Communications in Nonlinear Science and Numerical Simulation, 2018
We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamilton... more We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a suitable normal form construction that allows to identify and approximate the periodic orbits which survive to the breaking of the resonant torus. Our algorithm allows to treat the continuation of approximate orbits which are at leading order degenerate, hence not covered by classical averaging methods. We discuss possible future extensions and applications to localized periodic orbits in chains of weakly coupled oscillators.
We prove the non-integrability of perturbed separable planar potentials by using the perturbed no... more We prove the non-integrability of perturbed separable planar potentials by using the perturbed normal variational equations, with a reasoning analogous to Ziglin's theorem. We apply the above criterion to a Hamiltonian that cannot be proved non-integrable by other known non-integrability criteria.
A wide class of partial differential equations have at least three conservation laws that remain ... more A wide class of partial differential equations have at least three conservation laws that remain invariant for certain solutions of them and especially for solitary wave solutions. These conservation laws can be considered as the energy, pseudomomentum and mass integrals of these solutions. We investigate the invariant relation between the energy and the pseudomomentum for solitary waves in two Boussinesq-type equations that come from the theory of elasticity and lattice models.
We prove that Hill's lunar problem does not possess a second analytic integral of motion, ind... more We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity ? of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that
We prove the non-integrability of perturbed separable planar potentials by using the perturbed no... more We prove the non-integrability of perturbed separable planar potentials by using the perturbed normal variational equations, with a reasoning analogous to Ziglin's theorem. We apply the above criterion to a Hamiltonian that cannot be proved non-integrable by other known non-integrability criteria.
In this paper, a simple criterion to prove non-integrability of symplectic, perturbed twist mappi... more In this paper, a simple criterion to prove non-integrability of symplectic, perturbed twist mappings in 2n-dimensions is developed for sufficiently small perturbations. In addition, an upper bound for the number of isolating integrals the system can possess is provided. A criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found. The evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the Jacobian matrix concludes the theoretical part. The theory is finally applied to a twist map perturbed by the Suris potential and also a four-dimensional map with physical interest.
A connection between the Ψ-series local expansion of the solution of perturbed ODE's and the eval... more A connection between the Ψ-series local expansion of the solution of perturbed ODE's and the evaluation of the Melnikov vector has been found by Goriely and Tabor. Following an analogous procedure we find a straightforward relation between the failure of the compatibility condition of the Painleve test and the absense of an analytic integral for a periodically perturbed Hamiltonian system whose unperturbed part does not necessarily possess a homoclinic loop.
Nonintregrability and continuation of periodic orbits
We compare the non integrability test for Hamiltonian systems we have proved for n>=3 with the co... more We compare the non integrability test for Hamiltonian systems we have proved for n>=3 with the continuation of periodic orbits of Poincare-Melnikov theory and prove that, if a system is completely nonintegrable, then the periodic orbits are continued on a dense set of resonant tori on which nonintegrability is established.
We prove that Hill's lunar problem does not possess a second analytic integral of motion, indepen... more We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity ω of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for ω in an open interval around zero. Then, by selecting suitable values of ω, b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two-body problem in a rotating frame.
The non-integrability of a two-degrees-of-freedom Hamiltonian is investigated by a method based o... more The non-integrability of a two-degrees-of-freedom Hamiltonian is investigated by a method based on a well-known theorem of Poincar6. It is proved that the perturbed system is non-integrable for values of the perturbative parameter in an open interval around zero, if, on a dense set of resonant tori of the unperturbed system, the average value of the perturbative function, evaluated along the periodic orbits on each torus, depends on the particular orbit. An application to the separable quartic oscillator with a quadratic perturbation is made and it is shown that if the perturbation is non-separable in the same coordinates, the system is non-integrable.
The structure of the resonance zone in nearly integrable Hamiltonian systems is studied by a more... more The structure of the resonance zone in nearly integrable Hamiltonian systems is studied by a more general method than the pendulum approximation. This method applies to the case of a non-degenerate integrable part in the Hamiltonian. This problem may be overcome in a class of galactic-type polynomial potentials, in the case where the higher-order term is by itself integrable. An illustrative example is worked out.
A wide class of partial differential equations have at least three conservation laws that remain ... more A wide class of partial differential equations have at least three conservation laws that remain invariant for certain solutions of them and especially for solitary wave solutions. These conservation laws can be considered as the energy, pseudomomentum and mass integrals of these solutions. We investigate the invariant relation between the energy and the pseudomomentum for solitary waves in two Boussinesq-type equations that come from the theory of elasticity and lattice models.
Journal of the Mechanical Behavior of Materials, 2002
A wide class of partial differential equations has at least three conservation laws that remain i... more A wide class of partial differential equations has at least three conservation laws that remain invariant for certain solutions of them and especially for solitary wave solutions. These conservation laws can be considered as the energy, the pseudomomentum and the mass integrals of these solutions. We investigate the invariant relation between the energy and the pseudomomentum for solitary waves in the Boussinesq equation that comes from the theory of elasticity.
The Homogeneous Markov System as elastic medium. The three dimensional case
Every attainable structure of the so called continuous-time Homogeneous Markov System (HMS) with ... more Every attainable structure of the so called continuous-time Homogeneous Markov System (HMS) with fixed size and state space S={1,2,...,n} is considered as a particle of R^n and consequently the motion of the structure corresponds to the motion of the particle. Under the assumption that "the motion of every particle-structure at every time point is due to its interaction with its surroundings", R^n becomes a continuum. Then the evolution of the set of the attainable structures corresponds to the motion of the continuum. For the case of a three-state HMS it is stated that the concept of the two-dimensional isotropic elasticity can further interpret its evolution.
A Non-Integrability Test for Perturbed Hamiltonian Systems of Two Degrees of Freedom
NATO ASI Series, 1994
The non-integrability of Hamiltonian systems and especially nearly integrable ones has been exten... more The non-integrability of Hamiltonian systems and especially nearly integrable ones has been extensively investigated in the past and criteria for establishing this property in given Hamiltonians have appeared. Although these criteria do not always supply the answer to the question of non-integrability, they can often be applied to physical problems and may be divided into different categories, each one exploiting certain characteristic properties peculiar to integrable or non-integrable systems.
In this Letter we consider n degrees-of-freedom integrable Hamiltonian systems subjected to a non... more In this Letter we consider n degrees-of-freedom integrable Hamiltonian systems subjected to a non-Hamiltonian perturbation controlled by a small parameter ε. An obstruction to the analytic continuation of the integrals of motion of the unperturbed system with respect to ε is developed for sufficiently small perturbations. The theory is applied to a perturbed system of Morse oscillators.
An analytical description of magnetic islands is presented for the typical case of a single pertu... more An analytical description of magnetic islands is presented for the typical case of a single perturbation mode introduced to tokamak plasma equilibrium in the large aspect ratio approximation. Following the Hamiltonian structure directly in terms of toroidal coordinates, the well known integrability of this system is exploited, laying out a precise and practical way for determining the island topology features, as required in various applications, through an analytical and exact flux surface label. [
Journal of Physics A: Mathematical and General, 1990
By combining Darboux's results on the direct construction of integrable systems with results from... more By combining Darboux's results on the direct construction of integrable systems with results from the inverse problem of dynamics, we prove that if a planar potential admits a monoparametric family of conic sections with constant focal distance or a family of confocal parabolas, it is integrable and the second invariant is quadratic in the moments, while the existence of a family of straight lines, either parallel or interesecting at one point also suffices for integrability, the second invariant being linear in the momenta.
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Papers by E. Meletlidou