Table of contents

A loss in smoothness of a switching process decreases the accuracy of an adiabatic invariant. Here we show that for classical Hamiltonian systems the degree of smoothness can be observed in the signal.

We prove, with the assistance of rigorous computer calculations, that Widom's renormalization fixed point (1983), and any pair in its domain of attraction, has a Holder continuous invariant curve through the origin. We deduce that any complex analytic map attracted by Widom's fixed point has a Siegel disc bounded by a Holder continuous Jordan curve passing through a critical point of the map. It is known empirically that the stable manifold of Widom's fixed point has real codimension two. This would imply that the above holds on an open set of maps having a golden mean Siegel disc.

We study the stability of solitary waves of two coupled Boussinesq equations which model weakly nonlinear vibrations in a cubic lattice. A Hamiltonian formulation is presented. Known variational methods are observed to be incapable of establishing stability or detecting instability. Instead, the problem is linearized and studied using the Evans function, an analytic function whose zeros, when in the right half plane, correspond to discrete unstable eigenvalues. It is proved that if there is a linear exponential instability, then at transition a pair of complex conjugate eigenvalues emerges into the right half plane. The Evans function is computed numerically and we observe complex conjugate pairs of zeros crossing into the right half plane. The first pair that crosses does so in close agreement with the conclusions of Christiansen, Lomdahl and Muto (1990 Nonlinearity 4 477), which were drawn from numerically computed solutions of the initial-value problem. The instability mechanism in this system differs from that typical in finite-dimensional Hamiltonian systems, where transition to instability occurs via collisions of imaginary eigenvalues. Here, the transition involves resonance poles, which are the zeros of the Evans function in the left half plane, that cross the continuous spectrum and emerge as unstable eigenvalues.

We consider the Ginzburg-Landau equation, delta _tu= delta _x²u+u-u mod u mod ², with complex amplitude u(x, t). We first analyse the phenomenon of phase slips as a consequence of the local shape of u. We next prove a global theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth-order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.

A variational characterization is employed to find isolated vortices-steady solutions of the two-dimensional Euler equations-as relative equilibrium solutions. The steady states are parametrized by the values of invariant integrals of the dynamical system corresponding to the physical quantities of enstrophy, positive and negative circulation and linear and angular momentum. The variational principles can be treated analytically in special configurations but are solved numerically in other cases, where the optimization scheme by Eydeland solves for the free boundaries. We find continuous distributions of vorticity on supports that are not necessarily simply connected. For special choices of the parameter values isolated mono-, di-, tri- and quadrupolar vortices are found.

This is an attempt to study mathematically billiards with moving boundaries. We assume that the boundary remains closed, regular and strictly convex, deforming periodically in time, in the normal direction. We describe the associated billiard diffeomorphism and the corresponding invariant measure. We discuss the stability of 2-periodic orbits and investigate the boundedness of the velocity in some precise examples. Finally, we present the Hamiltonian formalism and the symplectic structure, considering that a moving billiard is a billiard with rigid boundary on an augmented configuration space, with a singular metric.

We consider discrete invertible dynamical systems L with the property that the kth iterate L^k possesses (reversing) symmetries mat are not possessed by L. A map U is called a (reversing) k-symmetry of L if k is the smallest positive integer for which U is a (reversing) symmetry of L^k. In this paper we discuss the particular case that L possesses a cyclic reversing k-symmetry group. We derive a decomposition property of maps that possess a cyclic reversing k-symmetry group and we classify the occurrence of such groups in invertible dynamical systems. We discuss the occurrence of nonsimultaneously linearizable nonisomorphic reversing k-symmetry groups in maps possessing cyclic reversing k-symmetry groups, illustrated by an example of a diffeomorphism on the plane R². We also construct examples of diffeomorphisms with cyclic reversing k-symmetry groups on the circle S¹, on the two-torus T², and on the cylinder S¹*R.

Let Gamma be a finite group acting on Rⁿ and let x₀ be an initial point for a Gamma -equivariant map f:Rⁿ to Rⁿ. The question of determining the symmetries of the omega -limit set omega _f(x₀) is discussed by Barany et al. and Dellnitz et al. The methods introduced therein are based on the notion of a symmetry detective. Detectives replace the question of determining the symmetries of the set omega _f(x₀) by the easier question of determining the symmetries of a point in an associated space W. The detective theorem of Barany et al. has a limitation in that its implementation tacitly assumes that omega _f(x₀) contains a point of trivial isotropy; this assumption is explicit in Dellnitz et al. In this paper we extend the ideas of these authors to present sufficient conditions for an equivariant polynomial phi :Rⁿ to W to be a detective, even when the omega -limit set is contained in a proper fixed-point subspace. We show that W need only satisfy the conditions given by Barany et al. and Dellnitz et al. while the map phi has to satisfy certain conditions in addition to the ones listed by these authors. We also present a density theorem for such detectives and we show that the detective for rings of p coupled cells (nearest-neighbour coupling) with D_p symmetry first given by Barany et al. is a detective for all (SBR) attractors.

Dynamical systems with symmetry can possess different conjugate attractors simultaneously. If a parameter is varied then these may collide at reflection hyperplanes. Since such a collision typically leads to an increase in the symmetry of the dynamics, this phenomenon is called a symmetry increasing bifurcation. We show that there are restrictions on the type of symmetry increasing bifurcations that can occur. Moreover, we give an algebraic criterion which guarantees that certain collisions of attractors are admissible in the sense that they can be constructed in an appropriate dynamical system. Finally, for two-dimensional mappings we classify all the admissible symmetry increasing bifurcations at reflection hyperplanes.

A standard use of shadowing lemmas in dynamical systems theory is to prove density of periodic points. When there is symmetry present it is natural to investigate the density of periodic points of a given symmetry. We obtain several results in this direction. As an application, we prove a result about compact group extensions of hyperbolic and nonuniformly hyperbolic dynamical systems.

We apply J Mather's theory (1991) on minimizing measures to the case of positive definite autonomous Lagrangians L:TM to R. We show that the minimal action function beta (h)=min( integral Ld mu mod rho ( mu )=h), has radial derivative, rho ( mu ) in H₁(M,R) is the homology vector of the invariant probability mu . As one consequence, for any supporting domain S of beta , the closure of the union of the support of any minimizing measures with homology in S is contained in a fixed energy level. It is also shown that there is a one-to-one correspondence between minimizing measures for Lagrangians associated to mechanical systems (kinetic minus potential energy) and minimizing measures for the geodesic problem on a fixed energy level.

In this paper we will investigate the relevance of a stable family of relative equilibria in a dissipative Hamiltonian system with symmetry. We are interested in relative equilibria of the Hamiltonian system, whose stability follows from the fact that they are local extrema of the energy-momentum function which is a combination of the Hamiltonian and a conserved quantity of the Hamiltonian system, induced by the momentum map related to the symmetry group. Although the dissipative perturbation is equivariant under the action of the symmetry group, it will destroy the conservation law associated with the symmetry group. We will specify its dissipative properties in terms of the induced time behaviour of the momentum map and quasi-static attractive properties of the relative equilibria. By analysing the time behaviour of the previously mentioned energy-momentum function we derive sufficient conditions such that solutions of the dissipative system which are initially close to a relative equilibrium can be approximated by a (long) curve of relative equilibria. At the end we illustrate the method by analysing the example of a rigid body in a rotational symmetric field with dissipative rotation-like perturbation added.

The non-trivial zeros of the Riemann zeta-function have been conjectured to be pairwise distributed like the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) of random matrix theory. They therefore behave like the energy levels of quantum systems whose classical limit is strongly chaotic and non-invariant with respect to time-reversal. We show that this analogy extends directly to higher-order statistics. Starting with an explicit formula relating the zeros to the prime numbers (the analogue of the Gutzwiller trace formula of quantum chaology), we demonstrate that the 3-point and 4-point zero correlation functions are asymptotically equivalent to the corresponding GUE results. Our method centres around a Hardy-Littlewood conjecture concerning the distribution of the primes. The calculation generalises a previous study of 2-point correlations and involves the introduction of several new techniques. These will form the basis of a demonstration, to be described in another paper, that the equivalence extends to the general n-point correlation function.

We consider periodic pertubations of a family of autonomous vector fields which has a degenerate homoclinic orbit. We show that the set of parameters for which homoclinic points exist is foliated by Whitney umbrellas.

The general solutions of both the Duffing equation and the Lorenz system admit meromorphic Laurent series representations about movable poles only for certain values of their respective parameters. In all other parameter regimes, the general solutions contain movable logarithmic branch points. Series expansions about such points are termed psi-series and constitute formal general solutions. In this article, we prove the convergence of the psi-series solutions of the Duffing equation and the Lorenz system, thus establishing that the formal solutions are actual solutions.

We prove that the perturbative expansion for the KAM invariant tori of the Thirring model (with interaction depending also on the action variables) is convergent by using techniques usual in quantum field theory like the multiscale decomposition and the tree expansion. The proof follows the ideas of Eliasson (1988) and extends the results found in the case of an action-independent interaction potential by Gallavotti (1994). The connection with the methods of quantum field theory is emphasized through the introduction of a particular resummation of the perturbative series.

Considered herein are the generalized Korteweg-de Vries equations with a homogeneous dissipative term appended. Solutions of these equations that start with finite energy decay to zero as time tends toward infinity. We present an asymptotic form which renders explicit the relative strengths of the dissipative, dispersive, and nonlinear effects in this decay.