An Improved Criterion for the Unique Existence of the Limit Cycle of a Liénard-type System with One Parameter

Hayashi, Makoto

doi:10.1007/s12346-021-00463-z

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2022

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“…If x 1 > p and λ 1 ≥ λ * , then we see from the Theorem in [10] that the orbit y B (x) must intersect the half segment l = {(x, y) | x = p, y ≤ λ 1 F (x)}. Thus, we see from Lemma 3, Lemma 5 and the uniqueness of solution orbits that the orbit y B (x) must round clockwise outside Ω and cannot stay in the domain {(x, y) | p < x, y ∈ R}.…”

Section: Main Results and Proofsmentioning

confidence: 96%

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A Survey for a Generalized Li\'enard Equation Without the Massera Condition

Hayashi¹

2022

Int. J. of Appl. Math.

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Massera's method [13] is powerful for the criterion of the unique existence of limit cycles of a classical Liénard equation as is seen in the Van der Pol equation. However, by the reason of which the condition of f (x) is strong (it is the condition that there does not exist the solutions satisfying the equation f ′ (x) = 0 except x = 0), we cannot apply it to many examples. Our purpose is to give several theorems such as can be applied to the generalized Liénard equation without Massera's condition. As an application, the unique existence of limit cycles of the Duff-Levinson system which is well-known in [3] or [15] is discussed as an improvement of [7].

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Section: Main Results and Proofsmentioning

confidence: 96%

“…where λ is a positive real number. For instance, see [7]- [10]. Equation ( 4) is equivalent to the following Liénard system:…”

Section: Main Results and Proofsmentioning

confidence: 99%