arXiv subjects

Bob Rink

Publications and source records attributed to Bob Rink.

34 records · Page 2Linked to original sources

arXiv2019

The parameterization method for center manifolds

In this paper, we present a generalization of the parameterization method, introduced by Cabré, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system.

math.DS↗

arXiv2017

Transversality in Dynamical Systems with Generalized Symmetry

We prove that a generic $k$-parameter bifurcation of a dynamical system with a monoid symmetry occurs along a generalized kernel or center subspace of a particular type. More precisely, any (complementable) subrepresentation $U$ is given a number $K_U$ and a number $C_U$. A $k$-parameter bifurcation can generically only occur along a generalized kernel isomorphic to $U$ if $k \geq K_U$. It can generically only occur along a center subspace isomorphic to $U$ if $k \geq C_U$. The numbers $K_U$ and $C_U$ depend only on the decomposition of $U$ into indecomposable subrepresentations. In particular, we prove that a generic one-parameter steady-state bifurcation occurs along one absolutely indecomposable subrepresentation. Likewise, it follows that a generic one-parameter Hopf bifurcation occurs along one indecomposable subrepresentation of complex or quaternionic type, or along two isomorphic absolutely indecomposable subrepresentations. In order to prove these results, we show that the set of endomorphisms with generalized kernel (or center subspace) isomorphic to $U$ is the disjoint union of a finite set of conjugacy invariant submanifolds of codimension $K_U$ and higher (or $C_U$ and higher). The results in this article hold for any monoid, including non-compact groups.

math.DS↗

arXiv2016

Projection blocks in homogeneous coupled cell networks

We introduce a special subset of the graph of a homogeneous coupled cell network, called a projection block, and show that the network obtained from identifying this block to a single point can be used to understand the generic bifurcations of the original network. This technique is then used to describe the bifurcations in a generalized feed-forward network, in which the loop can contain more than one cell.

math.DS↗

arXiv2016

Center manifolds of coupled cell networks

Dynamical systems with a network structure can display anomalous bifurcations as a generic phenomenon. As an explanation for this it has been noted that homogeneous networks can be realized as quotient networks of so-called fundamental networks. The class of admissible vector fields for these fundamental networks is equal to the class of equivariant vector fields of the regular representation of a monoid. Using this insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to explain the difference in generic bifurcations between three example networks with identical spectral properties and identical robust synchrony spaces.

math.DS↗

arXiv2014

Graph fibrations and symmetries of network dynamics

Dynamical systems with a network structure can display collective behaviour such as synchronisation. Golubitsky and Stewart observed that all the robustly synchronous dynamics of a network is contained in the dynamics of its quotient networks. DeVille and Lerman have recently shown that the original network and its quotients are related by graph fibrations and hence their dynamics are conjugate. This paper demonstrates the importance of self-fibrations of network graphs. Self-fibrations give rise to symmetries in the dynamics of a network. We show that every homogeneous network admits a lift with self-fibrations and that every robust synchrony in this lift is determined by the symmetries of its dynamics. These symmetries moreover impact the global dynamics of network systems and can be used to explain and predict generic scenarios for synchrony breaking. We also discuss networks with interior symmetries and nonhomogeneous networks.

math.DS↗

arXiv2013

Continuity of the Peierls barrier and robustness of laminations

We study the Peierls barrier for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start with the case of a fixed local potential and derive an estimate for the difference of the periodic Peierls barrier and the Peierls barrier of a general rotation number in a given point. A similar estimate was obtained by Mather in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that the Peierls barrier is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers for a given rotation number, is open in the C1-topology.

math.DS↗

arXiv2013

The laminations of a crystal near an anti-continuum limit

The anti-continuum limit of a monotone variational recurrence relation consists of a lattice of uncoupled particles in a periodic background. This limit supports many trivial equilibrium states that persist as solutions of the model with small coupling. We investigate when a persisting solution generates a so-called lamination and prove that near the anti-continuum limit the collection of laminations of solutions is homeomorphic to the (N-1)-dimensional simplex, with N the number of distinct local minima of the background potential. This generalizes a result by Baesens and MacKay on twist maps near an anti-integrable limit.

math.DS↗

arXiv2013

Coupled cell networks and their hidden symmetries

Dynamical systems with a coupled cell network structure can display synchronous solutions, spectral degeneracies and anomalous bifurcation behavior. We explain these phenomena here for homogeneous networks, by showing that every homogeneous network dynamical system admits a semigroup of hidden symmetries. The synchronous solutions lie in the symmetry spaces of this semigroup and the spectral degeneracies of the network are determined by its indecomposable representations. Under a condition on the semigroup representation, we prove that a one-parameter synchrony breaking steady state bifurcation in a coupled cell network must generically occur along an absolutely indecomposable subrepresentation. We conclude with a classification of generic one-parameter bifurcations in monoid networks with two or three cells.

math.DS↗

arXiv2012

Amplified Hopf bifurcations in feed-forward networks

In a previous paper, the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like $λ^{1/2}$, $λ^{1/6}$, $λ^{1/18}$, etc. Such amplified Hopf branches were previously found by others in a subclass of feed-forward networks with three cells, first under a normal form assumption and later by explicit computations. We explain here how these bifurcations arise generically in a broader class of feed-forward chains of arbitrary length.

math.DS↗

arXiv2012

Coupled cell networks: semigroups, Lie algebras and normal forms

We introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra. This implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself. Networks without the semigroup property will support normal forms with a more general network architecture, but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network. We explain how to compute Lie brackets and normal forms of coupled cell networks and we characterize the SN-decomposition that determines the normal form symmetry. This paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid.

math.DS↗

arXiv2012

A dichotomy theorem for minimizers of monotone recurrence relations

Variational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel-Kontorova model for a ferromagnetic crystal. For such problems, Aubry-Mather theory establishes the existence of "ground states" or "global minimizers" of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a nontrivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and nonphysical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps.

math.DS↗

arXiv2011

Ghost circles in lattice Aubry-Mather theory

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multidimensional counterpart of monotone twist maps. They often admit a variational structure, so that the solutions are the stationary points of a formal action function. Classical Aubry-Mather theory establishes the existence of a large collection of solutions of any rotation vector. For irrational rotation vectors this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the gradient flow of the formal action function and we prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called "ghost circle". The existence of ghost circles is first proved for rational rotation vectors and Morse action functions. The main technical result is a compactness theorem for ghost circles, based on a parabolic Harnack inequality for the gradient flow, which implies the existence of ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry-Mather set has a gap, then this gap must be parametrized by minimizers, or contain a non-minimizing solution.

math.DS↗

arXiv2011

On the destruction of minimal foliations

Monotone variational recurrence relations such as the Frenkel-Kontorova lattice, arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number. They are the action minimizers that constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected or a Cantor set. A connected Aubry-Mather set is called a minimal foliation. In the case of twist maps, it describes an invariant circle, while in solid state physics it corresponds to a continuum of ground states. A Cantor Aubry-Mather set is called a minimal lamination. In this paper we prove that when the rotation number of a minimal foliation is either rational or easy to approximate by rational numbers, then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation. This generalizes a theorem of Mather for twist maps to general recurrence relations.

math.DS↗

arXiv2005

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.

nlin.SI↗

arXiv2001

Direction reversing travelling waves in the Fermi-Pasta-Ulam chain

This paper considers the famous Fermi-Pasta-Ulam chain with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is completely integrable. We study how the level sets of the integrals foliate the phase space. Our study reveals all the integrable structure in the low energy domain of the chain. If the number of particles in the chain is even, then this foliation is singular. The method of singular reduction shows that the system has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between travelling waves. Thus we discover a new class of solutions which can be described as direction reversing travelling waves. They remarkably show an interesting interaction of normal modes in the absense of energy transfer. These solutions can easily be observed numerically.

nlin.SI↗

arXiv2000

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 it in lower-order resonant Hamiltonian systems. So the FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries.

nlin.CD↗