An Exponential Estimate of the Time of Stability of Nearly-Integrable Hamiltonian Systems

Nekhoroshev, N. N.

doi:10.1070/rm1977v032n06abeh003859

Cited by 662 publications

(497 citation statements)

References 14 publications

(14 reference statements)

Supporting

Mentioning

477

Contrasting

Unclassified

Order By: Relevance

“…2] and also [DG01]; the ideas were initially developed in [Nek77]). To study the behavior of the trajectories of H in the region close to a resonance of multiplicity n, with 1 ≤ n < N (associated to a module of resonances M ⊂ Z N ), one carries out some steps of normalizing transformation, in order to minimize the nonresonant terms of the Fourier expansion in the angular variables ϕ.…”

Section: Motivationmentioning

confidence: 99%

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

Delshams

Gutiérrez

Pacha

2013

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and sufficient condition for the transversality, in terms of a Riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop. The key point of our approach is to write the invariant manifolds in terms of generating functions, which are solutions of the Hamilton-Jacobi equation. In some examples, we show that it is enough to analyse the phase portrait of the Riccati equation without solving it explicitly. Finally, we consider an analogous problem in a perturbative situation. If the invariant manifolds of the unperturbed loop coincide, we have a problem of splitting of separatrices. In this case, the Riccati equation is replaced by a Mel nikov potential defined as an integral, providing a condition for the existence of a perturbed loop and its transversality. This is also illustrated with a concrete example.

show abstract

Section: Motivationmentioning

confidence: 99%

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

Delshams

Gutiérrez

Pacha

2013

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…When expanding a rigorous finite order procedure into a series, even in a very benign example, like periodic averaging of analytic ordinary differential equationṡ x = f (x, t/ε), convergence of the expansion cannot be expected in general. Nevertheless, beyond a finite asymptotic expansion, there are exponential estimates in several aspects, early examples are by Nekhoroshev, and Neishtadt; see, e.g., [Nek79,Nei84,LM88]. These then yield upper estimates on all kinds of effects created by the periodic nonautonomous structure.…”

Section: Introductionmentioning

confidence: 99%

Exponential Estimates in Averaging and Homogenisation

Matthies

2006

Analysis, Modeling and Simulation of Multiscale Problems

View full text Add to dashboard Cite

Abstract. Many partial differential equations with rapid spatial or temporal scales have effective descriptions which can be derived by homogenisation or averaging. In this article we deal with examples, where quantitative estimates of the error is possible for higher order homogenisation and averaging. In particular, we provide theorems, which allow homogenisation and averaging beyond all orders by giving exponential estimates of appropriately averaged and homogenised descriptions.Methods include iterated averaging transformations, optimal truncation of asymptotic expansions and highly regular solutions (Gevrey regularity). Prototypical examples are reaction-diffusion equations with heterogeneous reaction terms or rapid external forcing, nonlinear Schrödinger equations describing dispersion management, and second-order linear elliptic equations.

show abstract

“…Note that this result is not sufficient to obtain uniform stability estimates, as in Nekhoroshev Theorem below. More precise normal form results are given in [9] and [11].…”

Section: Symplectic Diffeomorphisms and Normalmentioning

confidence: 99%

Hamiltonian Systems: Stability and Instability Theory

Bernard

2006

Encyclopedia of Mathematical Physics

View full text Add to dashboard Cite

The solar system has long appeared to astronomers and mathematicians as a model of stability. On the other hand, statistical mechanics relies on the assumption that large assemblies of particles form highly unstable systems (at the microscopic scale). Yet all these physical situations are described, at least to a certain degree of approximation, by Hamiltonian systems.One may hope that Hamiltonian systems can be classified in two different categories, stable and unstable ones. However, the situation is much more complicated and both stable and unstable behaviors cohabit in typical systems. Even our examples are not perfect paradigms of stability and instability. Indeed, it is now clear from numerical as well as theoretical points of views that some instability is present over long time-scales in the solar systems, so that for example future collisions between planets can not be completely ruled out in view of our present understanding. On the other hand, unexpected patterns of stability have been discovered in systems involving a large number of particles.Understanding the impact of stable and unstable effects in Hamiltonian systems has been considered since Poincaré as one of the most important questions in dynamical systems. In the present text, we will discuss model Hamiltonian systems of the formwhere (q, p) ∈ T d × U , with U a bounded open subset of R n . Recall that the equations of motion areThe textbook [1] is a good general introduction on Hamiltonian systems. We will always denote by ω(p) the frequency map ∂ p h(p), which plays a crucial role. Here, as is obvious in (2), the action variables p are preserved under the evolution in the unperturbed case ǫ = 0. We will try to explain what is known on the evolution of these action variables for the perturbed system. As we will see, in many situations, these variables are extremely stable. For example, KAM theorem implies that, for a positive measure of initial conditions (q 0 , p 0 ) the trajectory (q(t), p(t)) satisfies p(t) − p (0) Cǫ for all times. Examples show that some initial conditions may lead to unstable trajectories, that is trajectories such that p(t) − p(0) 1/C for some t (depending on ǫ) and some fixed constant C independent of ǫ. However, this is, as we will see, possible only for very large time t (meaning that t as a function of ǫ has to go to infinity very quickly when ǫ −→ 0). The main questions here are to understand in what situation instability is or is not possible, and what kind of evolutions can have the actions variable p. Another important question is to estimate the speed (as a function of the parameter ǫ) of the evolutions of p.

show abstract

An Exponential Estimate of the Time of Stability of Nearly-Integrable Hamiltonian Systems

Cited by 662 publications

References 14 publications

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

Exponential Estimates in Averaging and Homogenisation

Hamiltonian Systems: Stability and Instability Theory

Contact Info

Product

Resources

About