Linear stability analysis of the figure-eight orbit in the three-body problem

We show that the well-known figure-eight orbit of the three-body problem is linearly stable. Building on the strong amount of symmetry present, the monodromy matrix for the figure-eight is factored so that its stability can be determined from the first twelfth of the orbit. Using a clever change of coordinates, the problem is then reduced to a 2×2 matrix whose entries depend on solutions of the associated linear differential system. These entries are estimated rigorously using only a few steps of a Runge–Kutta–Fehlberg algorithm. From this, we conclude that the characteristic multipliers are distinct and lie on the unit circle. The methods and results presented are applicable to a wide range of Hamiltonian systems containing symmetric periodic solutions.

An empirical approach for delayed oscillator stability and parametric identification

This paper investigates a semi-empirical approach for determining the stability of systems that can be modelled by ordinary differential equations with a time delay. This type of model is relevant to biological oscillators, machining processes, feedback control systems and models for wave propagation and reflection, where the motion of the waves themselves is considered to be outside the system model. A primary aim is to investigate the extension of empirical Floquet theory to experimental or numerical data obtained from time-delayed oscillators. More specifically, the reconstructed time series from a numerical example and an experimental milling system are examined to obtain a finite number of characteristic multipliers from the reduced order dynamics. A secondary goal of this work is to demonstrate a benefit of empirical characteristic multiplier estimation by performing system identification on a delayed oscillator. The principal results from this study are the accurate estimation of delayed oscillator characteristic multipliers and the utilization the empirical results for parametric identification of model parameters. Combining these results with previous research on an experimental milling system provides a particularly relevant result—the first approach for identifying all model parameters for stability prediction directly from the cutting process vibration history.

Stability Analysis of the Rocking Block: Numerical Investigations on Analytical Stability Boundaries

The present paper illustrates some recently developed techniques to evaluate stability features, such as characteristic multipliers and Lyapunov’s exponents, for a problem with strong discontinuities. The problem analyzed concerns the plane dynamics of a rigid block simply supported on a harmonically moving rigid ground. In previous papers [1–3] many aspects have been investigated and the following results have been carried out: 1. the general procedure to approach numerically the problem has been outlined; 2. a rigorous method has been established to calculate characteristic multipliers and Lyapunov’s exponents at the instants of discontinuities; 3. the stability boundaries of symmetric sub-harmonic responses have been drafted by means of closed-form analytical methods based on various hypotheses of linearization. In this paper the new capacities allowed by the techniques and methods pointed out at the previous points 1., 2. and 3. are exploited with the aims: • to investigate numerically in the occurrence of dissipative impacts the features of dynamic responses across the upper boundary of a stability range (i.e. that of the (1,3) sub-harmonic response) included in a larger one (i.e. that of the (1,1) response); • to analyze the responses attainable with a value of the restitution coefficient equal to 1 to describe the impulsive phases (namely for non-dissipative impacts); in previous works, these responses have been classified as quasi-periodic or chaotic [4]. By means of the new techniques proposed and implemented, it has been possible to classify and analyze more deeply such presumed quasi-periodic or chaotic responses and at the same time to clarify the role played by the initial conditions.