On the number of crossings in the spectrum of a hermitian matrix which depends on a real parameter

Intersection of the eigenvalues i(h) of an n-dimensional hermitian matrix A + hB (h being a real parameter) is discussed. An upper limit for the number of intersections is derived in terms of the rank of the Gramian of the symmetrized products of order 0, 1, …, n — 1 of A and B

The construction of one unstable manifold for the dissipative Henon mapping

The unstable manifold of a saddle point of the Henon mapping is constructed analytically via a contraction mapping, for a range of parameter values where the second fixed point is a stable node. One invariant piece of this manifold connects the saddle with the second fixed point. Rigorous error bounds are derived for the each step of the iterative procedure. It is demonstrated that an algebraic approximation with known accuracy can be given of the unstable manifold

Periodic travelling waves in a non-integrable one-dimensional lattice

The existence of a one-parameter family of periodic solutions representing longitudinal travelling waves is established for a one-dimensional lattice of identical particles with nearest-neighbour interaction. The potential is not given in closed form but is specified by only a few global properties. The lattice is either infinite or consists ofN particles on a circle with fixed circumference. Waves with low energy are sinusoidal and their properties are studied using bifurcation methods. Waves of high energy, however, are of solitary type, i.e. the excitation is strongly localized

Level crossing and the space of operators commuting with the Hamiltonian

The space of n-dimensional hermitean matrices that commute with a given hermitean matrix A + hB, h being a real parameter, is discussed. In particular a basis in this space is constructed consisting of polynomials in h of the lowest possible total degree. The sum of the degrees of the elements of this minimal basis equals 1/2n((n-1) - q , q being the number of linearly independent linear relations between the symmetrized products of A and B of order 0,…,n − 1. These linear relations determine the values of h for which crossing occurs, the total number of crossings for each value, and in some cases the order of the different crossings. A discussion of the noncrossing rule concludes this paper.\u

Asymptotic equality of the isolated and the adiabatic susceptibility

Many-particle systems with a hamiltonian of the form H = A + hB, h being a parameter, are discussed. In particular, for a certain class of these systems, a criterion is derived for the asymptotic equality of the isolated and the adiabatic susceptibility or, equivalently, for the ergodicity of B. This criterion states that, for sufficiently large particle number, any hermitian operator polynomial in h of any degree J that commutes with H(h) can be written as a linear combination of the powers H0, …, HJ with polynomial coefficients

Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction

We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping

Bounds for bounded motion around a perturbed fixed point

We consider a dissipative map of the plane with a bounded perturbation term. This perturbation represents e.g. an extra time dependent term, a coupling to another system or noise. The unperturbed map has a spiral attracting fixed point. We derive an analytical/numerical method to determine the effect of the additional term on the phase portrait of the original map, as a function of the δ bound on the perturbation. This method yields a value δ c such that for δδ c the orbits about the attractor are certainly bounded. In that case we obtain a largest region in which all orbits remain bounded and a smallest region in which these bounded orbits are captured after some time (the analogue of 'basin' and 'attractor respectively')

Transient periodic behaviour related to a saddle-node bifurcation

The authors investigate transient periodic orbits of dissipative invertible maps of R2. Such orbits exist just before, in parameter space, a saddle-node pair is formed. They obtain numerically and analytically simple scaling laws for the duration of the transient, and for the region of initial conditions which evolve into transient periodic orbits. An estimate of this region is then obtained by the construction-after extension of the map to C2-of the stable manifolds of the two complex saddles in C2 that bifurcate ino the real saddle-node pai

On the influence of heat conduction on paramagnetic dispersion and absorption curves

Starting from the model of Casimir and Du Pré, which has been refined by Eisenstein, an expression for the differential paramagnetic susceptibility of a paramagnetic material, placed in a (gaseous or liquid) bath, is derived. This expression contains among others the coefficient of heat conduction and heat capacity of the surroundings and a heat resistance between sample and bath.\ud \ud Numerical analysis and a short comparison with experiments show that in the temperature region of liquid helium the above-mentioned quantities have a remarkable influence on the dispersion and absorption curves