Geometric Numerical Integration Applied to The Elastic Pendulum at Higher Order Resonance

In this paper 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.Comment: 15 pages, 6 figure

Construction of Integrals of Higher-Order Mappings

We find that certain higher-order mappings arise as reductions of the integrable discrete A-type KP (AKP) and B-type KP (BKP) equations. We find conservation laws for the AKP and BKP equations, then we use these conservation laws to derive integrals of the associated reduced maps.Comment: appear to Journal of the Physical Society of Japa

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

WIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY

Abstract. In this paper we present an analysis of a system of coupled oscillators suggested by atmospheric dynamics. 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 has an invariant manifold which exists for all values of the parameters. This invariant manifold 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 ourselves 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 the energy-preserving system and its bifurcations. Using these results, we provide the condition for having a Hopf bifurcation of one of the two equilibria of the perturbed system. We also numerically follow the periodic solution created via the Hopf bifurcation and find a sequence of Period-Doubling and Fold bifurcations, also a Torus bifurcation. Keywords: High-order resonances, singular perturbation, bifurcation. 1

Hamiltonian Systems with Widely Separated Frequencies

In this paper 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: \Sigmai! 1 and \Sigmai! 2 , with 0 ! ! 2 =! 1 1 or ! 2 =! 1 AE 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

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 determining the size 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. Keywords. Hamiltonian mechanics, higher-order resonance, normal forms, symmetry, elastic pendulum. AMS clasification. 34E05, 70H33, 70K30 1 Introduction Symmetries play an essential part in studying the theory and applications of dynamical systems. For a..

Application of Lagrange Multiplier Method for Computing Fold Bifurcation Point in A Two-Prey One Predator Dynamical System

We propose by means of an example of applications of the classical Lagrange Multiplier Method for computing fold bifurcation point of an equilibrium ina one-parameter family of dynamical systems. We have used the fact that an equilibrium of a system, geometrically can be seen as an intersection between nullcline manifolds of the system. Thus, we can view the problem of two collapsing equilibria as a constrained optimization problem, where one of the nullclines acts as the cost function while the other nullclines act as the constraints