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In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: : where ''x'' is the position coordinate—which is a function of the time ''t'', and ''μ'' is a scalar parameter indicating the nonlinearity and the strength of the damping. ==History== The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips.〔Cartwright, M.L., ("Balthazar van der Pol" ), ''J. London Math. Soc.'', 35, 367-376, (1960).〕 Van der Pol found stable oscillations, which he called relaxation-oscillations〔Van der Pol, B., "On relaxation-oscillations", ''The London, Edinburgh and Dublin Phil. Mag. & J. of Sci.'', 2(7), 978-992 (1926).〕 and are now known as limit cycles, in electrical circuits employing vacuum tubes. When these circuits were driven near the limit cycle they become entrained, i.e. the driving signal pulls the current along with it. Van der Pol and his colleague, van der Mark, reported in the September 1927 issue of Nature 〔Van der Pol, B. and Van der Mark, J., “Frequency demultiplication”, ''Nature'', 120, 363-364, (1927).〕 that at certain drive frequencies an irregular noise was heard. This irregular noise was always heard near the natural entrainment frequencies. This was one of the first discovered instances of deterministic chaos.〔Kanamaru, T., ("Van der Pol oscillator" ), ''Scholarpedia'', 2(1), 2202, (2007).〕 The Van der Pol equation has a long history of being used in both the physical and biological sciences. For instance, in biology, Fitzhugh〔FitzHugh, R., “Impulses and physiological states in theoretical models of nerve membranes”, ''Biophysics J'', 1, 445-466, (1961).〕 and Nagumo〔Nagumo, J., Arimoto, S. and Yoshizawa, S. "An active pulse transmission line simulating nerve axon", ''Proc. IRE'', 50, 2061-2070, (1962).〕 extended the equation in a planar field as a model for action potentials of neurons. The equation has also been utilised in seismology to model the two plates in a geological fault,〔Cartwright, J., Eguiluz, V., Hernandez-Garcia, E. and Piro, O., "Dynamics of elastic excitable media", ''Internat. J. Bifur. Chaos Appl. Sci. Engrg.'', 9, 2197–2202, (1999).〕 and in studies of phonation to model the right and left vocal fold oscillators. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Van der Pol oscillator」の詳細全文を読む スポンサード リンク
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