Nonlinear systems with fast parametric oscillations.

*(English)*Zbl 0531.34024This paper discusses a fairly standard method using small parameters with the method of averaging [cf., for example Chapter 5 in J. K. Hale ”Ordinary Differential Equations” (1969; Zbl 0186.409] and applies it to study the effects of rapidly oscillatory time dependence in certain systems of ordinary differential equations on the stability properties of solutions of the equations without such oscillatory time dependence. Two types of specific examples are discussed; the first contains three socalled classical equations of the theory of oscillations: specifically, a Duffing equation, a Rayleigh equation, and a van der Pol equation. The second type are equations arising in certain chemical reactions. Since the Duffing equation discussed is of the form ẍ\(+a\dot x-bx+cx^ 3=0\), where a, b, and c are positive constants, it would be of interest to have mentioned an oscillatory system described by this of this type of equation; most standard examples would lead to an equation of this form but with \(b<0\).

Reviewer: G.Seifert

##### MSC:

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

##### Keywords:

small parameters; method of averaging; Duffing equation; Rayleigh equation; van der Pol equation
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\textit{R. Bellman} et al., J. Math. Anal. Appl. 97, 572--589 (1983; Zbl 0531.34024)

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