No abstract

The derivative nonlinear Schrödinger ͑DNLS͒ equation, describing propagation of circularly polarized Alfvén waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model ͑equal dampings of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase͒, no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic relaxation oscillations that are absent for zero growth rate. This hard transition in phase-space behavior occurs for left-hand ͑LH͒ polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable, with damping less than about ͑unstable wave frequency͒ 2 /4ϫion cyclotron frequency. The structural stability of the transition was explored by going into a fully 3-wave model ͑different dampings of daughter waves, four-dimensional flow͒; both models differ in significant phase-space features but keep common features essential for the transition.

Abstract. Previously a new dynamic instability that affected electrodynamic tethers in inclined orbits was studied, with a simple one-bar model that neglected the contribution of the tether lateral dynamics. The \u8f exibility of the tether (lateral dynamics), however, plays an important role in the overall motion of the system as shown by numerical simulations of a bare-tether generator in a circular inclined orbit. The same analytical techniques of the previous work are now applied to investigate the dynamics and stability of an electrodynamic tether system modeled by two articulated bars that account for the lowest lateralmodes of the tether. The analysis,which can be directly extended to any electrodynamic tether system, has been focused on two particular, but important cases: the combination of a conductive and a nonconductive leader tether (as in the PropulsiveSmall Expendable Deployment System) and a homogeneous, conductive tether. The lateral dynamics is extremely rich, with skip rope motion, instability peaks, and chains of bifurcations for different regions of the parameter space. The same energy pumpingmechanism that destabilizes the rigid model (one bar) is found to drive an even faster instability of the lateral modes. Damping, which has not been included in the analysis, could change this unstable behavior

Nonlinearly coupled, damped oscillators at 1:1 frequency ratio, one oscillator being driven coherently for efficient excitation, are exemplified by a spherical swing with some phase-mismatch between drive and response. For certain damping range, excitation is found to succeed if it lags behind, but to produce a chaotic attractor if it leads the response. Although a period-doubhng sequence, for damping increasing, leads to the attractor, this is actually born as a hard (as regards amplitude) bifurcation at a zero growth-rate parametric line; as damping decreases, an unstable fixed point crosses an invariant plane to enter as saddle-focus a phase-space domain of physical solutions. A second hard bifurcation occurs at the zero mismatch line, the saddle-focus leaving that domain. Times on the attractor diverge when approaching either fine, leading to exactly one-dimensional and noninvertible limit maps, which are analytically determined.

A hard-in-amplitude transition to chaos in a class of dissipative flows of broad applicability is presented. For positive values of a parameter F, no matter how small, a fully developed chaotic attractor exists within some domain of additional parameters, whereas no chaotic behavior exists for F < 0. As F is made positive, an unstable fixed point reaches an invariant plane to enter a phase half-space of physical solutions; the ghosts of a line of fixed points and a rich heteroclinic structure existing at F = 0 make the limits t --* +oc, F ~ +0 non-commuting, and allow an exact description of the chaotic flow. The formal structure of flows that exhibit the transition is determined. A subclass of such flows (coupled oscillators in near-resonance at any 2 : q frequency ratio, with F representing linear excitation of the first oscillator) is fully analysed.