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SUMMARYThis paper presents a new technique for studying the stability properties of dynamic systems modeled by delay-differential equations (DDEs) with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the 'infinite-dimensional Floquet transition matrix U '. Two different formulas for the computation of the approximate U , whose size is determined by the number of polynomials employed, are given. The first one uses the direct integral form of the original system in state space form while the second uses a convolution integral (variation of parameters) formulation. Additionally, a variation on the former method for direct application to second-order systems is also shown. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. An extension of the method to the case where the delay and parametric periods are commensurate is also shown. Stability charts are produced for several examples of time-periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance-modulated turning. The results indicate that this method is an effective way to study the stability of time-periodic DDEs.

Accurate 3D quantum state-to-state reaction probabilities and collision lifetimes for the H+O2→OH+O combustion reaction for total angular momentum J=0 are reported. The reaction probabilities are dominated by resonances, many of which overlap. The total reaction probability is not enhanced by vibrational or rotational excitation of the reactants. The first accurate final state distributions are presented, and probability is found to spread out into all open product vibrational and rotational channels, with a rather strong tendency to produce highly excited product rotational states. The first calculated collision lifetimes are presented, and resonances with lifetimes of 1–10 ps are found at most energies. Whether the reaction behaves statistically is discussed, and future extensions needed are also discussed.

This paper presents a tracking control scheme for spacecraft formation flying with a decentralized collision avoidance scheme, using a virtual leader state trajectory. The configuration space for a spacecraft is the Lie group SE (3), which is the set of positions and orientations in three-dimensional Euclidean space. A virtual leader trajectory, in the form of attitude and orbital motion of a virtual satellite, is generated offline. Each spacecraft tracks a desired relative configuration with respect to the virtual leader in an autonomous manner, to achieve the desired formation. The relative configuration between a spacecraft and the virtual leader is described in terms of exponential coordinates on SE(3). A continuous-time feedback tracking control scheme is designed using these exponential coordinates and the relative velocities. A Lyapunov analysis guarantees that the spacecraft asymptotically

We report accurate 3D quantum reaction probabilities for the H+O2 combustion reaction and find that they are resonance dominated and rise linearly above threshold. The reaction probability is not enhanced by vibrational or rotational excitation of the reactants but does increase above the threshold for vibrationally excited products.