James D. Meiss

Transport in Hamiltonian systems

Abstract We develop a theory of transport in Hamiltonian systems in the context of iteration of area-preserving maps. Invariant closed curves present complete barriers to transport, but in regions without such curves there are still invariant Cantor sets named cantori, which appear to form partial barriers. The flux through the gaps of the cantori is given by Mathers differences in action. This gives useful bounds on transport between regions, and for one parameter families of maps it provides a universal scaling law when a curve has just broken. The bounds and scaling law both agree well with numerical experiment of Chirikov and help to explain an apparent disagreement with results of Greene. By dividing the irregular components of phase space into regions separated by the strongest partial barriers, and assuming that the motion is mixing within these regions, we present a global picture of transport, and indicate how it can be used, for example, to predict confinement times and to explain longtime tails in the decay of correlations.

Markov tree model of transport in area-preserving maps

Abstract Transport in an area-preserving map with a mixture of regular and chaotic regions is described in terms of the flux through invariant cantor sets called cantori. A model retaining a discrete set of cantori approaching a boundary circle gives the Markov chain description of Hanson, Cary and Meiss. The inclusion of cantori surrounding island chains, and islands about islands, etc. gives a Markov tree model with a slower decay rate. The survival probability distribution is shown to decay asymptotically as a power law. The decay exponent agrees reasonably well with the computations of Karney and of Chirikov and Shepelyanski.

Solitary drift waves in the presence of magnetic shear

The two‐component fluid equations describing electron‐drift and ion‐acoustic waves in a nonuniform magnetized plasma are shown to possess nonlinear two‐dimensional solitary wave solutions. In the presence of magnetic shear, radiative shear damping is exponentially small in Ls/Ln for solitary drift waves, in contrast to linear waves.

Resonances in area-preserving maps

Abstract A resonance for an area-preserving map is a region of phase space delineated by “partial separatrices”, curves formed from pieces of the stable and unstable manifold of hyperbolic periodic points. Each resonance has a central periodic orbit, which may be elliptic or hyperbolic with reflection. The partial separatrices have turnstiles like the partial barriers formed from cantori. In this paper we show that the areas of the resonances, as well as the turnstile areas, can be obtained from the actions of homoclinic orbits. Numerical results on the scaling of areas of resonances with period and parameter are given. Computations show that the resonances completely fill phase space when there are no invariant circles. Indeed, we prove that the collection of all hyperbolic cantori together with their partial barriers occupies zero area.

Shear-Alfvén dynamics of toroidally confined plasmas

Abstract Recent developments in the stability theory of toroidally confined plasmas are reviewed, with the intention of providing a picture comprehensible to non-specialists. The review considers a class of low-frequency, electromagnetic disturbances that seem especially pertinent to modern high-temperature confinement experiments. It is shown that such disturbances are best unified and understood through consideration of a single, exact fluid moment: the shear-Alfven law. Appropriate versions of this law and its corresponding closure relations are derived, essentially from first principles, and applied in a variety of mostly, but not exclusively, linear contexts. Among the specific topics considered are: flux coordinates (including Hamada coordinates), the Newcomb solubility condition, Shafranov geometry, magnetic island evolution, reduced MHD and its generalizations, drift-kinetic electron response, classical tearing, twisting, and kink instabilities, pressure-modified tearing instability (Δ-critical), collisionless and semi-collisional tearing modes, the ballooning representation in general geometry, ideal ballooning instability. Mercier criterion, near-axis expansions, the second stability region, and resistive and kinetic ballooning modes. The fundamental importance of toroidal topology and curvature is stressed.

Algebraic Decay in Self-Similar Markov Chains

A continuous-time Markov chain is used to model motion in the neighborhood of a critical invariant circle for a Hamiltonian map. States in the infinite chain represent successive rational approximants to the frequency of the invariant circle. For the case of a noble frequency, the chain is self-similar and the nonlinear integral equation for the first passage time distribution is solved exactly. The asymptotic distribution is a power law times a function periodic in the logarithm of the time. For parameters relevant to the critical noble circle, the decay proceeds ast−4.05.

Targeting chaotic orbits to the Moon through recurrence

Abstract Transport times for a chaotic system are highly sensitive to initial conditions and parameter values. In a previous paper, we presented a technique to find rough orbits (epsilon chains) that achieve a desired transport rapidly. The strategy is to build the epsilon chain from segments of a long orbit — the point is that long orbits have recurrences in neighborhoods where faster orbits must also pass. If a local hyperbolicity condition is satisfied, then a nearby shadow orbit may be constructed with significantly smaller errors. In this paper, we modify the technique to find real orbits, in configuration space, of the restricted three body problem. We find a chaotic Earth-Moon transfer orbit that achieves ballistic capture and that requires 38% less total velocity boost than a comparable Hohmann transfer orbit.

SCATTERING OF REGULARIZED-LONG-WAVE SOLITARY WAVES

Abstract The Lagrangian density for the regularized-long-wave equation (also known as the BBM equation) is presented. Using the trial function technique, ordinary differential equations that describe the time dependence of the position of the peaks, amplitudes, and widths for the collision of two solitary waves are obtained. These equations are analyzed in the Born and “equal-width” approximations and compared with numerical results obtained by direct integration utilizing the split-step fast Fourier-transform method. The computations show that collisions are inelastic and that production of solitary waves may occur.

Converse KAM theory for symplectic twist maps

The authors derive a criterion for the non-existence of invariant Lagrangian graphs for symplectic twist maps of an arbitrary number of degrees of freedom, interpret it geometrically, and apply it to a four-dimensional example.

Linear stability of periodic orbits in lagrangian systems

Abstract The orbits of lagrangian systems are given by paths with stationary action. We derive a formula for the linear stability of periodic orbits in discrete-time one-degree-of-freedom lagrangian systems, in terms of the second variation of the action about the orbit in the space of periodic paths.