Kolmogorov-Arnold-Moser Research Articles

Quantum analog of the Kolmogorov-Arnold-Moser theorem in the anisotropic Dicke model and its possible implications in the hybrid Sachdev-Ye-Kitaev models

The classical Kolmogorov-Arnold-Moser (KAM) theorem provides the underlying mechanism for the stability of the solar system under some small chaotic perturbations. Despite many previous efforts, any quantum version of the KAM theorem remains elusive. In this work, we provide a quantum KAM theorem in the context of the anisotropic Dicke model, which is the most important quantum optics model. It describes a single mode of photons coupled to $N$ qubits with both a rotating wave (RW) term and a counter-RW (CRW) term. As the ratio of the CRW over the RW term increases from zero to one, the systems evolves from quantum integrable to quantum chaotic. We establish a quantum KAM theorem to characterize such an evolution quantitatively by both large-$N$ expansion and random matrix theory and find agreement from the two complementary approaches. Connections and differences between the Dicke models and Sachdev-Ye-Kitaev (SYK) or hybrid SYK models are examined. A possible quantum KAM theorem in terms of other quantum chaos criteria such as the quantum Lyapunov exponent is also discussed.

Quasi-periodic breathers in Newton’s cradle

We consider the parameterized Newton’s cradle lattice with Hertzian interactions in this paper. The positive parameters are {βn : |n| ≤ b} with a fixed integer b ≥ 0, and the Hertzian potential is V(x)=11+α|x|1+α for a fixed real number α > α* ≔ 12b + 25. Corresponding to a large Lebesgue measure set of (βj)|j|≤b∈R+2b+1, we show the existence of a family of small amplitude, linearly stable, quasi-periodic breathers for Newton’s cradle lattice, which are quasi-periodic in time with 2b + 1 frequencies and localized in space with rate 1|n|1+α as |n| ≫ 1. To overcome obstacles in applying the Kolmogorov–Arnold–Moser (KAM) method due to the finite smoothness of V, especially when α is not an integer and to obtain a sharp estimate of the localization rate of the quasi-periodic breathers, the proof of our result uses the Jackson–Moser–Zehnder analytic approximation technique but with refined estimates on error bounds, depending on the smoothness and dimension, which provide crucial controls on the convergence of KAM iterations.

Invariant tori for multi-dimensional integrable Hamiltonians coupled to a single thermostat

This paper demonstrates sufficient conditions for the existence of Kolmogorov-Arnol’d-Moser (KAM) tori in a singly thermostated, integrable Hamiltonian system with n degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2—which include the Nosé thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends theorem 3.2 of Legoll et al (2009 Nonlinearity 22 1673–94) to prove that a ‘typical’ singly thermostated, integrable, real-analytic Hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable Hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion.

Universal ratio in random matrix theory and chaotic-to-integrable transition in type-I and type-II hybrid Sachdev-Ye-Kitaev models

We investigate chaotic to integrable transition in two types of hybrid SYK models, which contain both $q=4$ SYK with interaction $J$ and $q=2$ SYK with an interaction $K$ in type-I or ${(q=2)}^{2}$ SYK with an interaction $\sqrt{K}$ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $N$ even and $N$ odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio, which is the ratio of the next-nearest-neighbor (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study chaotic to integrable transitions (CIT) in both types of hybrid SYK models. Some preliminary results on possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem and its dual version in the quantum chaotic side are given. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $1/N$ expansion in the large $N$ limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at $N=\ensuremath{\infty}$, $1/N$ expansions at a finite $N$, and the CIT characterized by the RMT at a finite $N$: the former focus on the ground state and its low-energy excitations (edge states in the Fock space), the latter on excited states (bulk states in the Fock space). We also discuss eigenstate thermalization hypothesis (ETH)'s power on a quantum chaotic bulk state and its inability to encode the edge states. Comments on some previously related works are given. Some future perspectives, especially the failure of the Zamoloddchikov's c-theorem in 1d CFT RG flow are outlined.

On the non-existence of stepped-pressure equilibria far from symmetry

The Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502) has been successful in the construction of equilibria in 3D configurations that contain a mixture of flux surfaces, islands and chaotic magnetic field lines. In this model, the plasma is sliced into sub-volumes separated by ideal interfaces, and in each volume the magnetic field is a Beltrami field. In the cases where the system is far from possessing a continuous symmetry, such as in stellarators, the existence of solutions to a stepped-pressure equilibrium with given constraints, such as a multi-region relaxed MHD minimum energy state, is not guaranteed but is often taken for granted. Using SPEC, we have studied two different scenarios in which a solution fails to exist in a slab with analytic boundary perturbations. We found that with a large boundary perturbation, a certain interface becomes fractal, corresponding to the break up of a Kolmogorov–Arnold–Moser (KAM) surface. Moreover, an interface can only support a maximum pressure jump while a solution of the magnetic field consistent with the force balance condition can be found. An interface closer to break-up can support a smaller pressure jump. We discovered that the pressure jump can push the interface closer to being non-smooth through force balance, thus significantly decreasing the maximum pressure it can support. Our work shows that a convergence study must be performed on a SPEC equilibrium with interfaces close to break-up. These results may also provide insights into the choice of interfaces and have applications in finding out the maximum pressure a machine can support.

Quasi-periodic solutions for the generalized Benjamin–Bona–Mahony​ equation

A generalized Benjamin–Bona–Mahony (gBBM) equation ut−uxxt+ux+u4ux=0,subject to the periodic boundary condition is studied in this paper. Based on a new infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem with normal frequencies of finite limit-points, it is shown that the above gBBM equation admits plenty of time-quasi-periodic solutions with two frequencies of high modes.

Nonequilibrium Dynamics and Weakly Broken Integrability.

Motivated by dynamical experiments on cold atomic gases, we develop a quantum kinetic approach to weakly perturbed integrable models out of equilibrium. Using the exact matrix elements of the underlying integrable model, we establish an analytical approach to real-time dynamics. The method addresses a broad range of timescales, from the intermediate regime of prethermalization to late-time thermalization. Predictions are given for the time evolution of physical quantities, including effective temperatures and thermalization rates. The approach provides conceptual links between perturbed quantum many-body dynamics and classical Kolmogorov-Arnold-Moser theory. In particular, we identify a family of perturbations which do not cause thermalization in the weakly perturbed regime.

Finite-size coherent particle structures in high-Prandtl-number liquid bridges

Clustering of small rigid spherical particles in particle accumulation structures (PAS) of various shapes is found with highly resolved numerical simulations in a thermocapillary liquid bridge at a Prandtl number of 68. The intricate Kolmogorov--Arnold--Moser (KAM) structure of the fluid flow is unraveled for several Reynolds numbers for flows periodic in time and space. For these flows, a rich variety of periodic and quasiperiodic attractors for particles nearly density-matched to the fluid is found near closed streamlines and KAM tori approaching the free surface closely. A large parametric study finds the PAS dependence on particle size, particle-to-fluid density ratio, and Reynolds number.

Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems.

We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics. To prove this result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.

Investigations on Dynamical Stability in 3D Quadrupole Ion Traps

We firstly discuss classical stability for a dynamical system of two ions levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled oscillators. We obtain the solutions of the coupled system of equations that characterizes the associated dynamics. In addition, we supply the modes of oscillation and demonstrate the weak coupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov–Arnold–Moser (KAM) torus. In an attempt to better describe dynamical stability of the system, we introduce a model that characterizes dynamical stability and the critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the associated Hamilton function. The ion distribution can be described by means of numerical modeling, based on the Hamilton function we assign to the system. The approach we introduce is effective to infer the parameters of distinct types of traps by applying a unitary and coherent method, and especially for identifying equilibrium configurations, of large interest for ion crystals or quantum logic.

A KAM Theorem for Two Dimensional Completely Resonant Reversible Schrödinger Systems

In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrodinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrodinger systems on two dimensional torus.

On the Dynamics of a Gravitational Dipole

The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a neighborhood of its nominal mode. The orbit of the center of mass is assumed to be circular or elliptic with small eccentricity. Consideration is given both to planar and arbitrary spatial deviations of the gravitational dipole from its position corresponding to the nominal mode. The analysis is based on the classical Lyapunov and Poincaré methods and the methods of Kolmogorov – Arnold – Moser (KAM) theory. The necessary calculations are performed using computer algorithms. An analytic representation is given for conditionally periodic oscillations. Special attention is paid to the problem of the existence of periodic motions of the gravitational dipole and their Lyapunov stability, formal stability (stability in an arbitrarily high, but finite, nonlinear approximation) and stability for most (in the sense of Lebesgue measure) initial conditions.

Lagrangian transport in two‐dimensional time‐periodic cavity flow

AbstractThe Lagrangian transport in a laminar incompressible flow in a two‐dimensional square cavity driven by a harmonic tangential oscillation of a single cavity wall is investigated numerically for a range of Reynolds (Re) and Strouhal (Str) numbers. The topology of fluid trajectories is described by means of stroboscopic projections, which reveal the co‐existence of chaotic trajectories and regular Kolmogorov‐Arnold‐Moser (KAM) tori. The higher the frequency of the lid oscillation the more regular the fluid motion becomes and the size of the KAM tori increases. For low frequencies the KAM tori are strongly stretched along instantaneous streamlines of the flow, while for high frequencies they resemble streamlines of a mean flow.

Comparing Finite-Time Lyapunov Exponents and Lagrangian Descriptors for identifying phase space structures in a simple two-dimensional, time-periodic double-gyre model

This paper compares the advantages, limitations, and computational considerations of using Finite-Time Lyapunov Exponents (FTLEs) and Descriptors (LDs) as tools for identifying barriers and mechanisms of fluid transport in two-dimensional time-periodic flows. These barriers and mechanisms of transport are often referred to as Lagrangian Coherent Structures, though this term often changes meaning depending on the author or context. This paper will specifically focus on using FTLEs and LDs to identify stable and unstable manifolds of hyperbolic stagnation points, and the Kolmogorov-Arnold-Moser (KAM) tori associated with elliptic stagnation points. The background and theory behind both methods and their associated phase space structures will be presented, and then examples of FTLEs and LDs will be shown based on a simple, periodic, time-dependent double-gyre toy model with varying parameters.

Response solutions to the quasi-periodically forced systems with degenerate equilibrium: a simple proof of a result of W Si and J Si and extensions

We give a simple proof of the existence of response solutions in some quasi-periodically forced systems with degenerate fixed points. The same questions were answered by Si and Si (2018 Nonlinearity 31 2361–18) using two versions of Kolmogorov–Arnold–Moser (KAM) theory. Our method is based on reformulating the existence of response solutions as a fixed point problem in appropriate spaces of smooth functions. By algebraic manipulations, the fixed point problem is transformed into a problem dealt with contraction mapping principle. Compared to the KAM method, the present method does not incur a loss of regularity. That is, the solutions we obtain have the same regularity as the forcing. Moreover, the method here applies when problems are only finitely differentiable. It also weakens slightly the non-resonance conditions on the forcing frequencies. Since the method is based on the contraction mapping principle, we also obtain automatically smooth dependence on parameters and, when studying complex versions of the problem we discover the new phenomenon of monodromy. We also present results for higher dimensional systems, but for higher dimensional systems, the concept of degenerate fixed points is much more subtle than in one dimensional systems. To illustrate the power of the method, we also consider two problems not studied by Si and Si (2018 Nonlinearity 31 2361–18): the forcing with zero average and second order oscillators. We show that in the zero average forcing case, the solutions are qualitatively different, but for the second order oscillators are remarkably similar.

Response solutions to harmonic oscillators beyond multi–dimensional brjuno frequency

This paper focuses on the quasi–periodically forced nonlinear harmonic oscillators \begin{document}$ \begin{equation*} \ddot{x}+\lambda^{2}x = \epsilon f(\omega t,x), \end{equation*} $\end{document} where $ \lambda \in \mathcal{O} $, a closed interval not containing zero, the forcing term $ f $ is real analytic, and the frequency vector $ \omega \in \mathbb{R}^d \, (d \geq 2) $ is beyond Brjuno frequency, which we call as Liouvillean frequency. For the given class of the frequency $ \omega\in\mathbb{R}^{d}, $ which will be given later, we prove the existence of real analytic response solutions (the response solution is the quasi–periodic solution with the same frequency as the forcing) for the above equation. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite–dimensional harmonic oscillator systems with Liouvillean frequency.

Attractors for the motion of a finite-size particle in a two-sided lid-driven cavity

Abstract

Toppling Pencils-Macroscopic Randomness from Microscopic Fluctuations.

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for , the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol’d-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

KAM theory for the reversible perturbations of 2D linear beam equations

In the present paper, we prove an infinite dimensional reversible Kolmogorov-Arnold-Moser (KAM) theorem. As an application, we study the existence of KAM tori for a class of two dimensional (2D) non-Hamiltonian completely resonant beam equations with derivative nonlinearities. The Birkhoff normal form theory is also used since there are no external parameters in the equations.

Lagrangian chaos in steady three-dimensional lid-driven cavity flow.

Steady three-dimensional flows in lid-driven cavities are investigated numerically using a high-order spectral-element solver for the incompressible Navier-Stokes equations. The focus is placed on critical points in the flow field, critical limit cycles, their heteroclinic connections, and on the existence, shape, and dependence on the Reynolds number of Kolmogorov-Arnold-Moser (KAM) tori. In finite-length cuboidal cavities at small Reynolds numbers, a thin layer of chaotic streamlines covers all walls. As the Reynolds number is increased, the chaotic layer widens and the complementary KAM tori shrink, eventually undergoing resonances, until they vanish. Accurate data for the location of closed streamlines and of KAM tori are provided, both of which reach very close to the moving lid. For steady periodic Taylor-Görtler vortices in spanwise infinitely extended cavities with a square cross section, chaotic streamlines occupy a large part of the flow domain immediately after the onset of Taylor-Görtler vortices. As the Reynolds number increases, the remaining KAM tori vanish from the Taylor-Görtler vortices, while KAM tori grow in the central region further away from the solid walls.