Jeroen S.W. Lamb

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• PhD
• Professor (Full) at Imperial College London

We numerically study bifurcations of attractors of the Hénon map with additive bounded noise with spherical reach. The bifurcations are analysed using a finite-dimensional boundary map. We distinguish between two types of bifurcations: topological bifurcations and boundary bifurcations. Topological bifurcations describe discontinuous changes of att...

We establish the existence of intermittent two-point dynamics and infinite stationary measures for a class of random circle endomorphisms with zero Lyapunov exponent, as a dynamical characterisation of the transition from synchronisation (negative Lyapunov exponent) to chaos (positive Lyapunov exponent).

We propose a notion of random horseshoe for one-dimensional random dynamical systems with non-uniform expansion, and we provide sufficient conditions that guarantee their abundant existence, which is shown to hold for a class of non-uniformly expanding random circle endomorphisms with bounded additive noise.

We propose a notion of random horseshoe and prove density of random horseshoes and existence of a random young tower with exponential tail for non uniformly expanding random dynamical systems with additive noise

We develop a powerful and general method to provide arbitrarily accurate rigorous upper and lower bounds for Lyapunov exponents of stochastic flows. Our approach is based on computer-assisted tools, the adjoint method and established results on the ergodicity of diffusion processes. We do not require any structural assumptions on the stochastic sys...

We study the asymptotic behaviour of stationary densities of one-dimensional random diffeomorphisms, at the boundaries of their support, which correspond to deterministic fixed points of extremal diffeomorphisms. In particular, we show how this stationary density at a boundary depends on the underlying noise distribution, as well as the linearisati...

In a class of heterogeneous random networks, where each node dynamics is a random dynamical system, interacting with neighbor nodes via a random coupling function, we characterize the hub behavior as the mean-field, subject to statistically controlled fluctuations. In particular, we prove that the fluctuations are small over exponentially long time...

We propose a notion of conditioned stochastic stability of invariant measures on repellers: we consider whether quasi-ergodic measures of absorbing Markov processes, generated by random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure in the zero-noise limit...

We consider transitions to chaos in random dynamical systems induced by an increase in noise amplitude. We show how the emergence of chaos (indicated by a positive Lyapunov exponent) in a logistic map with bounded additive noise can be analyzed in the framework of conditioned random dynamics through expected escape times and conditioned Lyapunov ex...

We consider invertible linear maps with additive spherical bounded noise. We show that minimal attractors of such random dynamical systems are unique, strictly convex and have a continuously differentiable boundary. Moreover, we present an auxiliary finite-dimensional deterministic boundary map for which the unit normal bundle of this boundary is g...

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu $ on M where the transition kernel ${\mathcal P}$ admits an eigenfunction $0\leq \eta \in L^1(M,\mu )$ . We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which ensure that $\eta (x) \mu (\mathrm {d} x)$ is a quas...

We consider transitions to chaos in random dynamical systems induced by an increase of noise amplitude. We show how the emergence of chaos (indicated by a positive Lyapunov exponent) in a logistic map with bounded additive noise can be analysed in the framework of conditioned random dynamics through expected escape times and conditioned Lyapunov ex...

The problem of detecting and quantifying the presence of symmetries in datasets is useful for model selection, generative modeling, and data analysis, amongst others. While existing methods for hard-coding transformations in neural networks require prior knowledge of the symmetries of the task at hand, this work focuses on discovering and character...

We propose a notion of random horseshoe for one-dimensional random dynamical systems. We prove the abundance of random horseshoes for a class of circle endomorphisms subject to additive noise, large enough to make the Lyapunov exponent positive. In particular, we provide conditions which guarantee that given any pair of disjoint intervals, for almo...

We study the problem of persistence of minimal invariant sets with smooth boundary for a certain class of discrete-time set-valued dynamical systems, naturally arising in the context of random dynamical systems with bounded noise. In particular, we introduce a single-valued map, the so-called boundary map, which has the property that a certain clas...

We consider a three-dimensional slow-fast system arising in fluid dynamics with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. We show that in presence of noise, the deterministic slow manifold can be viewed as an approximat...

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq \eta\in L^1(M,\mu)$. We find conditions on the transition densities of $\mathcal P$ with respect to $\mu$ which ensure that $\eta(x) \mu(\mathrm d x)$ is a quasi-ergodic m...

We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this...

We consider reversible vector fields in $\mathbb{R}^{2n}$ such that the set of fixed points of the involutory reversing symmetry is $n$-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that topological entropy is positive when the stable and unsta...

Time-reversal symmetry arises naturally as a structural property in many dynamical systems of interest. While the importance of hard-wiring symmetry is increasingly recognized in machine learning, to date this has eluded time-reversibility. In this paper we propose a new neural network architecture for learning time-reversible dynamical systems fro...

We establish the existence of a full spectrum of Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed the process conditioned to never being absorbed, the Q-process, into the framework of random dynamical systems, allowing us to study multiplicative ergodic properties. We show that the finite-t...

We establish existence and uniqueness of quasi-stationary and quasi-ergodic measures for almost surely absorbed discrete-time Markov chains under weak conditions. We obtain our results by exploiting Banach lattice properties of transition functions under natural regularity assumptions.

We consider SDEs driven by two different sources of additive noise, which we refer to as intrinsic and common. We establish almost sure existence and uniqueness of pullback attractors with respect to realisations of the common noise only. These common noise pullback attractors are smooth probability densities that depend only on (the past of) a com...

We consider a \(\mathbb{Z}_2\)-equivariant flow in \(\mathbb{R}^{4}\) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit \(\Gamma\). We provide criteria for the existence of stable and unstable invariant manifolds of \(\Gamma\). We prove that if these manifolds intersect transversely, creating a so-called sup...

We study the global topological structure and smoothness of the boundaries of $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of planar sets $E \subset \mathbb{R}^2$. We show that for a compact set $E$ and $\varepsilon > 0$ the boundary $\partial E_\varepsilon$ can be expressed as...

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network of two symmetrically linked star subnetworks of identical oscillators with shear and Kuramoto–Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength...

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network consisting of two symmetrically linked star subnetworks consisting of identical oscillators with shear and Kuramoto--Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-...

We study geometric and topological properties of singularities on the boundaries of $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 : \textrm{dist}(x, E) \leq \varepsilon \}$ of planar sets $E \subset \mathbb{R}^2$. We develop a novel technique for analysing the boundary and obtain, for a compact set $E$ and $\varepsilon > 0$, a...

We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent. This addresses an open problem posed by Kevin Lin and Lai-Sang Young, extending results by Qiudong Wang and Lai-Sang Young on periodically kic...

The emergence of noise-induced chaos in a random logistic map with bounded noise is understood as a two-step process consisting of a topological bifurcation flagged by a zero-crossing point of the supremum of the dichotomy spectrum and a subsequent dynamical bifurcation to a random strange attractor flagged by a zero crossing point of the Lyapunov...

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor wit...

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in...

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor wit...

Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists...

We provide a classification of random orientation-preserving homeomorphisms of $\mathbb{S}^1$, up to topological conjugacy of the random dynamical systems generated by i.i.d. iterates of the random homeomorphism. This classification covers all random circle homeomorphisms for which the noise space is a connected Polish space and an additional extre...

We study stochastic resonance in an over-damped approximation of the stochastic Duffing oscillator from a random dynamical systems point of view. We analyse this problem in the general framework of random dynamical systems with a nonautonomous forcing. We prove the existence of a unique global attracting random periodic orbit and a stationary perio...

We consider non-elementary T-points in reversible systems in R2n+1. We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.

Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists...

We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure. This leads to the study of a singularity which inherits certain structure from the Hamiltonian nature of the s...

We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent. This addresses an open problem posed by Kevin Lin and Lai-Sang Young, extending results by Qiudong Wang and Lai-Sang Young on periodically kic...

We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure. This leads to the study of a singularity which inherits certain structure from the Hamiltonian nature of the s...

We consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria. The constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated. Such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions, and called T-points in this context. We study co...

We develop the dichotomy spectrum for random dynamical system and demonstrate its use in the characterization of pitchfork bifurcations for random dynamical systems with additive noise. Crauel and Flandoli had shown earlier that adding noise to a system with a deterministic pitchfork bifurcation yields a unique attracting random fixed point with ne...

The study of the behavior of solutions of ODEs often benefits from deciding on a convenient choice of coordinates. This choice of coordinates may be used to "simplify" the functional expressions that appear in the vector field in order that the essential features of the flow of the ODE near a critical point become more evident. In the case of the a...

The study of the behavior of solutions of ODEs often benefits from deciding on a convenient choice of coordinates. This choice of coordinates may be used to "simplify" the functional expressions that appear in the vector field in order that the essential features of the flow of the ODE near a critical point become more evident. In the case of the a...

We show that resonance zones near an elliptic periodic point of a reversible map must, generically, contain asymptotically stable and asymptotically unstable periodic orbits, along with wild hyperbolic sets.

In this survey we discuss current directions of research in the dynamics of nonsmooth systems, with emphasis on bifurcation theory. An introduction to the state-of-the-art (also for non-specialists) is complemented by a presentation of the main open problems. We illustrate the theory by means of elementary examples. The main focus is on piecewise s...

We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are often satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower...

We prove Knudsen's law for a gas of particles bouncing freely in a two dimensional pipeline with serrated walls consisting of irrational triangles. Dynamics are randomly perturbed and the corresponding random map studied under a skew-type deterministic representation which is shown to be ergodic and exact. Comment: 21 pages, 5 figures

We study the existence and branching patterns of wave trains in the one-dimensional infinite Fermi–Pasta–Ulam (FPU) lattice. A wave train Ansatz in this Hamiltonian lattice leads to an advance–delay differential equation on a space of periodic functions, which carries a natural Hamiltonian structure. The existence of wave trains is then studied by...

Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stra...

In unfoldings of resonant homoclinic bellows interesting bifurcation phenomena occur: two suspensed Smale horseshoes can collide and disappear in saddle-node bifurcations (all periodic orbits disappear through saddle-node bifurcations, there are no other bifurcations of periodic orbits), or a suspended horseshoe can go through saddle-node and perio...

In this paper we employ equivariant Lyapunov-Schmidt procedure to give a clearer understanding of the one-to-one correspondence of the peri-odic solutions of a system of neutral functional differential equations with the zeros of the reduced bifurcation map, and then set up equivariant Hopf bifur-cation theory. In the process we derive criteria for...

We study steady-state bifurcation in reversible-equivariant vector fields. We assume an action on the phase space of a compact Lie group G with a normal subgroup H of index two, and study vector fields that are H-equivariant and have all elements of the complement GH as time-reversal symmetries. We focus on separable bifurcation problems that can b...

Abstract In this article we study the dynamical behavior near solitary wave solutions of lattice differential equations (LDEs). We are interested in the case, where the solitary wave profile induces a homoclinic solution of the associated traveling-wave equation. Using exponential dichotomies we prove the existence of a C,- function. This approach...

We show that in the neighbourhood of relative equilibria and relative periodic solutions, coordinates can be chosen so that the equations of motion, in normal form, admit certain additional equivariance conditions up to arbitrarily high order. In particular, normal forms for relative periodic solutions effectively reduce to normal forms for relativ...

We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real principal eigenvalues and connecting orbits that approach the equilibria along principal directions. We show how suspensions of subs...

We consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria. The constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated. In the literature such heteroclinic cycles have been associated with the termination of a branch of homoclinic solutions, and have been named T-po...

We consider a coupled cell network of differential equations with finite symmetry group Γ, where Γ permutes cells transitively. We show how the structure of the coupled cell network, represented by a directed graph whose vertices represent individual cells and edges represent couplings, can be taken into account in the bifurcation analysis of a ful...

We consider nonresonant and weakly resonant Hopf bifurcation from periodic solutions and relative periodic solutions in dynamical systems with symmetry. In particular, we analyse phase-locking and irrational torus flows on the bifurcating relative tori.Results are obtained for systems with compact and noncompact symmetry groups. In the noncompact c...

We study bifurcations of homoclinic cycles in equivariant flows in R n , equivariant under the linear action of a finite group. The homoclinic cycles we consider are not robust, but of codimension one. Examples are given by several coexisting homoclinic loops, all related by symmetry, to a symmetric equilibrium. We assume real and semi-simple princ...

We study the dynamics near transverse intersections of stable and unstable manifolds of sheets of symmetric periodic orbits in reversible systems. We prove that the dynamics near such homoclinic and heteroclinic intersections is not C 1 structurally stable. This is in marked contrast to the dynamics near transverse intersections in both general and...

We study nonperiodic tilings of the line obtained by a projection method with an interval projection structure. We obtain a geometric characterisation of all interval projection tilings that admit substitution rules and describe the set of substitution rules for each such a tiling. We show that each substitution tiling admits a countably infinite n...

We study the dynamics near a symmetric Hopf-zero (also known as saddle-node Hopf or fold-Hopf) bifurcation in a reversible vector field in R3, with involutory an reversing symmetry whose fixed point subspace is one-dimensional. We focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between t...

In this paper we study codimension-one Hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields. Such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point. The eigenvalue movements near such a degeneracy typicall...

We present a large class of parallelogram tilings that have two important features in common with the Ammann-Beenker tiling: they can be constructed by canonical projection from Z4 ⊂ R4 to a plane, and they admit substitution rules.

For reversible two-dimensional diffeomorphisms we establish a new type of Newhouse regions (regions of structural instability density). We prove that in these regions there exists a dense set of diffeomorphisms having, simultaneously, infinitely many stable, infinitely many unstable, and infinitely many elliptic type periodic orbits.

We study the existence of periodic solutions in the neighbourhood of symmetric (partially) elliptic equilibria in purely reversible Hamiltonian vector fields. These are Hamiltonian vector fields with an involutory reversing symmetry R. We contrast the cases where R acts symplectically and anti-symplectically. In case R acts anti-symplectically, gen...

We study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries.In previous work, we focused primarily on codimension one bifurcations. In this paper, we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifu...

In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However the theory gives a uniform m...

We give explicit differential equations for the dynamics of Hamiltonian systems near relative equilibria. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic structure and symmetry properties of the Hamiltoni...

. We present results on generic separable" bifurcations of equilibria of equivariant dynamical systems with nite symmetry groups and timereversing symmetries. These bifurcation problems are reduced to standard equivariant bifurcation problems using equivariant transversality theory. For each symmetry-breaking subgroup the dimension of the bifurcati...

This paper addresses the question whether normal forms of smooth reversible vector fields in R4 at an elliptic equilibrium possess a formal Hamiltonian structure. In the non-resonant case we establish a formal conjugacy between reversible and Hamiltonian normal forms. In the case of non-semi-simple 1:1 resonance and p:q resonance with p+q>2 we esta...

We review recent developments in the theory for generic bifurcation from periodic and relative periodic solutions in equivariant dynamical systems. 1 Introduction Equivariant bifurcation theory is concerned, to a large extent, with local bifurcation theory for vector fields that are equivariant with respect to the action of a compact Lie group Gamm...

We discuss the behavior of rotating and traveling waves – such as traveling fronts and pulses – in partial differential equations with continuous symmetry groups under perturbations that break the continuous symmetries but preserve some discrete (lattice) symmetries. Our theory provides a rigorous and universal framework for understanding pinning,...

Let Γ ⊂ O(n) be a finite group acting orthogonally on . We say that Γ is a reversing symmetry group of a homeomorphism, diffeomorphism or flow ( or ) if Γ has an index two subgroup whose elements commute with ft and for all elements and all t, ft ○ ϱ(x) = ϱ ○ f−t(x). We give necessary group and representation theoretic conditions for subgroups of r...

We show how transfer matrix models on chains that are selfsimilar (renormalizable) with respect to a substitution rule can be transformed from multi-site models in which transfer matrices depend on the nature of a finite number of neighbouring sites, to on-site models in which transfer matrices depend on the nature of one site only. We present suff...

Relative periodic solutions are ubiquitous in dynamical systems with continuous symmetry. Recently, Sandstede, Scheel and Wulff derived a center bundle theorem, reducing local bifurcation from relative periodic solutions to a finite-dimensional problem. Independently, Lamb and Melbourne showed how to systematically study local bifurcation from isol...

In this paper we address some global dynamical features in area-preserving dynamical systems on the plane2 which obstruct the presence of a time-reversal symmetry that is a reflection. We show that in the case of flows, reversibility and non-reversibility are non-generic properties. In the case of maps we show that reversibility is a non-generic pr...

In this paper we classify the structure of linear reversible systems (vector fields) on n that are equivariant with respect to a linear representation of a compact Lie group H. We assume the time-reversal symmetry R also acts linearly and is such that the group G that is generated by H and R is again a compact Lie group. The main tool for the class...

A diffeomorphism is called a (reversing) k-symmetry of a dynamical system in represented by the diffeomorphism if k is the smallest positive integer for which U is a (reversing) symmetry of (the k-times iterate of f), i.e. . In this paper we show how k-symmetry naturally arises in the context of return maps of flows with spacetime symmetries. We d...

. Discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries. We present a systematic approach to the study of local bifurcation from discrete rotating waves. The approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group...

: Dynamical systems may possess in addition to symmetries that leave the equations of motion invariant, reversing symmetries that invert the equations of motion. Such dynamical systems are called (weakly) reversible. Some consequences of the existence of reversing symmetries for dynamical systems with discrete time (mappings) are discussed. A rever...

A map U : IR d 7! IR d is called a (reversing) k-symmetry of the dynamical system represented by the map L : IR d 7! IR d if k is the smallest positive integer for which U is a (reversing) symmetry of the kth iterate of L. We study generic local bifurcations of fixed points that are invariant under the action of a compact Lie group Sigma that is a...

We study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves....

In this paper we survey the topic of time-reversal symmetry in dynamical systems. We begin with a brief discussion of the position of time-reversal symmetry in physics. After defining time-reversal symmetry as it applies to dynamical systems, we then introduce a major theme of our survey, namely the relation of time-reversible dynamical sytems to e...

In this paper, we discuss some recent developments in the understanding of generic bifurcation from periodic solutions with spatiotemporal symmetries. We focus mainly on the theory for bifurcation from isolated periodic solutions in dynamical systems with a compact symmetry group. Moreover, we discuss how our theory justifies certain heuristic assu...

In studies of the dynamics of a kicked pendulum, webs along which stochastic diffusion takes place, have been observed to occur at resonance. The author deals with stochastic webs with crystallographic symmetries. Via an analysis of the (reversing) symmetries of the equations of motion, symmetry properties of stochastic webs are revealed. Furthermo...

In this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typical properties with the quasiperiodic Fibonacci chain. For a subclass we explicitly construct the atomic surface. Moreover, scaling properties of the energy spectrum are discussed in relation to the dynamics of trace maps.

A map is called a (reversing) k-symmetry of the dynamical system represented by the map if k is the smallest positive integer for which U is a (reversing) symmetry of the kth iterate of L. We study generic local bifurcations of fixed points that are invariant under the action of a compact Lie group that is a reversing k-symmetry group of the map L,...

In a microscopic crystallographic model with incommensurate phases, the magnetoelastic DIFFOUR model, phase transitions at the paraphase boundary are connected to bifurcations of a symplectic mapping. Special attention is paid to the ferro-antiferro transition at the paraphase boundary and its implication on the incommensurate parts of the phase di...

We show that if a quasiperiodic two-symbol sequence obtained by the canonical projection method has an infinite number of predecessors with respect to a substitution rule 0305-4470/31/18/001/img1, then 0305-4470/31/18/001/img1 is an invertible substitution rule. Vice versa, we show that every quasiperiodic two-symbol sequence that has an infinite n...