Existence of limit cycles for generalized Liénard equations

Alsholm, Preben K.

doi:10.1016/0022-247x(92)90387-s

Cited by 28 publications

(42 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Consequently at most k small limit cycles can bifurcate by Hopf from these centers, when we perturb them inside the class of all Liénard systems of degree m = 2k + 1 or 2k + 2, see Zuppa [22], and also Blows and Lloyd [2]. Third, López and López-Ruiz [13] have studied the Liénard systems (1) in what they call the strongly nonlinear regime. In this regime they show that the conjecture is true when m is odd.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 97%

“…Second, it is known that systems (1) have a center at the origin if and only if a i = 0 for all i's odd, and that these a i with i odd are the Liapunov constants of systems (1). Consequently at most k small limit cycles can bifurcate by Hopf from these centers, when we perturb them inside the class of all Liénard systems of degree m = 2k + 1 or 2k + 2, see Zuppa [22], and also Blows and Lloyd [2].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…We remark that it is not necessary that F (x) be an odd function as in the papers [1,14]. Theorem 3.…”

Section: Corollarymentioning

confidence: 96%

See 2 more Smart Citations

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

Chen

Llibre

Zhang

2007

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

Section: Introduction and Statement Of The Resultsmentioning

confidence: 97%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

Chen

Llibre

Zhang

2007

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…A bound on the value of ν for which no limit cycles exist is known [24]. The criterion for ν → ∞ also indicates that no limit cycles exist in this regime.…”

Section: Examplesmentioning

confidence: 99%

Variational approach to a class of nonlinear oscillators with several limit cycles

Depassier

Mura

2001

Phys. Rev. E

View full text Add to dashboard Cite

We study limit cycles of nonlinear oscillators described by the equationẍ + νF (ẋ) + x = 0 with F an odd function. Depending on the nonlinearity, this equation may exhibit different number of limit cycles. We show that limit cycles correspond to relative extrema of a certain functional.Analytical results in the limits ν → 0 and ν → ∞ are in agreement with previously known criteria.For intermediate ν numerical determination of the limit cycles can be obtained.

show abstract

“…Therefore, the zero solution of system (14) is asymptotically stable. Hence, the equilibrium X =Ẋ = 0 of Eq.…”

Section: Theorem 3 Let the Control In System (10) Be Defined By The mentioning

confidence: 97%

On the asymptotic stability of equilibria of nonlinear mechanical systems with delay

Aleksandrov

Zhabko

2013

Diff Equat

View full text Add to dashboard Cite

We consider some classes of nonlinear mechanical systems with retarded argument. It is assumed that, in the absence of delay, the systems in question have asymptotically stable equilibria. We analyze how the delay affects the stability of these equilibria. The Lyapunov function method and Razumikhin's approach are used to derive conditions under which asymptotic stability is preserved for arbitrary delay values. We suggest a method for stabilizing strongly nonlinear conservative systems by constructing a delay feedback control depending only on the generalized coordinates.

show abstract

Existence of limit cycles for generalized Liénard equations

Cited by 28 publications

References 3 publications

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

Variational approach to a class of nonlinear oscillators with several limit cycles

On the asymptotic stability of equilibria of nonlinear mechanical systems with delay

Contact Info

Product

Resources

About