Introduction
Most vibrations are nonlinear and transient in nature, occur suddenly and randomly, have varying magnitudes and are difficult to analyze by linearizing and applying standard methods, as each case differs from the other. It is now possible to study nonlinear vibrations analytically using differential equations due to Duffing, van der Pol, Mathieu, etc., or graphically using integral or response curves or phaseportraits (Stoker, 1966; Nayfeh and Mook, 1979; Mickens, 1981; Thompson and Steward, 1989; Srinivasan, 1995). Interesting studies were made on nonlinear oscillations from the solution of equations of motion related to undamped free vibrations, forced vibrations, damped free vibrations and damped forced vibrations. These are characterized by the frequency and amplitude of oscillations. Rao (1992), Sarma et al. (1995, 1997a, b), Sarma and Rao (1998), Potti et al. (1999), Swamy et al. (2003), Muthurajan et al. (2005) and Tiwari et al. (2005) have obtained solutions of the equations of motion of a conservative system by several methods.
The problem of the determination of limitcycles is fundamental in the theory of oscillations of nonlinear nonconservative systems. The limitcycle is a closed integral curve in the phaseplane, which corresponds to a periodic solution of the equation of motion. It has the important that all integral curves in its neighborhood spiral toward it from both outside and inside. The problem can be solved by direct methods only in a few cases. It is a very difficult task to identify the presence of limitcycles for a given differential equation.
Burnette and Mickens (1996) have examined the stability of limitcycles on
a modified van der Pol equation and proposed a criterion for identification
of the stationary values of the amplitude corresponding to the actual limitcycles
of a general nonlinear differential equation, which is represented by the harmonic
oscillator equation, with addition of a small nonlinear term. This study demonstrates
the identification of stable limitcycles of the modified van der Pol equation
in the higher order averaging method from the time derivative of the amplitude
function. It also explains the inadequacy of the criterion as proposed by Burnette
and Mickens (1996).
Analysis
Burnette and Mickens (1996) have considered a modified van der Pol equation:
to provide a criterion for determining the actual limitcycles that occur in
a general nonlinear differential equation:
for which the averaging method of KrylovBogoliubovMitropolsky (KBM) can be applied. Here F is a polynomial function of its arguments.
According to the generalized method of KBM, the nth order approximation to the solution of Eq. 2 is (Mickens, 1981).
Where u_{1}, u_{2},.……. u_{n1} are periodic functions of Ψ with a period 2 π. The quantities a (t) and ψ_{(t0} are defined by
Here K (a) is the time derivative of the amplitude function.
Applying the second approximation (n = 2) of the KBM averaging method to Eq. 1, Burnette and Mickens (1996) have obtained the amplitude Eq. 4, which is rewritten (after defining z = a^{2}) in the form:
For this case, Ā (z) is a cubic polynomial and one of the stationary values
of the amplitude is zero, which is reported as the unstable fixed point. They
have obtained other two stationary values of the amplitude by solving a quadratic
equation and identified them as a stable and unstable limit points. The behavior
of these values was also examined from a plot showing the variation of Ā
(z) with z. Stationary amplitudes can be found from the points of intersection
of the curve Ā (z) with the zaxis.
Though not explained clearly by Burnette and Mickens (1996), one can guess
from the plot that stable amplitudes correspond to the points where the curve
intersects the zaxis from the upper side and unstable amplitudes correspond
to the points where the curve intersects the zaxis from the lower side. This
phenomenon can also be explained from the positive and negative values of
at the stationary values of the amplitude. Stable stationary values of the amplitude
are those at which <
0, whereas unstable stationary values of the amplitude are those at which >
0.
Burnette and Mickens (1996) have observed that the stationary values of the amplitude corresponding to the unstable limit point increases without bound as ε → 0 (referred this to the spurious limitcycle), whereas the stationary values of the amplitude corresponding to the stable limit point is bounded as ε → 0 (referred this to the actual limitcycle). Based on this observation, they proposed a criterion for the determination of the actual limitcycles in the use of higher order averaging techniques. In that criterion, the actual limitcycles correspond to solutions of K (a) = 0 in Eq. 4 that are bounded as ε → 0. And suggested to ignore all other solutions correspond to spurious limitcycles, in the analysis of the properties of the solutions to Eq. 1. However, usage of this procedure to a general nonliner differential Eq. 2 requires expressions for the stationary values of the amplitude in terms of ε for applying the limit condition ε → 0. The task is involved, if one seeks a solution for Eq. 2 applying the higher order averaging method of KBM.
It is very interesting to note that the cubic Eq. 11 of
Burnette and Mickens (1996) reduces to a quadratic equation, if one applies
the limiting condition as ε → 0 and the stationary values of the amplitude
correspond to the first order approximation of KBM averaging method. That is
the reason why one of the stationary values of the amplitude from the cubic
Eq. 11 of Burnette and Mickens (1996) becomes infinity, when
ε = 0. In general, this criterion may not be convenient for identification
of the stationary values of the amplitude corresponding to the actual limitcycles
of a general nonlinear differential Eq. 2 through higher
order averaging methods of KBM.
In second order approximation (n = 2), the functions in Eq. 3
and 5 are obtained as
The amplitude relation (9) of Burnette and Mickens (1996) needs correction due to their erroneous expression for A_{2 }(a).
Defining z = a^{2} and using Eq. 8 and 9
in Eq. 4, one obtains
Where,
It should be noted that α_{1} and α_{2} are greater than zero for ε ≠ 0. Hence, the stationary values of the amplitude can be obtained from the three roots of Ā (z):
For the case ε = 0,
Which imply that
This corresponds to:
For a negligibly small ε, α_{2} in Eq. 12
becomes insignificant and the third stationary value of the amplitude will be
extremely large, whereas the second stationary value of the amplitude will be
close to that obtained from the first approximation of the averaging method.
Identification of stable limitpoint for Eq. 1 is done here
based on the positive and negative values of
at the stationary values of the amplitude as follows:
This leads to the conclusion that Eq. 1 has a single stable
limit cycle for the second approximation of the averaging method of KBM. For
sufficiently small value of ε, the first approximation of the averaging
method will give the properties of the differential Eq. 1.
The addition of terms of higher approximation does not modify the qualitative
character of the solution, but merely modify their quantitative nature slightly.
Results and Discussion
To demonstrate the identification of stable limitcycle of the modified van
der Pol Eq. 1 from the time derivative of the amplitude function,
the parameters in the differential equation are specified as:
Figure 1 and 2 show the variation of Ā
(z) and
with z. when β = 0, the amplitude equation corresponding the second approximation
reduces to that of first approximation, having two stationary v alues of the
amplitude. It can be seen from Fig. 1 that >
0 at z = 0 and
< 0 at z = 4, is the stable limitpoint.
Table 1: 
Variation of
with β at the stationary values of the amplitude, ,
for ε = 0.1 (one of the stationary amplitudes, ) 


Fig. 1: 
Variation of Ā (z) and
with z for ε = 0.1 and β = 0 . Solid line represents variation
of Ā (z) whereas broken line represents variation of .
Star refers to the squared value of the stationary amplitude 

Fig. 2: 
Variation of Ā (z) and
with z for ε = 0.1 and β = 0. Solid line represents variation
of Ā (z) whereas broken line represents variation of .
Star refers to the squared value of the stationary amplitude 
When β = 10, as expected, Ā (z) curve meets the zaxis at three
points.
at one of the stationary points of the amplitude is negative and hence it corresponds
to the stable limitpoint of differential Eq. 1. Table
1 gives the variation of
with β at the three stationary values of (z_{1}, z_{2}
and z_{3}) of the amplitude, .
It is found that z_{2} is the stable limitpoint for all values of ,
whereas, z_{1} and z_{3} are the unstable limitpoints.
It can be seen from Fig. 1 and 2 that the
stable stationary amplitudes correspond to the points where the Ā (z) curve
intersects the zaxis from the upper side (at the point of intersection, <
0). The unstable stationary amplitudes correspond to the points where the Ā
(z) curve intersects the zaxis from the lower side (at the point of intersection,
> 0).
Conclusions
The stable limitcycles of differential Eq. 1 in the higher order averaging method can be identified easily from the time derivative of the amplitude function and the sign of its derivative at the stationary values of the amplitude. Hence the new criterion of Burnette and Mickens (1996) may not possess any additional advantages in identifying the stablelimit point of a modified van der Pol equation.