Volume 6, Issue 1, March 1996
Index of content:
6(1996); http://dx.doi.org/10.1063/1.166153View Description Hide Description
The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting ‘‘Lagrangian’’ vector fields, i.e., the symplectic 2‐form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals.
6(1996); http://dx.doi.org/10.1063/1.166149View Description Hide Description
Exponentially small separatrix splitting for a pendulum with rapidly oscillating suspension point and for the standard Chirikov map is studied by means of a new averaging method, which is a continuous version of the Neishtadt averaging procedure. An asymptotic formula for the rate of the separatrix splitting is obtained.
6(1996); http://dx.doi.org/10.1063/1.166154View Description Hide Description
Recent results describing non‐trivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so‐called quasiattractors of these systems may exhibit rather non‐trivial features which are in a sharp distinction with that one could expect in analogy with hyperbolic or Lorenz‐like attractors. For instance, the impossibility of giving a finite‐parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors.
6(1996); http://dx.doi.org/10.1063/1.166155View Description Hide Description
Numerous physical systems with two competing frequencies exhibit frequency locking and chaos associated with quasiperiodicity. In this paper we review certain universal aspects of the quasiperiodic route to chaos by making use of the standard circle map. Particular attention is paid to the golden mean and silver mean with a view to comparison with experimental work.
6(1996); http://dx.doi.org/10.1063/1.166156View Description Hide Description
The billiard system of Benettin and Strelcyn [Phys. Rev. A 17, 773–785 (1978)] is generalized to a two‐parameter family of different shapes. Its boundaries are composed of circular segments. The family includes the integrable limit of a circular boundary, convex boundaries of various shapes with mixed dynamics, stadiums, and a variety of nonconvex boundaries, partially with ergodic behavior. The extent of chaos has been measured in two ways: (i) in terms of phase space volume occupied by the main chaotic band; and (ii) in terms of the Lyapunov exponent of that same region. The results are represented as a kind of phase diagram of chaos. We observe complex regularities, related to the bifurcation scheme of the most prominent resonances. A detailed stability analysis of these resonances up to period six explains most of these features. The phenomenon of breathing chaos [Nonlinearity 3, 45–67 (1990)]—that is, the nonmonotonicity of the amount of chaos as a function of the parameters—observed earlier in a one‐parameter study of the gravitational wedge billiard, is part of the picture, giving support to the conjecture that this is a fairly common global scenario.
6(1996); http://dx.doi.org/10.1063/1.166150View Description Hide Description
High accuracy experimental results on the nonlinear dynamical behaviour of a dripping faucet are presented. The distribution functions for droplet sizes and drip intervals together with return maps are studied for various dripping rates. Increasing this control parameter, chaotic behaviour is obtained and discussed.
6(1996); http://dx.doi.org/10.1063/1.166157View Description Hide Description
A method is presented for determining the initial conditions of classical orbits from the quantum spectra of the diamagnetic hydrogen atom. Each classical trajectory which is closed at the nucleus produces a sinusoidal fluctuation in the photoabsorptionspectrum. The amplitude of each orbit’s contribution appears in the Fourier transform of a spectrum computed at constant scaled energy. For a given initial state, closed‐orbit theory gives the dependence of this recurrence amplitude on the initial angle of an orbit. By comparing the recurrence amplitudes for different initial states, the initial conditions of closed classical orbits are determined from quantum spectra.
6(1996); http://dx.doi.org/10.1063/1.166151View Description Hide Description
The steady incompressible flow in a unit sphere introduced by Bajer and Moffatt [J. Fluid Mech. 212, 337 (1990)] is discussed. The velocity field of this flow differs by a small perturbation from an integrable field whose streamlines are almost all closed. The unperturbed flow has two stationary saddle points (poles of the sphere) and a two‐dimensional separatrix passing through them. The entire interior of the unit sphere becomes the domain of streamline chaos for an arbitrarily small perturbation. This phenomenon is explained by the nonconservation of a certain adiabatic invariant that undergoes a jump when a streamline crosses a small neighborhood of the separatrix of the unperturbed flow. An asymptotic formula is obtained for the jump in the adiabatic invariant. The accumulation of such jumps in the course of repeated crossings of the separatrix results in the complete breaking of adiabatic invariance and streamline chaos.
Catastrophic extinction, noise‐stabilized turbulence and unpredictability of competition in a modified Volterra–Lotka model6(1996); http://dx.doi.org/10.1063/1.166152View Description Hide Description
Spatial coexistence and competition among species is investigated through a modified Volterra–Lotka model which takes into account sexual breeding. This allows the population specific growth rate to depend on the population density. As a result of this modification the degeneracy inherent in the classical model is eliminated and qualitatively novel regimes are observed, as demonstrated by parametric analysis of the model. In the case where the corresponding parameters of competing species do not differ significantly the model can be reduced to a single Ginzburg–Landau type equation. The spatially distributed model is analyzed both in the absence and in the presence of noise mimicking inherent fluctuations in birth and death rates. It is shown that noise can qualitatively change the behavior of the system. Not only does it induce the formation of spatial patterns, but also switches on endless turbulent‐like rearrangement of the system. When initially unpopulated habitat is occupied by competing species even a very low‐intensity noise makes the final state of the system totally unpredictable and sensitive to any fluctuations.
6(1996); http://dx.doi.org/10.1063/1.166148View Description Hide Description
The electrical activity of the heart usually shows dynamical behavior which is neither periodic nor deterministically chaotic: The interbeat intervals seem to contain a random component. Although long term predictions are thus impossible, good predictions can be made for times smaller than one heart cycle. This fact is used in order to suppress measurement errors by a local geometric projection method which was originally developed for chaotic signals. The result constitutes evidence that techniques of time series analysis based on chaos theory can be useful despite the fact that very few natural phenomena have been actually established to be deterministically chaotic.