks these oscillations are transients. In this work, we examine these in a two-neuron network, and we show analytically that the duration of such...

The behavior of neural networks may be influenced by transmission delays and many studies have derived constraints on parameters such as connection weights and output functions which ensure that the asymptotic dynamics of a network with delay remains similar to that of the corresponding system without delay. However, even when the delay does not af...

We study the dynamics of a one parameter family of two degrees of freedom Hamiltonian systems that includes the Hénon-Heiles system. We show that several dynamical properties of this family, like the existence of large stochastic regions in certain parts of the phase space, are related to two canonical invariants that can be explicitly computed. Th...

We consider two degrees of freedom Hamiltonian systems with orbits homoclinic (bi-asymptotic) to saddle-center equilibria (related to pairs of pure real and pure imaginary eigenvalues). We show that two canonical invariant numbers determine whether the system is “diffusive” or not in a neighborhood of the homoclinic orbit.

Little attention has been paid in the past to the effects of interunit transmission delays (representing axonal and synaptic delays) on the boundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response ne...

A ring neural network is a closed chain in which each unit is connected unidirectionally to the next one. Numerical investigations indicate that continuous-time excitatory ring networks composed of graded-response units can generate oscillations when interunit transmission is delayed. These oscillations appear for a wide range of initial conditions...

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy le...

We consider 2-degrees of freedom Hamiltonian systems with an involutive symmetry and a pair of orbits bi-asymptotic (homoclinic) to a saddle-center equilibrium (related to pairs of pure real, ±ν, and pure imaginary eigenvalues, ±ω i). We show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one-dime...

Motivated by the collisionless shock problem in plasma, we study the stationary Vlasov-Maxwell system in one space dimension via a dynamical system approach. We construct flat-tail and oscillatory-tail solutions, as well as solitons in the presence of a nontrivial magnetic field. Flat-tail solutions connect different densities, pressures, and stren...

We show that the equation , x ∈ (0, π), α < -1, which models transversal nonlinear vibrations of a buckled beam, has invariant four-dimensional manifolds of solutions containing periodic orbits with transversal homoclinic orbits to them. The basic tool used in the proof is a theorem concerning two degrees of freedom Hamiltonian systems with saddle-...

The Helmholtz-Kirchhoff ODEs governing the planar motion ofN point vortices in an ideal, incompressible fluid are extended to the case where the fluid has impurities. In this case the resulting ODEs have an additional inertia-type term, so the point vortices are termed massive. Using an electromagnetic analogy, these equations also determine the be...

We consider canonical two degrees of freedom analytic Hamiltonian systems with Hamiltonian functionH=1/2[p 12+p 22]+U(q 1,q 2), where U(q1, q2) = 1/2[− v2q 12 + ω2q 22] +O(q 12 + q 22)3/2) and ∂q2 U(q1, 0) = 0. Under some additional, not so restrictive hypothesis, we present explicit conditions for the exisstence of transversal homoclinic orbits to...

We consider the parametrized family of equations∂ tt ,u-∂ xx u-au+∥u∥ 22α u=O,x∃(0,πL), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invarian...

We investigated the structure of the so-called first Hopf bifurcation surface associated to a differential equation with two time delays. A geometrical approach leading naturally to a number theoretic approach provides rigourous results which are corroborated by previous numerical and experimental (optical compound resonator) results.

É feito um estudo de um modelo macroeconômico de hiperinflação. O modelo é descrito por um sistema de duas equações diferenciais com 1 retardamento onde as variáveis dependentes são a inflação e a quantidade de moeda no mercado. O primeiro passofoi fazer uma análise de estabilidade linear dos pontos de equilíbrio e sua dependência com o parâmetro q...