Unstable Hamiltonian Systems Research Articles

Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation where , with n, d ⩾ 1, V i are Morse potentials, and ɛ is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations H 1. Indeed, the set of admissible H 1 is C ω dense and C 3 open (a fortiori, C ω open). Our perturbative technique for the genericity is valid in the C k topology for all k ∈ [3, ∞) ∪ {∞, ω}.

Gevrey genericity of Arnold diffusion in a priori unstable Hamiltonian systems

It is well known that under generic C r smooth perturbations, the phenomenon of global instability, known as Arnold diffusion, exists in a priori unstable Hamiltonian systems. In this paper, by using variational methods, we will prove that under generic Gevrey smooth perturbations, Arnold diffusion still exists in the a priori unstable Hamiltonian systems of two and a half degrees of freedom.

Noise-Induced Stabilization of Perturbed Hamiltonian Systems

Noise-induced stabilization is the phenomenon in which the addition of randomness to an unstable system of ordinary differential equations results in a stable system of stochastic differential equations. With stability defined as global stochastic boundedness, Hamiltonian systems can never be stabilized by the addition of noise that is constant in space. In this article, we investigate how to deterministically perturb a class of unstable Hamiltonian systems in such a way that the qualitative behavior is preserved, but that enables the systems to exhibit noise-induced stabilization.

Arnold diffusion in a neighborhood of strong resonances

The paper deals with nearly integrable multidimensional a priori unstable Hamiltonian systems. Assuming the Hamilton function is smooth and time-periodic, we study perturbations that are trigonometric polynomials in the “angle” variables in the first approximation. For a generic system in this class, we construct a trajectory whose projection on the space of slow variables crosses a small neighborhood of a strong resonance. We also estimate the speed of this crossing.

Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion

We consider models given by Hamiltonians of the formH(I,φ,p,q,t;ε)=h(I)+∑j=1n±(12pj2+Vj(qj))+εQ(I,φ,p,q,t;ε) where I∈I⊂Rd,φ∈Td, p,q∈Rn, t∈T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28,29,43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem.We show that, for 0<ε≪1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion.The method of proof follows closely the strategy of [28,29]. The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I∈I⊂Rd. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature.The present paper, however, differs substantially from [28,29]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail.More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [28,29].

Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems

We prove the existence of Arnold diffusion in a typical a priori unstable Hamiltonian system outside a small neighbourhood of strong resonances. More precisely, we consider a near-integrable Hamiltonian system with Hamiltonian H = H0 + εH1 + O(ε2), where the unperturbed Hamiltonian H0 is essentially the product of a one-dimensional pendulum and n-dimensional rotator. Coordinates y = (y1, …, yn) on the rotator space are first integrals in the unperturbed system and become slow variables after perturbation.The main result is as follows. Suppose that the time-periodic perturbation H1 is Cr-generic, is sufficiently large. A resonance , where is a frequency vector and , is called strong if |k| < C⋄. The constant C⋄ is determined by H0 and H1 and does not depend on ε. Let Q be a domain on the rotator space such that its closure is free from strong resonances and let γ be a smooth curve on Q. Then for any small ε > 0 there exists a trajectory whose projection to Q moves in a c| log ε|αε1/4-neighbourhood of γ is any constant and c = c(α) > 0) with average velocity along γ of order ε/| log ε|.

Instability for &lt;em&gt;a priori&lt;/em&gt; unstable Hamiltonian systems: A dynamical approach

In this article, we consider an a priori unstable Hamiltonian system with three degrees of freedom, for which we construct a drifting solution with an optimal time of instability. Such a result has been already proved by Berti, Bolle and Biasco using variational arguments, and by Treschev using his separatrix map theory. Our approach is new: it is based on a special type of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus, and it gives the first concrete application of this idea introduced by Moeckel in an abstract setting and further studied by Marco. Our method should also be useful in obtaining the optimal time of instability in the more difficult context of a priori stable Hamiltonian systems.

A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom and we apply the geometric mechanism for diffusion introduced in [A. Delshams, R. de la Llave, T.M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc. 179 (844) (2006), viii+141 pp.], and generalized in [A. Delshams, G. Huguet, Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems, Nonlinearity 22 (8) (2009) 1997–2077]. We provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits.The simplification of the hypotheses allows us to perform the straightforward computations along the proof and present the geometric mechanism of diffusion in an easily understandable way. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

An extended geometric criterion for chaos in the Dicke model

We extend HBLSL's (Horwitz, Ben Zion, Lewkowicz, Schiffer and Levitan) new Riemannian geometric criterion for chaotic motion to Hamiltonian systems of weak coupling of potential and momenta by defining the ‘mean unstable ratio’. We discuss the Dicke model of an unstable Hamiltonian system in detail and show that our results are in good agreement with that of the computation of Lyapunov characteristic exponents.

Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems

In this paper we consider the case of a general perturbation, for r large enough, of an a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom, and we provide explicit conditions on it, which turn out to be generic and are verifiable in concrete examples, which guarantee the existence of Arnold diffusion.This is a generalization of the result in Delshams et al (2006 Mem. Am. Math. Soc.) where the case of a perturbation with a finite number of harmonics in the angular variables was considered.The method of proof is based on a careful analysis of the geography of resonances created by a generic perturbation and it contains a deep quantitative description of the invariant objects generated by the resonances therein. The scattering map is used as an essential tool to construct transition chains of objects of different topology. The combination of quantitative expressions for both the geography of resonances and the scattering map provides, in a natural way, explicit computable conditions for instability.

Detecting order and chaos in three-dimensional Hamiltonian systems by geometrical methods

We use a geometrical method to distinguish between ordered and chaotic motion in three-dimensional Hamiltonian systems. We show that this method gives results in agreement with the computation of Lyapunov characteristic exponents. We discuss some examples of unstable Hamiltonian systems in three dimensions, giving, as a particular illustration, detailed results for a potential obtained from a Hamiltonian obtained from a Yang-Mills system.

Geometry of Hamiltonian Chaos

The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model when a transition is made to an associated manifold. We find, in this way, a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this associated manifold results in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions.

Evolution of slow variables in a priori unstable Hamiltonian systems

We study diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems. These systems are perturbations of integrable ones, which have a family of hyperbolic tori. We prove that in the case of two and a half degrees of freedom the action variable generically drifts (i.e. changes on a trajectory by a quantity of order one). Moreover, there exists a trajectory such that the velocity of this drift is ε/logε, where ε is the parameter of the perturbation.

Ensemble dynamics of chaos

AbstractSystems of deterministic chaos demonstrate that the ensemble and trajectory descriptions of dynamics are not equivalent. This is shown through explicit constructions of the generalized spectral decompositions of the time evolution operator for probability densities. This realizes part of the long‐standing goal of the Brussels–Austin groups directed by Ilya Prigogine to elucidate the dynamical foundations of irreversibility. A brief review of the main aspects of the work are given here along with a new derivation of the fractality of eigenstates in some systems and a discussion of a Lyapunov functional in an unstable Hamiltonian system. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Drift in phase space: a new variational mechanism with optimal diffusion time

We consider nonisochronous, nearly integrable, a priori unstable Hamiltonian systems with a (trigonometric polynomial) O( μ)-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T d =O((1/ μ)ln(1/ μ)) by a variational method which does not require the existence of “transition chains of tori” provided by KAM theory. We also prove that our estimate of the diffusion time T d is optimal as a consequence of a general stability result derived from classical perturbation theory.

Trajectories in a neighbourhood of asymptotic surfaces of a priori unstable Hamiltonian systems

We consider a priori unstable (initially hyperbolic) near-integrable Hamiltonian systems in a neighbourhood of asymptotic surfaces of a family of hyperbolic tori. Applying the idea of the anti-integrable limit, we construct a large class of chaotic trajectories.

The extension of classical dynamics for unstable Hamiltonian systems

Classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the Liouville equation. It is shown that for the class of large nonintegrable Poincaré systems (LPS), the two descriptions are not equivalent. Practically all dynamical systems studied in statistical mechanics belong to this class. The basic step is the extension of the Liouville operator L H outside the Hilbert space to functions singular in their Fourier transformation. This function space plays an important role in statistical mechanics as functions of the Hamiltonian, and therefore equilibrium distribution functions belong to this class. Physically, these functions correspond to situations characterized by “persistent interactions” as they are realized in macroscopic physics. Persistent interactions are introduced in contrast to “transient interactions” studied in quantum mechanics by the S-matrix approach (asymptotically free in and out states). The eigenvalue problem for the Liouville operator L H is solved in this generalized function space for LPS. We obtain a complex, irreducible spectral representation. Complex means that the eigenvalues are complex numbers, whose imaginary parts refer to the various irreversible processes such as relaxation times, diffusion.… Irreducible means that these representations cannot be implemented by trajectory theory. As the result, the dynamical group of evolution splits into two semigroups. Moreover, the laws of classical dynamics take a new form as they have to be formulated on the statistical level. They express “possibilities” and no more “certitudes”. Two examples of typical classical systems, i.e., interacting particles and anharmonic lattices are studied.