Table of contents

We consider a Leray-type regularization for the isentropic Euler equations for a γ-law gas, and we investigate the existence of smooth solutions for the regularized system. The technique we use is the weakly nonlinear geometrical optics (WNGO) asymptotic theory. The WNGO theory applied to our system of equations predicts shock formation in finite time for γ ≠ 1 and suggests existence of global smooth solutions for γ = 1. We also perform numerical computations and show that the WNGO predictions are correct.

We give an example of a C¹ flow on the two-dimensional torus for which the pointwise rotation set is not closed.

We prove that in three space dimensions a nonlinear wave equation u_tt − Δu = u^p, with p⩾ 7 being an odd integer, has a countable family of regular spherically symmetric self-similar solutions.

A method for the study of steady-state nonlinear modes for the Gross–Pitaevskii equation (GPE) is described. It is based on the exact statement about the coding of the steady-state solutions of GPE which vanish as x → +∞ by reals. This allows us to fulfil the demonstrative computation of nonlinear modes of GPE, i.e. the computation which allows us to guarantee that all nonlinear modes within a given range of parameters have been found. The method has been applied to GPE with quadratic and double-well potentials, for both repulsive and attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these cases are represented. The stability of these modes has been discussed.

We present a nonlinear Fourier transform pair in any number of dimensions. This pair cannot be expressed in closed form but is defined through the solution of certain linear integral equations. The derivation of this result is based on the formulation of a nonlocal d-bar problem. The above nonlinear Fourier transform pair can be used for the solution of the Cauchy problem of a large class of nonlinear evolution equations in any number of spatial dimensions. Unfortunately, these nonlinear equations are highly nonlocal. Nevertheless, for the case of two dimensions they do reduce to the Davey–Stewartson equation which is a prototypical example of an integrable nonlinear evolution partial differential equation.

For continuous self-maps of compact metric spaces, we initiate a preliminary study of stronger forms of sensitivity formulated in terms of 'large' subsets of . Mainly we consider 'syndetic sensitivity' and 'cofinite sensitivity'. We establish the following. (i) Any syndetically transitive, non-minimal map is syndetically sensitive (this improves the result that sensitivity is redundant in Devaney's definition of chaos). (ii) Any sensitive map of [0,1] is cofinitely sensitive. (iii) Any sensitive subshift of finite type is cofinitely sensitive. (iv) Any syndetically transitive, infinite subshift is syndetically sensitive. (v) No Sturmian subshift is cofinitely sensitive. (vi) We construct a transitive, sensitive map which is not syndetically sensitive.

We study a stochastic nonlocal partial differential equation, arising in the context of modelling spatially distributed neural activity, which is capable of sustaining stationary and moving spatially localized 'activity bumps'. This system is known to undergo a pitchfork bifurcation in bump speed as a parameter (the strength of adaptation) is changed; yet increasing the noise intensity effectively slowed the motion of the bump. Here we study the system from the point of view of describing the high-dimensional stochastic dynamics in terms of the effective dynamics of a single scalar 'coarse' variable, i.e. reducing the dimensionality of the system. We show that such a reduced description in the form of an effective Langevin equation characterized by a double-well potential is quantitatively successful. The effective potential can be extracted using short, appropriately initialized bursts of direct simulation, and the effects of changing parameters on this potential can easily be studied. We demonstrate this approach in terms of (a) an experience-based 'intelligent' choice of the coarse variable and (b) a variable obtained through data-mining direct simulation results, using a diffusion map approach.

We prove the existence of cocoon bifurcations for the Michelson system where and is a parameter, based on the theory given in (Dumortier et al2006 Nonlinearity19 305–28). The main difficulty lies in the verification of the (topological) transversality of some invariant manifolds in the system. The proof is computer-assisted and combines topological tools including covering relations and the smooth ones using the cone conditions. These new techniques developed in this paper will have broader applicability to similar global bifurcation problems.

The hydrodynamical limit of systems of free particles of finite size is considered. The assumption is that all particles move freely on the line and stick to each other upon collision, to form compound particles whose mass and size are the sum of the masses and sizes of the particles before collision and whose velocity after collision is determined by conservation of linear momentum.

A constructive proof of existence and uniqueness for such a limit is provided. The proof implies an explicit form of the solution.

For p ⩾ 2 and diffeomorphisms of with , if ρ(f) < 2sin (π/q) then f has no periodic orbit of order i with 1 ⩽ i ⩽ q. If ρ(f) < 1 or both ρ(f) < 2 and ρ(f⁻¹) < 2 hold, then f is isotopic to the identity relative to its fixed point set. When p = 2, for any ε > 0 there exist diffeomorphisms such that ρ(f) < 1 + ε or both ρ(f) < 2 + ε and ρ(f⁻¹) < 2 + ε hold, with arbitrarily high linking numbers. The geometric complexity of a diffeomorphism f of , equal to the minimal number of horizontal and vertical diffeomorphisms necessary to write an element of the conjugacy class of f as a product, which is at most 2 when ρ(f) < 1 or both ρ(f) < 2 and ρ(f⁻¹) < 2 hold, may then be arbitrarily high for diffeomorphisms such that ρ(f) < 1 + ε or both ρ(f) < 2 + ε and ρ(f⁻¹) < 2 + ε hold. But a diffeomorphism of with arbitrarily high geometric complexity may also be isotopic to the identity relative to its fixed point set, and a diffeomorphism f with geometric complexity 2 and a periodic orbit of order q ⩾ 2 has a fixed point x such that , where denotes the linking number of f between x and .

The embedding theorem of Takens and its extensions have provided the theoretical underpinning for a wide range of investigations of time series derived from nonlinear dynamical systems. The theorem applies when the dynamical system is sampled uniformly in time. There has, however, been increasing interest in situations where observations on the system are not uniform in time, and in particular where they consist of a series of inter-event ('interspike') intervals. Sauer has provided an embedding theorem for the case where these intervals are generated by an integrate-and-fire mechanism. Here we prove several embedding theorems pertaining to non-uniform sampling. We consider two situations: in the first, the observations consist of the values of some function on the state space of the system, with the times between successive observations being given by another function—the sampling interval function; in the second, the sampling times are generated in the same way, but now the observations consist only of the intersample intervals. We prove embedding theorems both when the sampling interval function is allowed to be fairly general, and when it corresponds to integrate-and-fire sampling. We point out that non-uniform sampling might lead to better reconstructions than uniform sampling, for certain kinds of time series.

If (X, f) is a dynamical system given by a compact metric space X and a continuous map f : X → X then by the functional envelope of (X, f) we mean the dynamical system (S(X), F_f) whose phase space S(X) is the space of all continuous selfmaps of X and the map F_f : S(X) → S(X) is defined by F_f(φ) = f ○ φ for any φ ∊ S(X). The functional envelope of a system always contains a copy of the original system.

Our motivation for the study of dynamics in functional envelopes comes from semigroup theory, from the theory of functional difference equations and from dynamical systems theory. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope. Special attention is paid to orbit closures, ω-limit sets, (non)existence of dense orbits and topological entropy.

A two-dimensional body moves through a rarefied medium; the collisions of the medium particles with the body are absolutely elastic. The body performs both translational and slow rotational motion. It is required to select the body, from a given class of bodies, such that the average force of resistance of the medium to its motion is maximal.

Numerical and analytical results concerning this problem are presented. In particular, the maximum resistance in the class of bodies contained in a convex body K is proved to be 1.5 times the resistance of K. The maximum is attained on a sequence of bodies with a very complicated boundary. The numerical study was made for somewhat more restricted classes of bodies. The obtained values of resistance are slightly lower, but the boundary of obtained bodies is much simpler, as compared with the analytical solutions.