On certain relaxation oscillations: Confining regions
Authors:
Peter J. Ponzo and Nelson Wax
Journal:
Quart. Appl. Math. 23 (1965), 215-234
MSC:
Primary 34.45
DOI:
https://doi.org/10.1090/qam/186875
MathSciNet review:
186875
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Abstract: Relaxation oscillations described by the generalized Liénard equation, ${d^2}x/d{t^2} + \mu f(x)dx/dt + g(x) = 0$, with $\mu \gg 1$, are investigated in the phase and Liénard planes. When $f(x)$, $g(x)$, and $F\left ( x \right ) = \int _0^x {f\left ( u \right )du}$ are subject to certain restrictions, a number of analytic curves can be obtained in these planes which serve as bounds on solution trajectories. Piece-wise connection of such bounding curves provide explicit annular regions with the property that solution trajectories on the boundary of an annulus move to the interior with increasing time, $t$. The Poincaré—Bendixson theorem then guarantees at least one periodic orbit within such an annulus. It is shown that the periodic orbits which are isolated by this means are unique within the annulus, hence orbitally stable. The maximum width of the annulus is of order ${\mu ^{ - 4/3}}$, and the amplitude bounds obtained for the periodic solution agree favorably with the known amplitude for the specific case of the van der Pol equation ${d^2}x/d{t^2} + \mu \left ( {{x^2} - 1} \right )dx/dt + x = 0$. The results are generalized to less restrictive $f(x)$, $g(x)$, and $F(x)$ than those first considered.
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A. A. Dorodnytsin, Asymptotic solution of van der Pol’s equation, Prikl. Mat. Mech. 11 (1947) pp. 313–328 Am. Math. Soc. Transl. no. 88
L. Cesari, Asymptotic behaviour and stability problems in ordinary differential equations, Ergebnisse der Mathematik, New Series, vol. 16, Academic Press Inc., New York, 1963 (2nd edition)
J. LaSalle, Relaxation oscillations, Q. Appl. Math. 7 (1949) 1–19
N. Levinson and O. K. Smith, A general equation for relaxation oscillations, Duke Math. Journal 9 (1942) 382–403
J. LaSalle and S. Lefschetz, Stability by Liapunov’s direct method, with applications, Academic Press, New York, 1961, p. 62
M. L. Cartwright, Forced oscillations in nonlinear systems, in Annals of Mathematics Studies, No. 20, edited by S. Lefschetz, Princeton University Press, Princeton, 1950, pp. 152–153
A. A. Dorodnytsin, Asymptotic solution of van der Pol’s equation, Prikl. Mat. Mech. 11 (1947) pp. 313–328 Am. Math. Soc. Transl. no. 88
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© Copyright 1965
American Mathematical Society