Time-series rate of convergence to quasi-periodic oscillations
Abstract
We propose three algorithms that can fairly accurately estimate the degree of convergence to the limit cycle using time-series generated by systems that converge to a quasi-periodic oscillation and consider their applicability ranges. As a proof-of-concept, a trivial two-dimensional case is studied. A practically important three-dimensional case is considered. Generalization of the algorithm to the space of any number of dimensions is presented. An example of these algorithms was used for estimating the Van-der-Pol system convergence.
About the Authors
A. V. Bespalov
ITMO University
Russian Federation
Russian Federation
Saint Petersburg
E. V. Vilkova
ITMO University
Russian Federation
Russian Federation
Saint Petersburg
References
1. Jenkins A., Self-oscillation, Phys. Rep., 525, P. 167–222 (2013).
2. Lazarus A., Barois T., Perisanu S., Poncharal P. Simple modeling of self-oscillations in nanoelectromechanical systems. Applied Physics Letters, 96(19), P. 193114 (2010).
3. Hodges D. H., Pierce A. Introduction to structural dynamics and aeroelasticity. Cambridge (2002).
4. Hoque M. E. Active flutter control, LAP Lambert Academic Publishing. Germany (2010).
5. Lukin K. A., Maksymov P. P. Terahertz self-oscillations in reverse biased P-N junctions, in Proc Physics and Engineering of Microwaves, Millimeter and Submillimeter Waves and Workshop on Terahertz Technologies, 2007. MSMW ’07. The Sixth International Kharkov Symposium, Kharkov, 25–30 June, 2007, P. 201–203.
6. Muljadi E., Sallan J., Sanz M. Butterfield C. P. Investigation of self-excited induction generators for wind turbine applications, in Proc. IEEE Industry Applications Conference 1999, vol. 1, P. 509–515.
7. Benettin G., Galgani L., Giorgilli A., Strelcyn J. M. Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them, part 2: numerical application. Meccanica, 15(1), P. 21–30 (1980).
8. Leonov G. A. Chaotic dynamics and classical theory of the dynamic stability, Research center “Regular and chaotic dynamics”, Izhevsk, 186 p.
9. Malinestky G., Potapov A., Podlazov A. Nonlinear dynamics: approaches, results, hopes (synergetics: “from past to future”). KomKniga, Moscow (2006), 280 p.
10. Bespalov A., Chistyakov Y. On the local stability estimation using first Lyapunov exponent calculation, in Proc. Eurocon 2009, Saint Petersburg, 2009, P. 1985–1990.
11. Rosenstein M. T., Collins J. J., De Luca C. J A. Practical method for calculating largest Lyapunov exponents from small data sets Source, Physica D. 65(1-2), P. 117–134 (1993).
12. Wolf A., Swift J. B., Swinney H. L. Determining Lyapunov exponents from a time series. Physica, P. 285– 317 (1985).
13. Gilbert J., Kergomard J., Ngoya E. Calculation of the steady-state oscillations of a clarinet using the harmonic balance technique, J. Acoust. Aoc. Amer., 86(1), P. 35–41 (1989).
14. Nakhla M., Vlach J. A piecewise harmonic balance technique for determination of periodic response of nonlinear systems”. IEEE Transactions on Circuits and Systems, IEEE Transactions on Circuits and Systems, 1976, CAS-23, P. 85–91.
15. Smirnov D. Characterization of weak coupling between self-oscillation systems from short time series: Technique and applications. Journal of Communications Technology and Electronics, 51(6), P. 534–544 (2006).
Review
For citations:
Bespalov A.V., Vilkova E.V. Time-series rate of convergence to quasi-periodic oscillations. Nanosystems: Physics, Chemistry, Mathematics. 2014;5(3):354-362.
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