Stability in Hamiltonian systems : applications to the restricted three-body problem

The N-body problem is a classical famous problem which has attracted a lot of attention. It consists of describing the complete behavior of all solutions of the equations of motions for a given initial condition. Still related to this kind of problem Euler in 1772 describe the three-body problem in his eort to study the motion of the moon. Later on Jacobi in 1836 brought forward the main mathematical interest in an even more specic part of the three body problem, namely the one which is reduced to a conservative two degrees of freedom problem. This has somehow brought up an extensive study on mechanics. Despite of all this eort of great mathematicians, in general the N-body problem for N> 2 is still unsolved

Higher order resonance in two degree of freedom Hamiltonian system

This paper reviews higher order resonance in two degrees of freedom Hamilto- nian systems. We consider a positive semi-definite Hamiltonian around the origin. Using normal form theory, we give an estimate of the size of the domain where interesting dynamics takes place, which is an improvement of the one previously known. Using a geometric numerical integration approach, we investigate this in the elastic pendulum to find additional evidence that our estimate is sharp. In an extreme case of higher order resonance, we show that phase interaction between the degrees of freedom occurs on a short time-scale, although there is no energy interchange

Higher-Order Resonances in Dynamical Systems

This thesis is a collection of studies on higher-order resonances in an important class of dynamical systems called coupled oscillators systems. After giving an overview of the mathematical background, we start in Chapter 1 by presenting a study on resonances in two degrees of freedom, autonomous, Hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the H´enon-Heiles family of Hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4 : 1-resonance as the most prominent after the 2 : 1-resonance and which explains why the 3 : 1- resonance is neglected. In chapter 2 of this thesis we study the performance of a symplectic numerical integrator based on the splitting method. This method is applied to a subtle problem i.e. higher order resonance of the elastic pendulum. In order to numerically study the phase space of the elastic pendulum at higher order resonance, a numerical integrator which preserves qualitative features after long integration times is needed. We show by means of an example that our symplectic method offers a relatively cheap and accurate numerical integrator. In chapter 3 we study two degree of freedom Hamiltonian systems and applications to nonlinear wave equations. Near the origin, we assume that near the linearized system has purely imaginary eigenvalues: ±i[omega]1 and ±i[omega]2 with 0 <[omega]2/[omega]1«1 or w2/w1»1, which is interpreted as a perturbation of a problem with double zero eigenvalues. Using the averaging method, we compute the normal form and show that the dynamics differs from the usual one for Hamiltonian systems at higher order resonances. Under certain conditions, the normal form is degenerate which forces us to normalize to higher degree. The asymptotic character of the normal form and the corresponding invariant tori is validated using KAM theorem. This analysis is then applied to widely separated mode-interaction in a family of nonlinear wave equations containing various degeneracies. In chapter 4 we present an analysis of a system of coupled-oscillators. We make two assumptions for our system. The first assumption is that the frequencies of the characteristic oscillations are widely separated, and the second is that the nonlinear part of the vector field preserves the distance to the origin. Using the first assumption, we prove that the reduced normal form of our system, exhibits an invariant manifold which, exists for all values of the parameters and cannot be perturbed away by including higher order terms in the normal form. Using the second assumption, we view the normal form as an energy-preserving three-dimensional system which is linearly perturbed. Restricting our selves to a small perturbation, the flow of the energy-preserving system is used to study the flow in general. We present a complete study of the flow of energy-preserving system and the bifurcations in it. Using these results, we provide the condition for having a Hopf bifurcation of one of the two equilibria. We also numerically follow the periodic solution created via the Hopf bifurcation and find a sequence of period-doubling and fold bifurcations, and also a torus (or Naimark-Sacker) bifurcation. This chapter is a sequel to the study of the previous chapter, where a system of coupled oscillators with widely separated frequencies and energy-preserving quadratic nonlinearity is studied. However, in this paper we are more concerned with the energy-preserving nature of the nonlinearity. We also study a singularly perturbed conservative system in R [exp. n], which is a generalization of our system, and derive a condition for the existence of nontrivial equilibrium of such a system. Returning to the original system we start with for a different set of parameter values compared with those in [?]. Numerically, we find interesting bifurcations and dynamics such as torus (Naimark-Sacker) bifurcation, chaos and heteroclinic-like behaviou

Stability of axial orbits in galactic potentials

We investigate the dynamics in a galactic potential with two reflection symmetries. The phase-space structure of the real system is approximated with a resonant detuned normal form constructed with the method based on the Lie transform. Attention is focused on the stability properties of the axial periodic orbits that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of bifurcations and compare them with numerical results available in the literature.Comment: 20 pages, accepted for publication on Celestial Mechanics and Dynamical Astronom

Quantitative predictions with detuned normal forms

The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.Comment: Accepted for publication on Celestial Mechanics and Dynamical Astronom

BOGDANOV-TAKENS BIFURCATIONS IN THREE COUPLED OSCILLATORS SYSTEM WITH ENERGY PRESERVING NONLINEARITY

looked at pdf abstract DOI : http://dx.doi.org/10.22342/jims.18.2.113.73-8

Symmetry and resonance in Hamiltonian systems

In this paper we study resonances in two degrees of freedom, autonomous, hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the Henon-Heiles family of hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4 : 1-resonance as the most prominent after the 2 : 1-resonance and which explains why the 3 : 1-resonance is neglected

On a multiple timescales perturbation approach for a stefan problem with a time-dependent heat flux at the boundary

In this paper, a classical Stefan problem is studied. It is assumed that a small, time-dependent heat influx is present at the boundary, and that the initial values are small. By using a multiple timescales perturbation approach, it is shown analytically (most likely for the first time in the literature) how the moving interface and its stability are influenced by the time-dependent heat influx at the boundary and by the initial conditions. Accurate approximations of the solution of the problem are constructed, which are valid on long timescales. The constructed approximations turn out to agree very well with solutions of problems for which similarity solutions are available (in numerical form).</p

On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Mathematical Physic