On certain relaxation oscillations: Confining regions

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Abstract
Relaxation oscillations described by the generalized Liénard equation, d 2 x / d t 2 + μ f ( x ) d x / d t + g ( x ) = 0 {d^2}x/d{t^2} + \mu f(x)dx/dt + g(x) = 0 , with μ ≫ 1 \mu \gg 1 , are investigated in the phase and Liénard planes. When f ( x ) f(x) , g ( x ) g(x) , and F ( x ) = ∫ 0 x f ( u ) d u 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 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 μ − 4 / 3 {\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 + μ ( x 2 − 1 ) d x / d t + x = 0 {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 ) f(x) , g ( x ) g(x) , and F ( x ) F(x) than those first considered.

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