Discrete and Continuous Dynamical Systems
October 2004 , Volume 11 , Issue 4
Special issue on 'Hamiltonian Systems and Applications'
Guest editors: Amadeu Delshams and Antonio Giorgilli
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We describe a method for studying the existence and the linear stability of branches of periodic solutions for a dynamical system with a parameter. We apply the method to the planar restricted 3-body problem extending the results of [A]. More precisely, we prove the existence of some continuous branches of periodic orbits with the energy or the masses of the primaries as parameters, and provide an approximation of the orbits with rigorous bounds. We prove the linear stability or instability of the orbits.
We consider an example of singular or weakly hyperbolic Hamiltonian, with $3$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model consists of an integrable Hamiltonian possessing a $2$-dimensional hyperbolic invariant torus with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation of order $\mu=\varepsilon^p$. We choose $\omega$ as the golden vector. Our aim is to obtain asymptotic estimates for the splitting, proving the existence of transverse intersections between the perturbed whiskers for $\varepsilon$ small enough, by applying the Poincaré-Melnikov method together with a accurate control of the size of the error term.
The good arithmetic properties of the golden vector allow us to prove that the splitting function has 4 simple zeros (corresponding to nondegenerate critical points of the splitting potential), giving rise to 4 transverse homoclinic orbits. More precisely, we show that a shift of these orbits occurs when $\varepsilon$ goes across some critical values, but we establish the continuation (without bifurcations) of the 4 transverse homoclinic orbits for all values of $\varepsilon\to0$.
We consider a singular or weakly hyperbolic Hamiltonian, with $n+1$ degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model consists of an integrable Hamiltonian possessing an $n$-dimensional hyperbolic invariant torus with fast frequencies $\omega/\sqrt\varepsilon$ and coincident whiskers, plus a perturbation of order $\mu=\varepsilon^p$. The vector $\omega$ is assumed to satisfy a Diophantine condition.
We provide a tool to study, in this singular case, the splitting of the perturbed whiskers for $\varepsilon$ small enough, as well as their homoclinic intersections, using the Poincaré--Melnikov method. Due to the exponential smallness of the Melnikov function, the size of the error term has to be carefully controlled. So we introduce flow-box coordinates in order to take advantage of the quasiperiodicity properties of the splitting. As a direct application of this approach, we obtain quite general upper bounds for the splitting.
This paper describes the global flow of homogeneous polynomial potentials of degree 3 for negative and positive energy. For the negative energy case a blow up of McGehee type is enough to get the complete picture of the flow. In the positive energy case, McGehee blow up fails to give global information about the flow, but comparing with a separable case we are able to obtain all the possible asymptotic behavior of solutions, whenever the coefficients of the normal form of the potential are positive.
In this note we compare the frequencies of the motion of the Trojan asteroids in the Restricted Three-Body Problem (RTBP), the Elliptic Restricted Three-Body Problem (ERTBP) and the Outer Solar System (OSS) model. The RTBP and ERTBP are well-known academic models for the motion of these asteroids, and the OSS is the standard model used for realistic simulations.
Our results are based on a systematic frequency analysis of the motion of these asteroids. The main conclusion is that both the RTBP and ERTBP are not very accurate models for the long-term dynamics, although the level of accuracy strongly depends on the selected asteroid.
We revisit the celebrated model of Fermi, Pasta and Ulam with the aim of investigating, by numerical computations, the trend towards equipartition in the thermodynamic limit. We concentrate our attention on a particular class of initial conditions, namely, with all the energy on the first mode or the first few modes. We observe that the approach to equipartition occurs on two different time scales: in a short time the energy spreads up by forming a packet involving all low--frequency modes up to a cutoff frequency $\omega_c$, while a much longer time is required in order to reach equipartition, if any. In this sense one has an energy localization with respect to frequency. The crucial point is that our numerical computations suggest that this phenomenon of a fast formation of a natural packet survives in the thermodynamic limit. More precisely we conjecture that the cutoff frequency $\omega_c$ is a function of the specific energy $\epsilon = E/N$, where $E$ and $N$ are the total energy and the number of particles, respectively. Equivalently, there should exist a function $\epsilon_c(\omega)$, representing the minimal specific energy at which the natural packet extends up to frequency $\omega$. The time required for the fast formation of the natural packet is also investigated.
Fermi-Pasta-Ulam lattice is a classical mechanical system of an infinite number of discrete particles on a line. Each particle is assumed to interact with the nearest left and right neighbors only. We construct travelling waves in the system assuming that the potential has a singularity at zero. The waves appear near the hard ball limit.
We give a computer-assisted proof for the existence of a renormalization group fixed point (Hamiltonian) with non-trivial scaling, associated with the breakup of invariant tori with rotation number equal to the golden mean.
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