Table of contents

In this paper we are concerned with hierarchies of rational solutions and associated polynomials for the second Painlevé equation (P_II) and the equations in the P_II hierarchy which is derived from the modified Korteweg–de Vries hierarchy. These rational solutions of P_II are expressible as the logarithmic derivative of special polynomials, the Yablonskii–Vorob'ev polynomials. The structure of the roots of these Yablonskii–Vorob'ev polynomials is studied and it is shown that these have a highly regular triangular structure. Further, the properties of the Yablonskii–Vorob'ev polynomials are compared and contrasted with those of classical orthogonal polynomials. We derive the special polynomials for the second and third equations of the P_II hierarchy and give a representation of the associated rational solutions in the form of determinants through Schur functions. Additionally the analogous special polynomials associated with rational solutions and representation in the form of determinants are conjectured for higher equations in the P_II hierarchy. The roots of these special polynomials associated with rational solutions for the equations of the P_II hierarchy also have a highly regular structure.

This paper aims to exploit the geometrical and dynamical properties of general type-K competitive systems. We prove that there is a defined countable family of disjoint invariant n−1 cells that attract all non-convergent persistent trajectories in the type-K competitive system. For strongly type-K competitive systems, we prove that there exists an invariant n−1 manifold attracting all nontrivial orbits if the system is dissipative. We also establish the index theory of equilibria for general Kolmogorov systems. Combining such an index theory and the global attractivity, we classify the three-dimensional type-K Lotka–Volterra systems. There are four classes. We exploit the Hopf bifurcation and the global stability for positive equilibrium.

We study the visible parts of subsets of n-dimensional Euclidean space: a point a of a compact set A is visible from an affine subspace K of ⁿ, if the line segment joining P_K(a) to a only intersects A at a (here P_K denotes projection onto K). The set of all such points visible from a given subspace K is called the visible part of A from K.

We prove that if the Hausdorff dimension of a compact set is at most n−1, then the Hausdorff dimension of a visible part is almost surely equal to the Hausdorff dimension of the set. On the other hand, provided that the set has Hausdorff dimension larger than n−1, we have the almost sure lower bound n−1 for the Hausdorff dimensions of visible parts.

We also investigate some examples of planar sets with Hausdorff dimension bigger than 1. In particular, we prove that for quasi-circles in the plane all visible parts have Hausdorff dimension equal to 1.

In this paper we study stationary distributions for randomly forced Burgers and Hamilton–Jacobi equations in ^d in the case when the forcing potentials have a large global maximum and a small global minimum in a compact part of ^d. We also study the structure of minimizing trajectories for corresponding random Lagrangian systems.

This note concerns reaction–diffusion processes which display remarkable behaviour. Everywhere the concentration, density or temperature exceeds some critical level until at some moment in time it decreases to the critical level at one point in space. At this instant, the complete profile immediately drops to the critical level at every point in space, and then remains there.

Given an analytic family of vector fields in ² having a saddle point, we study the asymptotic development of the time function along the union of the two separatrices. We obtain a result (depending uniformly on the parameters) which we apply to investigate the bifurcation of critical periods of quadratic centres.

A point x in [0,1] is represented as a binary expansion, i.e. it is identified with an infinite binary sequence of 0 and 1. Given a map T satisfying 0⩽T(x)⩽1 for 0⩽x⩽1, we iterate the map T until the first n bits in x recur as the first n bits in the K_nth iterate T^K_n(x) for some K_n = K_n(x). We call K_n(x) the nth recurrence time of x. More precisely, put E_n,j = [ (j−1)/2ⁿ,j/2ⁿ), 1⩽j⩽2ⁿ, and let E_n(x) be one of the intervals E_n,j containing x. Then K_n(x) = min{j⩾1:T^j(x)∊E_n(x)}. For higher dimensional cases we define the recurrence time using subcubes instead of subintervals. For a wide class of T including Hénon mappings we present two conjectures: first, if T is ergodic and has positive entropy, then the sequence of averages of ( log ₂K_n)/n monotonically converges to the Hausdorff dimension as n→∞. Second, the values of K_nP_n are exponentially distributed as n→∞ where P_n(x) is the measure of E_n(x). To support our conjectures computer simulations are presented.

In this paper we study one-parameter families (f_μ)_{μ∊[−1,1]} of two-dimensional diffeomorphisms unfolding critical saddle-node horseshoes (say at μ = 0) such that f_μ is hyperbolic for negative μ. We describe the dynamics at some isolated secondary bifurcations that appear in the sequel of the unfolding of the initial saddle-node bifurcation.

We construct two classes of open sets of such arcs. For the first class, we exhibit a collection of parameter intervals I_n, I_n⊂(0,1], converging to the saddle-node parameter, I_n→0, such that the topological entropy of f_μ is a constant h_n in I_n and h_n is an increasing sequence. So, for parameters in I_n, the topological entropy is upper bounded by the entropy of the initial saddle-node diffeomorphisms. This illustrates the following intuitive principle: a critical cycle of an attracting saddle-node horseshoe is a destroying dynamics bifurcation. In the second class, the entropy of f_μ does not depend monotonically on the parameter μ.

Finally, when the saddle-node horseshoe is not an attractor, we prove that the entropy may increase after the bifurcation.

We establish the existence and local uniqueness of two classes of multi-bump, self-similar, blowup solutions for the cubic nonlinear Schrödinger equation close to the critical dimension d = 2. Our results for one class of orbits build on the earlier discovery of these orbits via numerical simulation and via asymptotic analysis, providing a proof of their existence. The second class of multi-bump orbits is new. These multi-bump orbits, many of which are thought to be unstable, appear to serve as guides for how different types of initially non monotone data might blow up.

These self-similar solutions are governed by a nonlinear, nonautonomous ordinary differential equation (ODE); and, when linearized, this ODE exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically. In between, the behaviour changes type; this region is called the midrange. For the solutions of the full ODE that we construct, all but one of the bumps—the exception being the central bump at the origin—lie in the midrange. The main steps in the proof involve (i) tracking a pair of manifolds of solutions of the governing ODE that satisfy the conditions at the origin and the asymptotic conditions, respectively, to a common point in the midrange, and (ii) showing that these intersect transversally. Geometric singular perturbation theory, adiabatic Melnikov theory, and the exchange lemma are used to analyse the dynamics in the midrange.

Let Q:[1,∞)→[1,∞) be a strictly increasing function with n⩽Q(n) for all sufficiently large integers n. Fix a positive integer N with N⩽⌊10^Q(n)/10^Q(n−1)⌋ for all sufficiently large integers n (here ⌊x⌋ denotes the integer part of x), and define by = { ∑_n=1^∞x_n/10^Q(n)| x_n=0,1,..., min(⌊10^Q(n)/10^Q(n−1)⌋,N)−1 for  n}. We determine the exact Hausdorff dimension function of for a large class of functions Q including Q(t) = Γ(t+1) for t⩾1. As an application of our results we exhibit a large class of dimension functions h for which the h-dimensional Hausdorff measure ^h() of the set of Liouville numbers is positive, i.e., such that 0 < ^h().

Recently Bandt, Keller and Pompe (2002 Entropy of interval maps via permutations Nonlinearity15 1595–602) introduced a method of computing the entropy of piecewise monotone interval maps by counting permutations exhibited by initial pieces of orbits. We show that for topological entropy this method does not work for arbitrary continuous interval maps. We also show that for piecewise monotone interval maps topological entropy can be computed by counting permutations exhibited by periodic orbits.

We consider the statistical behaviour of independent identically distributed compositions of a finite set of Euclidean isometries of ⁿ. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalize immediately to Markov chains.

We also give simple geometric criteria for orbits to grow linearly or sublinearly with probability one and for nondegeneracy (nonsingular covariance matrix) in the statistical limit theorems.

Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.

In this paper, we first recall the definition of a family of root-finding algorithms known as König's algorithms. We establish some local and some global properties of those algorithms. We give a characterization of rational maps which arise as König's methods of polynomials with simple roots. We then estimate the number of non-repelling cycles König's methods of polynomials may have. We finally study the geometry of the Julia sets of König's methods of polynomials and produce pictures of parameter spaces for König's methods of cubic polynomials.

We investigate the conditions to be imposed on the rate of mixing of a dynamical system in order to ensure the finiteness of some or all moments of hitting and return times to a given subset of its phase space. This amounts to studying the tail of the distribution of hitting and return times to this subset. Examples illustrate our results.

The butterfly-like Lorenz attractor is one of the best known images of chaos. The computations in this paper exploit symbolic dynamics and other basic notions of hyperbolicity theory to take apart the Lorenz attractor using periodic orbits. We compute all 111011 periodic orbits corresponding to symbol sequences of length 20 or less, periodic orbits whose symbol sequences have hundreds of symbols, the Cantor leaves of the Lorenz attractor, and periodic orbits close to the saddle at the origin. We derive a method for computing periodic orbits as close as machine precision allows to a given point on the Lorenz attractor. This method gives an algorithmic realization of a basic hypothesis of hyperbolicity theory—namely, the density of periodic orbits in hyperbolic invariant sets. All periodic orbits are computed with 14 accurate digits.

In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system u_1,t = u_1,xx − u^α₁u^β₂, u_2,t = du_2,xx + u^α₁u^β₂ with non-negative initial data, where the exponents α,β satisfy 1<α,β<2, α+β = 3 and the constant d>0 is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as |x|→∞. We show the exact large time behaviour of solutions which is universal.

We study the coexistence of phases in a two-species model whose free energy is given by the scaling limit of a system with long range interactions (Kac potentials) that are attractive between particles of the same species and repulsive between different species.

We consider an iterated function system (with probabilities) of isometries on an unbounded metric space (X,d). Under suitable conditions it is proved that the random orbit {Z_n}_n⩾0 of the iterations corresponding to an initial point Z₀∊X `escapes to infinity' in the sense that P(Z_n∊K)→0, as n→∞ for every bounded set K⊂X. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.

We study the equations governing the motion of second grade fluids in a bounded domain of ^d, d = 2,3, with Navier-slip boundary conditions with and without viscosity (averaged Euler equations). We show global existence and uniqueness of H³ solutions in dimension two. In dimension three, we obtain local existence of H³ solutions for arbitrary initial data and global existence for small initial data and positive viscosity. We close by finding Liapunov stability conditions for stationary solutions for the averaged Euler equations similar to the Rayleigh–Arnold stability result for the classical Euler equations.

We study upper bounds of the number of zeros of Abelian integrals of polynomial 1-forms of degree n over the compact level curves Γ_h of the elliptic Hamiltonian of degree four, having a `figure-of-eight loop'. Upper bounds are given for Γ_h lying inside the figure-of-eight loop and for Γ_h lying outside the figure-of-eight loop. The bounds are linearly dependent on n.

We consider a scalar nonlinear symmetric model of a suspension bridge. We study periodic solutions and prove existence and multiplicity results. The main tool is the bifurcation theory combined with the properties of the Fucík spectrum.

In this paper, a general predator–prey system is considered. By utilizing simple but crucial changes of variables, the system is reduced to a generalized Liénard system where a wealth of existing methods and results are applicable. A new uniqueness theorem of limit cycle for generalized Liénard system is obtained. Some conditions of known uniqueness theorems and nonexistence theorems are modified so that they are more easily applied. As an application, criteria for the uniqueness of limit cycles and global stability of a unique positive equilibrium of the general predator–prey system are derived, which include some results of Kuang and Freedman (1988 Math. Biosci.88 67–84). Several examples are given to illustrate our results.