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This paper deals with large scale aspects of Hill's equation (sic) + (a + bp(t)) x = 0, where p is periodic with a fixed period. In particular, the interest is the asymptotic radial density of the stability domain in the (a, b)-plane. It turns out that this density changes discontinuously in a certain direction and exhibits and interesting asymptotic fine structure. Most of the paper deals with the case where p is a Morse function with one maximum and one minimum, but also the discontinuous case of square Hill's equation is studied, where the density behaves differently.

This paper deals with the problem of estimation of the topological entropy for non-autonomous systems of differential equations via the second (direct) Lyapunov method. The main result of the paper is illustrated by examples concerning the Lorenz system and Duffing oscillator.

Historical evidence is reviewed to show that what Ed Lorenz meant by the iconic phrase 'the butterfly effect' is not at all captured by the notion of sensitive dependence on initial conditions in low-order chaos. Rather, as presented in his 1969 Tellus paper, Lorenz intended the phrase to describe the existence of an absolute finite-time predicability barrier in certain multi-scale fluid systems, implying a breakdown of continuous dependence on initial conditions for large enough forecast lead times. To distinguish from 'mere' sensitive dependence, the effect discussed in Lorenz's Tellus paper is referred to as 'the real butterfly effect'. Theoretical evidence for such a predictability barrier in a fluid described by the three-dimensional Navier-Stokes equations is discussed. Whilst it is still an open question whether the Navier-Stokes equation has this property, evidence from both idealized atmospheric simulators and analysis of operational weather forecasts suggests that the real butterfly effect exists in an asymptotic sense, i.e. for initial-time atmospheric perturbations that are small in scale and amplitude compared with (weather) scales of interest, but still large in scale and amplitude compared with variability in the viscous subrange. Despite this, the real butterfly effect is an intermittent phenomenon in the atmosphere, and its presence can be signalled a priori, and hence mitigated, by ensemble forecast methods. © 2014 IOP Publishing Ltd and London Mathematical Society.

We study the class of open continuous-time mechanical particle systems introduced in the paper by Khanin and Yarmola [Ergodic Properties of Random Billiards Driven by Thermostats. Commun. Math. Phys. 320, no. 1, 121-147 (2013)]. Using the discrete-time results from that paper we demonstrate rigorously that, in continuous time, a unique steady state exists and is sub-exponentially mixing. Moreover, all initial distributions converge to the steady state and, for a large class of initial distributions, convergence to the steady state is sub-exponential. The main obstacle to exponential convergence is the existence of slow particles in the system.

The reconstruction theorem deals with dynamical systems that are given by a map T : X → X of a compact metric space X together with an observable f : X → R from X to the real line R . In 1981, by use of Whitney’s embedding theorem, Takens proved that if T : M → M is a (two-sided) diffeomorphism on a compact smooth manifold M with dim M = d , for generic (T, f) there is a bijection between elements x ∈ M and corresponding sequence ( f T j ( x ) ) j = 0 2 d , and moreover, in 2002 Takens proved a generalised version for endomorphisms. In natural sciences and physical engineering, there has been an increase in importance of fractal sets and more complicated spaces, and also in mathematics, many topological and dynamical properties and stochastic analysis of such spaces have been studied. In the present paper, by use of some topological methods we extend the Takens’ reconstruction theorems of compact smooth manifolds to reconstruction theorems of ‘non-invertible’ dynamical systems for a large class of compact metric spaces, which contains PL-manifolds, manifolds with branched structures and some fractal sets, e.g. Menger manifolds, Sierpiński carpet and Sierpiński gasket and dendrites, etc.

The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of the fundamental examples of so-called semitoric systems. Pelayo & Vu Ngoc have given a classification of semitoric systems in terms of five symplectic invariants. Three of these invariants have already been partially calculated in the literature for a certain parameter range, together with the linear terms of the so-called Taylor series invariant for a fixed choice of parameter values. In the present paper we complete the classification by calculating the polygon invariant, the height invariant, the twisting-index invariant, and the higher-order terms of the Taylor series invariant for the whole family of systems. We also analyse the explicit dependence of the coefficients of the Taylor series with respect to the three parameters of the system, in particular near the Hopf bifurcation where the focus-focus point becomes degenerate., 59 pages, 16 figures

This paper investigates a family of nonlinear oscillators at Hopf bifurcation, driven by a small quasi-periodic forcing. In particular, we are interested in the situation that at bifurcation and for vanishing forcing strength, the driving frequency and the normal frequency are in k : 1 or k : 2 resonance. For small but non-vanishing forcing strength, a semi-global normal form system is found by averaging and applying a van der Pol transformation. The bifurcation diagram is organized by a codimension 3 singularity of nilpotent-elliptic type. A fairly complete analysis of local bifurcations is given; moreover, all the non-local bifurcation curves predicted by Dumortier et al (1991 Bifurcations of Planar Vector Fields (Lecture Notes in Mathematics vol 1480) (Berlin: Springer)), excepting boundary bifurcations, are found numerically.

This paper presents results from a flow visualization study of the wake structures behind solid spheres rising or falling freely in liquids under the action of gravity. These show remarkable differences to the wake structures observed behind spheres held fixed. The two parameters controlling the rise or fall velocity (i.e., the Reynolds number) are the density ratio between sphere and liquid and the Galileo number., Comment: 9 pages, 8 figures. Higher resolution on demand. To appear in Nonlinearity January 2005

Let $K\subset \mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that K can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of K as well as the set of elements in K with unique codings using the machinery of Mauldin and Williams (1988 Trans. Am. Math. Soc. 309 811–29). We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of K with multiple codings. Secondly, in the setting of β-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. Motivated by this application, we prove that the set of all the unique codings is a subshift of finite type if and only if it is a sofic shift. This equivalent condition was not mentioned by de Vries and Komornik (2009 Adv. Math. 221 390–427, theorem 1.8). Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the survivor set can be identified with a subshift of finite type. The third application partially answers a problem posed by Alcaraz Barrera (2014 PhD Thesis University of Manchester).

Impacting systems are found in a great variety of mechanical constructions and they are intrinsically nonlinear. In this paper it is shown how near-grazing systems, i.e. systems in which the impacts take place at low speed, can be described by discrete mappings. The derivation of this mapping for a harmonic oscillator with a stop is dealt with in detail. It is found that the resulting mapping for rigid obstacles is somewhat different from those presented earlier in the literature. The derivations are extended to systems with a compliant obstacle. We find that the map for impacts with a compliant obstacle is very similar to the one describing collisions with a rigid obstacle. A notable difference is a change of scale of the bifurcation parameter. We illustrate our findings in the limit of large damping, where the mechanism of period adding can be analysed exactly. The relevance of our results to experiments on practical impact systems is indicated.

In this paper we present two approaches to estimate the Hausdorff dimension of an invariant compact set of a dynamical system: the method of characteristic exponents (estimates of Kaplan-Yorke type) and the method of Lyapunov functions. In the first approach, using Lyapunov's first method we exploit characteristic exponents to obtain such an estimate. A close relationship with uniform asymptotic stability is hereby established. A second bound for the Hausdorff dimension of an invariant compact set is obtained by exploiting Lyapunov's direct method and thus relies on the use of Lyapunov functions.

It is known that the multifractal spectrum of harmonic measure plays a central role in the geometric function theory. Recently Smirnov and Beliaev introduced a new class of random fractals that are called random conformal snowflakes. They proved that the multifractal spectrum of snowflakes is related to the spectral radius of a particular integral operator. In this paper we exploit this connection to estimate the multifractal spectrum of snowflakes.

In this paper, we extend the work of Rubinstein, Sternberg and Keller (1989 SIAM J. Appl. Math. 49 116-33). We consider chemical reactions, phase transitions or other processes governed by a semilinear reaction-diffusion equation (with Neumann boundary conditions) for u(x, t, ∈) defined for t > 0 and x ∈ Ω̄ ⊂ ℝn by ut = ∈∇ · (k(x)∇u) + ∈-1Vu(x, u) x ∈ Ω where ∈ is a small parameter and V is a bistable potential for u; here V and k depend on x ∈ Ω and V is even in u. Here one of the stable minimizers is pointwise positive, and the fact that V is even in u then gives that the other stable minimizer is negative. The reaction rate ∈-1Vu (u) is large, while the diffusion coefficient is small. If the initial condition u(x, 0) = φ(x) is positive in the open domain Ω1, negative in the open domain Ω2 (with Ω1 ∩ Ω2 = ∅), and zero on a surface Γ∈ ⊂ Ω, with Ω1 ∪ Ω2 ∪ Γ∈ = Ω, then u rapidly tends to the positive stable state on Ω1, and to the negative stable state on Ω2; an interface of width O(∈) develops at Γ∈. Then each interface moves on a longer O(1/∈) timescale, either towards a stable equilibrium position Γ∈ near Γ0 for ∈ small, or away from unstable equilibrium positions. Here Γ0 is the limit curve that arises from {Γ∈} as ∈ → 0. The equilibrium locations for Γ0 are calculated from a geometric geodesic condition, together with their local stability. Simple formulae for these are derived, which depend only on the x variation in V and k for ∈ small.

Exchange-driven growth is a process in which pairs of clusters interact and exchange a single unit of mass. The rate of exchange is given by an interaction kernel $K(j,k)$ which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field kinetic equations of this process for the first time. We find two different classes of behaviour depending on whether $K(j,k)$ is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying $K(j,k)\leq Cjk$. This result is optimal in the sense that we show for a large class of initial conditions with kernels satisfying $K(j,k)\geq Cj^{\beta}$ ($\beta>1)$ the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for $K(j,k)\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ ($\mu,\nu\leq2,$ $\mu+\nu\leq3),$ while existence is lost for $K(j,k)\geq Cj^{\beta}$ ($\beta>2).$ In the intermediate regime $3, Comment: Mistakes in the uniqueness proofs are fixed. Some typos are corrected. Some references are added