Detection of water leaks in supply pipes using continuous wave sensor operating at 2.45 GHz

An innovative sensor technique has been developed to detect water leaks in supply pipes, especially in cases where the existing prevailing methods do not offer reliable solution, such as in PVC pipelines or for minor leaks. The presented system uses a Continuous Wave (CW) Doppler sensing unit operating at 2.45 GHz, consisting of a low power transmitter, a homodyne receiver and a digital signal processing unit. The operation principle is the detection of the Doppler frequency shift of the reflected electromagnetic wave by slightly moving water that leaks out of a pipe. A first prototype has already been developed and tested in test sites as well as in real water leaking conditions. The system has additionally been compared to the commercially available water leaking methods, verifying its reliability and accuracy in the detection of water leaks.     

Analytical invariant manifolds near unstable points and the structure of chaos

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry, Proc Lond Math Soc ser 2, 27:151–170, 1926; Moser, Commun Pure Appl Math 9:673, 1956 and 11:257, 1958; Moser, Giorgilli, Discret Contin Dyn Syst 7:855, 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al., Phys D 29:181, 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice the study of the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain, the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as (i) the order of truncation of the series increases, and (ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.