arXiv subjects

Hans Koch

Publications and source records attributed to Hans Koch.

14 recordsLinked to original sources

arXiv2022

Universal supercritical behavior for some skew-product maps

We consider skew-product maps over circle rotations $x\mapsto x+α$ (mod 1) with factors that take values in SL(2,R) In numerical experiments with $α$ the inverse golden mean, Fibonacci iterates of almost Mathieu maps with rotation number $1/4$ and positive Lyapunov exponent exhibit asymptotic scaling behavior. We prove the existence of such asymptotic scaling for "periodic" rotation numbers and for large Lyapunov exponent. The phenomenon is universal, in the sense that it holds for open sets of maps, with the scaling limit being independent of the maps. The set of maps with a given periodic rotation number is a real analytic codimension $1$ manifold in a suitable space of maps.

math.DS↗

arXiv2021

Asymptotic scaling and universality for skew products with factors in SL(2,R)

We consider skew-product maps over circle rotations $x\mapsto x+α$ (mod 1) with factors that take values in SL(2,R). This includes maps of almost Mathieu type. In numerical experiments, with $α$ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

math-ph↗

arXiv2020

A Hopf bifurcation in the planar Navier-Stokes equations

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

math.AP↗

arXiv2020

On trigonometric skew-products over irrational circle-rotations

We describe some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. Analogous results are believed to be true for the self-dual almost Mathieu maps, but they seem presently beyond reach.

math-ph↗

arXiv2019

Renormalization and universality of the Hofstadter spectrum

We consider a renormalization transformation $R$ for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of $R$ determine universal constants analogous to the critical exponents in the theory of phase transitions. Restricting to skew-product maps over a circle-rotations by the golden mean, we find several periodic orbits for $R$, and we conjecture that there are infinitely many. Interestingly, all scaling factors that have been determined to high accuracy appear to be algebraically related to the circle-rotation number. We present evidence that these values describe (among other things) local scaling properties of the Hofstadter spectrum.

math-ph↗

arXiv2019

Golden mean renormalization for the almost Mathieu operator and related skew products

Considering SL(2,R) skew-product maps over circle rotations, we prove that a renormalization transformation associated with the golden mean alpha has a nontrivial periodic orbit of length 3. We also present some numerical results, including evidence this period 3 describes scaling properties of the Hofstadter butterfly near the top of the spectrum at alpha, and scaling properties of the generalized eigenfunction for this energy.

math-ph↗

arXiv2019

Traveling wave solutions for the FPU chain: a constructive approach

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile $u$ of the wave. We consider solutions whose derivatives $u'$ need not be small, may change sign several times, but decrease at least exponentially. Our method of proof is computer-assisted. Unlike other methods, it does not require that the FPU potential has an attractive (positive) quadratic term. But we currently need to restrict the size of that term. In particular, our solutions in the attractive case are all supersonic.

math.DS↗

arXiv2018

Some breathers and multi-breathers for FPU-type chains

We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is spectrally stable. Symmetry properties are considered as well. In addition, we construct solutions that are close to (possibly infinite) sums of breather solutions.

math.DS↗

arXiv2017

Validated numerical solutions for some semilinear elliptic equations on the disk

Starting with approximate solutions of the equation $-Δu=wu^3$ on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions.

math.AP↗

arXiv2013

On Hamiltonian flows whose orbits are straight lines

We consider real analytic Hamiltonians whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q$. By a theorem of Moser, every polynomial Hamiltonian of degree 3 reduces to such a $q$-independent Hamiltonian via a linear symplectic change of variables. We show that such a reduction is impossible, in general, for polynomials of degree 4 or higher. But we give a condition that implies linear-symplectic conjugacy to another simple class of Hamiltonians. The condition is shown to hold for all nondegenerate Hamiltonians that are homogeneous of degree 4.

math.DS↗

arXiv2009

Modelling quasicrystals at positive temperature

We consider a two-dimensional lattice model of equilibrium statistical mechanics, using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 Wang tiles. This model has uncountably many ground state configurations, all of which are nonperiodic. The question addressed in this paper is whether nonperiodicity persists at low but positive temperature. We present arguments, mostly numerical, that this is indeed the case. In particular, we define an appropriate order parameter, prove that it is identically zero at high temperatures, and show by Monte Carlo simulation that it is nonzero at low temperatures.

cond-mat.stat-mech↗

arXiv2009

Period Doubling in Area-Preserving Maps: An Associated One Dimensional Problem

It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. We argue that the period doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Henon-like map: $$H^*(x,u)=(ϕ(x)-u,x-ϕ(ϕ(x)-u)),$$ where $ϕ$ solves $$ϕ(x)={2 \over λ} ϕ(ϕ(λx))-x.$$ We then give a ``proof'' of existence of solutions of small analytic perturbations of this one dimensional problem, and describe some of the properties of this solution. The ``proof'' consists of an analytic argument for factorized inverse branches of $ϕ$ together with verification of several inequalities and inclusions of subsets of $\field{C}$ numerically. Finally, we suggest an analytic approach to the full period doubling problem for area-preserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.

math.DS↗

arXiv2005

Most stable structure for hard spheres

The hard sphere model is known to show a liquid-solid phase transition, with the solid expected to be either face centered cubic or hexagonal close packed. The difference in free energy between the two structures is very small and various attempts have been made to determine which one is the more stable. We contrast the different approaches and extend one.

cond-mat.stat-mech↗

arXiv1998

A Renormalization Group for Hamiltonians: Numerical Results

We describe a renormalization group transformation that is related to the breakup of golden invariant tori in Hamiltonian systems with two degrees of freedom. This transformation applies to a large class of Hamiltonians, is conceptually simple, and allows for accurate numerical computations. In a numerical implementation, we find a nontrivial fixed point and determine the corresponding critical index and scaling. Our computed values for various universal constants are in good agreement with existing data for area-preserving maps. We also discuss the flow associated with the nontrivial fixed point.

chao-dyn↗