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On a Galoisian Approach to the Splitting of Separatrices
, 2003
"... For two degrees of freedom analytic Hamiltonian systems with a homoclinic orbit to a saddlecenter equilibrium point, we make the connection between two different approaches to nonintegrability: the (algebraic) Galois differential approach (a sophisticated version of Ziglin’s nonintegrability th ..."
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Cited by 7 (2 self)
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integrability theorem on the complex analytical variational equations associated to a particular integral curve) and a theorem of Lerman about the existence of transversal homoclinic orbits in the real part of the phase space. In order to accomplish this we use an interpretation given by GrottaRagazzo of Lerman’s
J. Differential Equations 197 (2004) 118–146 Conservative dynamics: unstable sets for saddlecenter loops
, 2003
"... We consider twodegreeoffreedom Hamiltonian systems with a saddlecenter loop, namely an orbit homoclinic to a saddlecenter equilibrium (related to pairs of pure real, 7n; and pure imaginary, 7oi; eigenvalues). We study the topology of the sets of orbits that have the saddlecenter loop as their ..."
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We consider twodegreeoffreedom Hamiltonian systems with a saddlecenter loop, namely an orbit homoclinic to a saddlecenter equilibrium (related to pairs of pure real, 7n; and pure imaginary, 7oi; eigenvalues). We study the topology of the sets of orbits that have the saddlecenter loop as their a and o limit set. A saddlecenter loop, as a periodic orbit, is a closed loop in phase space and the above sets are analogous to the unstable and stable manifolds, respectively, of a hyperbolic periodic orbit.
ARTICLE NO. 71 Convex Energy Levels of Hamiltonian Systems
, 2004
"... To Professor Jorge Sotomayor for his 60 th birthday We give a simple necessary and sufficient condition for a nonregular energy level of a Hamiltonian system to be strictly convex. We suppose that the Hamiltonian function is given by kinetic plus potential energy. We also show that this condition h ..."
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To Professor Jorge Sotomayor for his 60 th birthday We give a simple necessary and sufficient condition for a nonregular energy level of a Hamiltonian system to be strictly convex. We suppose that the Hamiltonian function is given by kinetic plus potential energy. We also show that this condition holds for several Hamiltonian functions, including the HénonHeiles one.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 4, 353–382 (2004) ARTICLE NO. 67 Plane Fields Related to Vector Fields on 3Manifolds
, 2003
"... This paper is dedicated to Prof. Jorge Sotomayor. This paper is a small collection of results about the topology of nonsingular plane fields which are either transverse or tangent to nonsingular volume preserving vector fields on 3manifolds. Emphasis is given to contact plane distributions and t ..."
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This paper is dedicated to Prof. Jorge Sotomayor. This paper is a small collection of results about the topology of nonsingular plane fields which are either transverse or tangent to nonsingular volume preserving vector fields on 3manifolds. Emphasis is given to contact plane distributions and to restrictions of Hamiltonian vector fields to hypersurfaces in symplectic 4manifolds.
Multitransition Homoclinic and Heteroclinic Solutions of the Extended FisherKolmogorov Equation
 J. Diff. Eq
, 1996
"... this paper we will concentrate on two types of potentials, doublewell and periodic. The two primary examples we have in mind are ..."
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Cited by 21 (9 self)
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this paper we will concentrate on two types of potentials, doublewell and periodic. The two primary examples we have in mind are
Parabolic resonances and instabilities
 Chaos
, 1997
"... A parabolic resonance is formed when an integrable twodegreesoffreedom Hamiltonian system possessing a circle of parabolic xed points is perturbed. It is proved that its occurrence is generic for one parameter families (codimension one phenomenon) of nearintegrable, two d.o.f. Hamiltonian syste ..."
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Cited by 11 (7 self)
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A parabolic resonance is formed when an integrable twodegreesoffreedom Hamiltonian system possessing a circle of parabolic xed points is perturbed. It is proved that its occurrence is generic for one parameter families (codimension one phenomenon) of nearintegrable, two d.o.f. Hamiltonian systems. Numerical experiments indicate that the motion near a parabolic resonance exhibits a new type of chaotic behavior which includes instabilities in some directions and long trapping times in others. Moreover, in a degenerate case, near a
at parabolic resonance, large scale instabilities appear. A model arising from an atmospherical study is shown to exhibit
at parabolic resonance. This supplies a simple mechanism for the transport of particles with small (i.e. atmospherically relevant) initial velocities from the vicinity of the equator to high latitudes. A modication of the model which allows the development of atmospherical jets unfolds the degeneracy, yet traces of the
at instabilities are clearly observed. 1
Attractive Periodic Sets in DiscreteTime Recurrent Networks (with Emphasis on FixedPoint Stability and Bifurcations in TwoNeuron Networks)
, 2001
"... We perform a detailed fixedpoint analysis of twounit recurrent neural networks with sigmoidshaped transfer functions. Using geometrical arguments in the space of transfer function derivatives, we partition the network statespace into distinct regions corresponding to stability types of the fixed ..."
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Cited by 9 (0 self)
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We perform a detailed fixedpoint analysis of twounit recurrent neural networks with sigmoidshaped transfer functions. Using geometrical arguments in the space of transfer function derivatives, we partition the network statespace into distinct regions corresponding to stability types of the fixed points. Unlike in the previous studies, we do not assume any special form of connectivity pattern between the neurons, and all free parameters are allowed to vary. We also prove that when both neurons have excitatory selfconnections and the mutual interaction pattern is the same (i.e., the neurons mutually inhibit or excite themselves), new attractive fixed points are created through the saddlenode bifurcation. Finally, for an Nneuron recurrent network, we give lower bounds on the rate of convergence of attractive periodic points toward the saturation values of neuron activations, as the absolute values of connection weights grow.
Anais da Academia Brasileira de Ciências (2002) 74(1): 25–31 (Annals of the Brazilian Academy of Sciences)
, 2001
"... www.scielo.br/aabc About periodic and quasiperiodic orbits of a new type for twist maps of the torus ..."
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www.scielo.br/aabc About periodic and quasiperiodic orbits of a new type for twist maps of the torus
Nonlinear Science © 1994 SpringerVerlag New York Inc. On the Motion of TwoDimensional Vortices with Mass
, 1992
"... Summary. The HelmholtzKirchhoff ODEs governing the planar motion of N 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 inertiatype term, so the point vortices are termed massive. Using an el ..."
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Summary. The HelmholtzKirchhoff ODEs governing the planar motion of N 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 inertiatype term, so the point vortices are termed massive. Using an electromagnetic analogy, these equations also determine the behavior of columns of charges in an external magnetic field. Using the symmetries, we reduce the four degrees of freedom system of two "massive " vortices to two degrees of freedom. We exhibit an integrable case and a nonintegrable one, according to choices of parameters. Nonintegrability is verified using a recent result obtained independently b Lerman and by Mielke, Holmes, and O'Reilly. Finally, we discuss the behavior of solutions as the masses of the vortices tend to zero, using for initial conditions a point of the trajectory of the HelmholtzKirchhoff equations. Key words: vortex dynamics, dynamics of particles in fluids, Hamiltonian systems, singular perturbations, transversal homoclinic orbits.
Results 1  10
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38