Table of contents

We study the generalized Rényi entropies which were introduced in the physics literature. The proper defintion of these entropies needs, in our opinion, further clarification. We show that the Rényi entropies do not contain any new information. On the other hand, we introduce the notion of the correlation entropies, which are not invariants of the dynamical systems in the measure-theoretic sense, but they are invariant under transformations of the state space with bounded distortion. With these correlation entropies we provide a formal definition which can serve as a basis for the results reported in the physics literature.

We study classical solutions of the vector O(3) sigma model in (2 + 1) dimensions, spontaneously broken to . The model possesses Skyrmion-type solutions as well as stable domain walls which connect different vacua. We show that different types of waves can propagate on the wall, including waves carrying a topological charge. The domain wall can also absorb Skyrmions and, under appropriate initial conditions, it is possible to emit a Skyrmion from the wall.

The mathematical question of the existence of structure for shock waves in planar motion of plasma is reduced to finding heteroclinic orbits of a system of five ordinary differential equations, which depend on five viscosity parameters. It is shown that the system is gradient-like and has four rest points. An explicit bound is obtained for the set of the bounded complete solutions, and an isolating neighbourhood for this system is constructed. Then using Conley theory we prove that the fast and slow shock waves, of arbitrary strength, always possess structure. Using a limiting technique, the switch-on and switch-off shock structures are obtained as a topological limit of the fast and slow shock structures, respectively. Furthermore, we show that most of the intermediate shocks, in the absence of the electric field, admit structure. The stability of all of these structures is also shown.

The q-difference Painlevé VI equation is investigated on the basis of Hirota forms. It is shown that it admits solutions expressed by Casorati determinants whose entries are given by basic hypergeometric functions.

We find all null conformal Killing vectors in with the metric . We classify one and two-dimensional totally null algebras generated by such vectors. Reductions of the self-dual Yang-Mills equations are obtained for gauge fields invariant with respect to these algebras.

PACS Numbers: 0210, 1130

Hamiltonian systems with rapidly oscillating explicit dependence on time are considered. The wavelength of this oscillation is treated as a small parameter and it is shown how to remove the time dependence up to some order in the small parameter by means of a canonical transformation presented in the form of an asymptotic series. The result has applications for the study of pulse propagation for high bit-rate transmission in optical fibres.

PACS number 4279Sz

This paper provides an analytical insight into the observed nonlinear behaviour of a simple widely used power electronic circuit (the buck converter) and draws parallels with a wider class of piecewise-smooth systems. After introducing the buck converter model and background, the most fascinating features of its dynamical behaviour are reviewed. So-called grazing and sliding solutions are discussed and their role in determining many of the buck converter's dynamical oddities is demonstrated. In particular, a local map is studied which explains how grazing bifurcations cause sharp turning points in the bifurcation diagram of periodic orbits. Moreover, these orbits are shown to accumulate onto a sliding trajectory through a `spiralling' impact adding scenario. The structure of such a diagram is derived analytically and is shown to be closely related to the analysis of homoclinic bifurcations. The results are shown to match perfectly with numerical simulations. The sudden jump to large-scale chaos and the fingered structure of the resulting attractor are also explained.

SO(d) gauged Skyrme models, in three and four dimensions respectively, are studied numerically. The soliton/instanton solutions in these models are stabilized by the degree of the maps rather than the Chern-Pontryagin indices, but some of these solutions carry the latter index too. These models feature asymptotic properties that are different from those of the corresponding Higgs models, mainly in that they exhibit catastrophic behaviour for some ranges of the coupling constants in the models.

We reformulate the notion of connectedness for compact metric spaces in a manner that may be implemented computationally. In particular, our techniques can distinguish between sets that are connected, have a finite number of connected components, have infinitely many connected components, or are totally disconnected. We hope that this approach will prove useful for studying structures in the phase space of dynamical systems.

We model the pulling of an elastic chain in a potential as a slow-fast system using dissipative dynamics, and prove that the dynamics is monotone. If the potential is periodic (the Frenkel-Kontorova model), we show the existence of the adiabatic (slow pulling) limit, describe completely the trajectory of the chain in the limit as a family of orbits of a twist diffeomorphism, and prove the existence of avalanches. Similarities with some self-organized criticality models are discussed.

PACS Number: 6390

Complete proofs are given for some claims of Middleton and of Floria and Mazo about the asymptotic behaviour of chains of balls and springs in a tilted periodic potential and generalizations, under gradient dynamics.

AMS classification number: 58F22

We develop a method to prove the existence of multisite localized periodic oscillations in networks of weakly coupled autonomous oscillators. First, we obtain a necessary condition and a sufficient condition for continuation of a degenerate periodic orbit. These two conditions are applied to the case of m-site periodic oscillations in the uncoupled limit, which are all degenerate and form an m-torus parametrized by phases of oscillations. The result is similar to Melnikov theory, i.e. some periodic orbits in the torus are continued for weak coupling when their relative phases are associated with zeros of two vector valued functions calculated from the solutions in the uncoupled limit. The continued periodic orbits are spatially exponentially localized for exponentially decaying coupling. This method is applied to both conservative and nonconservative oscillator networks. In particular, if oscillations in local dynamics are described by an action-angle form in Hamiltonian networks, the relative phase of the continued periodic orbit is associated with a nondegenerate critical point of a scalar function related to the effective action.

A class of non-compact billiards is introduced, namely the infinite step billiards, i.e. systems of a point particle moving freely in the domain , with elastic reflections on the boundary; here , and .

After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example . Playing an important role in this case are the so-called escape orbits, that is, orbits going to monotonically in the X-velocity. A fairly complete description of them is given. This enables us to prove some results concerning the topology of the dynamics on the billiard.

In semiclassical theories for chaotic systems, such as Gutzwiller's periodic orbit theory, the energy eigenvalues and resonances are obtained as poles of a non-convergent series . We present a general method for the analytic continuation of such a non-convergent series by harmonic inversion of the `time' signal, which is the Fourier transform of g(w). We demonstrate the general applicability and accuracy of the method on two different systems with completely different properties: the Riemann zeta function and the three-disk scattering system. The Riemann zeta function serves as a mathematical model for a bound system. We demonstrate that the method of harmonic inversion by filter-diagonalization yields several thousand zeros of the zeta function to about 12 digit precision as eigenvalues of small matrices. However, the method is not restricted to bound and ergodic systems, and does not require the knowledge of the mean staircase function, i.e. the Weyl term in dynamical systems, which is a prerequisite in many semiclassical quantization conditions. It can therefore be applied to open systems as well. We demonstrate this on the three-disk scattering system, as a physical example. The general applicability of the method is emphasized by the fact that one does not have to resort to a symbolic dynamics, which is, in turn, the basic requirement for the application of cycle expansion techniques.

In his well known paper `Singularities of vector fields', Takens made a topological classification of vector fields up to codimension 2 and introduced a semialgebraic stratification to distinguish the different cases; from dimensions he had to use the notion of `weak--equivalence'. In this paper we show how to classify singularities of vector fields on up to codimension 4 for the notion of equivalence. To separate the different cases we use a semianalytic stratification and show that a semialgebraic one is not possible, even for the notion of weak--equivalence. Up to codimension 3 the stratification is semialgebraic. We will always suppose that the vector fields are , although it will be clear that the results are valid for , with r sufficiently big. We provide a complete, but short, survey of the different techniques to be used, referring to the existing literature for precise calculations and pictures. We put much emphasis on the new results.

We study the dynamics of Lyapunov vectors in various models of one-dimensional distributed systems with spacetime chaos. We demonstrate that the vector corresponding to the maximum exponent is always localized and the localization region wanders irregularly. This localization is explained by interpreting the logarithm of the Lyapunov vector as a roughening interface. We show that for many systems, the `interface' belongs to the Kardar-Parisi-Zhang universality class. Accordingly, we discuss the scaling behaviour of finite-size effects and self-averaging properties of the Lyapunov exponents.

Upper bounds for the Hausdorff dimension of compact and invariant sets of diffeomorphisms are given using a singular value function of the tangent map and the topological entropy under the assumption, that there exists an equivariant splitting of the tangent bundle. This improves previous results for compact uniformly hyperbolic sets of diffeomorphisms satisfying an additional pinching condition. Furthermore, it is shown that the results can be extended to a special class of noninjective maps.

In this paper, we investigate the dynamical behaviour of the following one-dimensional dispersive viscoelasticity equation:

The nonlinear function is smooth, non-convex, unbounded and satisfies some general conditions of growth. By introducing equivalent norms, based on the Hamiltonian structure of the limit dispersive problem with , we can prove the existence of global absorbing balls, and a global attractor. Depending on the parameters , we make different transformations depending on whether is positive or negative; the latter case is the most interesting one as the dissipation is not strong enough to dominate dispersion. The semigroup S(t) generated by satisfies a spectral barrier property, and the existence of inertial manifolds is proved in both cases.

The billiard in a polygon is not always ergodic and never K-mixing or Bernoulli. Here we consider billiard tables by attaching disks to each vertex of an arbitrary simply connected, convex polygon. We show that the billiard on such a table is ergodic, K-mixing and Bernoulli.

We show that the radial d-dimensional sine-Gordon equation has localized metastable breather-like solutions. We analyse some properties of these solutions concentrating on their stability.

We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic fixed point in a flow on originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or `pleat', we may apply the theory of templates, or branched two-manifolds, to capture the topology of the flow. This analysis yields a class of flows which bifurcate from a Morse-Smale flow to a Smale flow containing periodic orbits of all knot and link types.

In this paper we consider the simplest model of quantum open systems, which is the case of one-body statistical physics with inflow boundary conditions at infinity. We develop a functional framework for some nonlinear dynamics, where the nonlinearities are (very) short-range perturbations of a given Hamiltonian. In this framework, one can prove the global in time existence and uniqueness of a solution for the time-dependent problem and the existence of steady states of which the form can be specified. This problem arises from the modelling of semiconductors and applications are reviewed at the end of the paper.

We establish results on the distribution of pre-images with respect to the absolutely continuous invariant measure for certain systems with indifferent periodic points. This proves to be somewhat more delicate than in the uniformly hyperbolic case. We apply these results to the one-dimensional Manneville-Pomeau equation and other one-dimensional examples. We also consider higher dimensional analogues.