390 research outputs found
Solving systems of transcendental equations involving the Heun functions
The Heun functions have wide application in modern physics and are expected
to succeed the hypergeometrical functions in the physical problems of the 21st
century. The numerical work with those functions, however, is complicated and
requires filling the gaps in the theory of the Heun functions and also,
creating new algorithms able to work with them efficiently.
We propose a new algorithm for solving a system of two nonlinear
transcendental equations with two complex variables based on the M\"uller
algorithm. The new algorithm is particularly useful in systems featuring the
Heun functions and for them, the new algorithm gives distinctly better results
than Newton's and Broyden's methods.
As an example for its application in physics, the new algorithm was used to
find the quasi-normal modes (QNM) of Schwarzschild black hole described by the
Regge-Wheeler equation. The numerical results obtained by our method are
compared with the already published QNM frequencies and are found to coincide
to a great extent with them. Also discussed are the QNM of the Kerr black hole,
described by the Teukolsky Master equation.Comment: 17 pages, 4 figures. Typos corrected, one figure added, some sections
revised. The article is a rework of the internal report arXiv:1005.537
Hybrid trajectory spaces
In this paper, we present a general framework for describing and studying hybrid systems. We represent the trajectories of the system as functions on a hybrid time domain, and the system itself by its trajectory space, which is the set of all possible trajectories. The trajectory space is given a natural topology, the compact-open hybrid Skorohod topology, and we prove the existence of limiting trajectories under uniform equicontinuity assumptions. We give a compactness result for the trajectory space of impulse differential inclusions, a class of nondeterministic hybrid system, and discuss how to describe hybrid automata, a widely-used class of hybrid system, as impulse differential inclusions. For systems with compact trajectory space, we obtain results on Zeno properties, symbolic dynamics and invariant measures. We give examples showing the application of the results obtained using the trajectory space approac
Systematic methodology for the global stability analysis of nonlinear circuits
A new methodology for the detection of Hopf, flip, and turning-point bifurcations in nonlinear circuits analyzed with harmonic balance (HB) is presented. It enables a systematic determination of bifurcation loci in terms of relevant parameters, such as input power, input frequency, and bias voltages, for instance. It does not rely on the use of continuation techniques and is able to globally provide the entire loci, often containing multivalued sections and/or disconnected curves, in a single simulation. The calculation of Hopf and flip bifurcations is based on the extraction of a small-signal admittance/impedance function from HB and the calculation of its zeros through a geometrical procedure. The method is ideally suited for the investigation of the global stability properties of power amplifiers and other nonlinear circuits. The turning-point locus, associated with either jump phenomena or synchronization, is obtained by taking into account the annihilation/generation of steady-state solutions that is inherent to this type of bifurcation. A technique is also presented for the exhaustive calculation of oscillation modes in multidevice oscillators and oscillators loaded with multiresonance networks. The new methodologies are illustrated through their application to a power amplifier and a multimode oscillator.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research projects TEC2014-60283-C3-1-R and TEC2017-88242-C3-1-R
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