An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms

Within‐session reproducibility of forced oscillometry

BACKGROUND: The forced oscillation technique (FOT) provides detailed information about the mechanics of the respiratory system, while requiring minimal co-operation by the patient. FOT may be abnormal in subjects with normal spirometry and appears to be more closely related to airway symptoms. It is, therefore, attractive in epidemiological studies, where a large number of different examinations are made in each subjects in a short period of time. Current technical standards recommend the mean of three consecutive measurements to be used, but there is limited information regarding within-session variability of FOT measurements.OBJECTIVE: The purpose of this study was to examine the within-session variability in FOT measurements in a large, population-based sample.METHODS: We performed three consecutive FOT measurements in 700 subjects using the impulse oscillometry system. The first measurement was compared to the mean of three measurements for resistance at 5 and 20 Hz (R5 and R20, respectively), R5-R20, reactance at 5 Hz (X5) and resonant frequency (fres ).RESULTS: The differences between the first and the mean of three measurements (median, interquartile range) were minimal, for example 0.002, -0.008 to 0.014 kPa L-1 s for R5 and -0.001, -0.008 to 0.005 kPa L-1 s for X5. Findings were numerically similar for men and women as well as for subjects with and without airflow obstruction at spirometry.CONCLUSIONS: We conclude that, whereas in clinical situations, three FOT measurements are to be preferred, a single measurement may suffice in epidemiological studies