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

Abstract

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.

The homotopy analysis method and the Liénard equation

“…A method that gives a sequence of algebraic approximations to the equation of each limit cycle can be found in [10], and a variational method showing that limit cycles correspond to relative extrema of certain functionals is explained in [14]. Here, we are interested in the application of another non-perturbative technique, the homotopy analysis method (HAM), to this problem.…”

<title>Survey of state-of-the-art vibration isolation research and technology for space applications</title>

“…25,22 Conditions for existence of a limit cycle have been shown in the case of both f and g are functions of position 25 and work exists on determining the numbers and locations of limit cycles. 26,27,28,29,30 Numerous authors have studied stability of Equation 2 as: 1) f and g being functions of position, 25,31,21 which has been generalized for n of this type of equations, 32,33 2) f and g being functions of time only, 34,35 3) f (t, x) = c(t)h(x) and g is a function of time and position, 36 and 4) general forms:ẍ + f (x,ẋ, t) |ẋ| αẋ + g(x) = 0 studied by Andreyev and Yurjeva 37 and a (t)ẍ + b (t) f (x,ẋ) + c (t) g (x) h (ẋ) = e (t, x,ẋ) by Athanassov. 38 A gap exists between existing semi-active vibration models and application of these results.…”

Maximum amplitude of limit cycles in Liénard systems

“…They include numerical integration [27] (see Table I), harmonic balance [28,29], variational analysis [30], the Melnikov method [31], empirical analyis [23], piecewise approximation [32], and bifurcation analysis [33]. Most of the methods (Melnikov, piece-wise approximation, bifurcation, and harmonic balance) are valid only for specific ranges of the strength parameter ε.…”

“…(1) have also been employed with success to approximate the amplitude of limit cycles in other Liénard systems where F (x) is an odd polynomial of degree greater than 3. Among these methods are the variational method of Depassier and Mura [30] and the empirical, nonperturbative method of Giacomini and Neukirch [23].…”