The Intrinsic Stochasticity of Near-Integrable Hamiltonian Systems

Under certain conditions, the dynamics of near-integrable Hamiltonian systems appears to be stochastic. This stochasticity (intrinsic stochasticity, or deterministic chaos) is closely related to the Kolmogorov-Arnold-Moser (KAM) theorem of the stability of near-integrable multiperiodic Hamiltonian systems. The effect of the intrinsic stochasticity attracts still growing attention both in theory and in various applications in contemporary physics. The paper discusses the relation of the intrinsic stochasticity to the modern ergodic theory and to the KAM theorem, and describes some numerical experiments on related astrophysical and high-temperature plasma problems. Some open questions are mentioned in Conclusion.     

Construction of Quasi-Periodic Breathers¶via KAM Technique

By developing a KAM theorem which involves an infinitely multiple normal frequency, it is shown that there are plenty of breathers, quasi-periodic in time and super-exponentially localized in space, for the networks of weakly coupled oscillators. This answers an open problem by Aubry [A2] in case the linearized system has no continuous spectrum. Received: 29 March 2001 / Accepted: 10 October 2001     

The KAM theory of systems with short range interactions,II

The proof of the results on the KAM theory of systems with short range interactions, stated in [6] is completed. Estimates on the decay of the interactions generated by the iterative procedure in the KAM theorem are proved, as well as the modification of the theorems of [2–3] needed for results.This work was completed while the author was at the Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN, 55455, USASupported in part by NSF Grant DMS-8403664     

Twistless KAM tori

A selfcontained proof of the KAM theorem in the Thirring model is discussed.Archived in mp_arc@math.utexas.edu#93-172;to get a TeX version, send an empty E-mail message     

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ? _x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them.     

A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces

In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi–periodic solutions and its local Birkhoff normal form. The applications to the higher dimensional beam equations and the higher dimensional Schrödinger equations with nonlocal smooth nonlinearity are also given in this paper.     

Proof of Nishida's Conjecture on Anharmonic Lattices

We prove Nishida's 1971 conjecture stating that almost all low-energetic motions of the anharmonic Fermi-Pasta-Ulam lattice with fixed endpoints are quasi-periodic. The proof is based on the formal computations of Nishida, the KAM theorem, discrete symmetry considerations and an algebraic trick that considerably simplifies earlier results. Supported by an EPSRC postdoctoral fellowship.     

KAM Theory in Configuration Space and Cancellations in the Lindstedt Series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasi-periodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where the KAM theorem is usually formulated by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser’s modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. In this paper, we revisit Moser’s theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates, and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.     

Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs

We consider infinite dimensional Hamiltonian systems. We prove the existence of “Cantor manifolds” of elliptic tori–of any finite higher dimension–accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are “branching” points of other Cantor manifolds of higher dimensional tori. We also answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a pre-assigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation.     

On the problem of stability for higher-order derivative Lagrangian systems

The problem of stability for dynamical systems whose Lagrangian function depends on the derivatives of a higher order than one is studied. The difficulty of this analysis arises from the indefiniteness of the Hamiltonian, so that the well-known Lagrange-Dirichlet theorem cannot be used and the methods of the canonical perturbation theory (KAM theory) must be employed. We show, with an example, that the indefiniteness of the energy does not forbid the stability.     

KAM for the Quantum Harmonic Oscillator

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. P?schel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schr?dinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schr?dinger equations with the harmonic potential and a quasi periodic in time potential.     

Symmetry and Resonance in Periodic FPU Chains

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This “Birkhoff–Gustavson normal form” retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect this in lower-order resonant Hamiltonian systems. So the periodic FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries. Received: 25 July 2000 / Accepted: 20 December 2000     

Differentiability at the Tip of Arnold Tongues for Diophantine Rotations: Numerical Studies and Renormalization Group Explanations

We study numerically the regularity of Arnold tongues corresponding to Diophantine rotation numbers of circle maps at the edge of validity of KAM theorem. This serves as a good test for the numerical stability of two different algorithms. We find empirically that Arnold tongues are only finitely differentiable at the tip. We also find several scaling properties of the Sobolev norms of the conjugacy near the breakdown. We also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the Sobolev regularity. We also uncover empirically some other patterns which require explanation.     

Ray dynamics in long-range deep ocean sound propagation

Recent results relating to ray dynamics in ocean acoustics are reviewed. Attention is focused on long-range propagation in deep ocean environments. For this class of problems, the ray equations may be simplified by making use of a one-way formulation in which the range variable appears as the independent (timelike) variable. Topics discussed include integrable and nonintegrable ray systems, action-angle variables, nonlinear resonances and the KAM theorem, ray chaos, Lyapunov exponents, predictability, nondegeneracy violation, ray intensity statistics, semiclassical breakdown, wave chaos, and the connection between ray chaos and mode coupling. The Hamiltonian structure of the ray equations plays an important role in all of these topics.     

KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions

In this paper, one-dimensional (1D) nonlinear wave equations

A remark on the KAM theorem applied to a four-vortex system

We consider a planar four-vortex system with unit intensities and apply the KAM theorem for two-dimensional tori with fixed frequency. We obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation number=(5—1)/2 and compare our result with numerical experiments.     

Critical behavior of a KAM surface: I. Empirical results

Kolmogorov-Arnol'd-Moser (KAM) surfaces are studied in the context of a perturbed two-dimensional twist map. In particular, we ask how a KAM surface can disappear as the perturbation parameter is increased. Following Greene, we use cycles to numerically construct the KAM curve and discover that at the critical coupling it shows structure at all length scales. Aspects of this structure are fitted by a scaling analysis; critical indices and scaling functions are determined numerically. Some evidence is presented which suggests that the results are universal.Supported in part by the Materials Research Laboratory Program of the National Science Foundation at the University of Chicago under grant No. NSF-MRL 7924007.Robert R. McCormick and National Science Foundation Fellow     

LIMIT-CYCLE STABILITY REVERSAL NEAR A HOPF BIFURCATION WITH AEROELASTIC APPLICATIONS

The objective of this paper is to present an analytical/numerical analysis of the phenomenon of limit-cycle stability reversal (from unstable to stable, and vice versa). A singular perturbation technique, the method of the normal form (in the asymptotic- expansion version), is utilized. The number of equations is then reduced to a “minimal set”, for which the results are in good agreement with those from the original equations. This minimal set is determined by the amplitude of the λ̂-points (a concept closely related to the small divisors in the KAM theory). This set is larger than that corresponding to the zero real-part eigenvalues (center-manifold theorem). The method is applied to a specific problem: an aeroelastic section with cubic free-play non-linearities where the parameter μ is the flight speed. Numerical studies have been performed to show the dependence of the Hopf bifurcation characteristics upon the structural and geometric properties of the wing section. Plots depicting amplitudes and frequency versus flight speed are presented.     

Breakdown of Stability of Motion in Superquadratic Potentials

Based on the KAM theory, investigation of the equation of motion of a classical particle in a one-dimensional superquadratic potential well, under the influence of an external time-periodic forcing, raised a hope that all the solutions are bounded. Indeed, due to the superquadraticity of the potential the frequency of oscillations of the solutions in the system tends to infinity as the amplitude increases. Therefore, because of this relationship between the frequency and the amplitude, intuitively one might expect that all resonances that could cause the accumulation of energy would be destroyed, and thus all solutions would stay bounded for all time. More formally, according to Moser's twist theorem, this could mean the existence of invariant tubes in the extended phase space and therefore would result in the boundedness of the solutions. Actually, the boundedness results have been established for a large class of superquadratic potentials, but in general, the above intuition turns out to be wrong. Littlewood showed it by creating a superquadratic potential in which an unbounded motion occurs in the presence of some particular piecewise constant forcing. Moreover, Littlewood's result holds for a larger class of forcings. Here it is proven for the continuous time-periodic forcing. For this purpose a new averaging technique for the forced motions in superquadratic potentials with rather weak assumptions on the differentiability of the potentials has been developed. Received: 1 February 1995 / Accepted: 15 March 1997