On the Period-Amplitude Relation by Reduction to Liénard Quadratic
Equation
SVETLANA ATSLEGA1,2, FELIX SADYRBAEV2
1Department of Mathematics,
Latvia University of Life Sciences and Technologies,
Liela street 2, Jelgava,
LATVIA
2Institute of Mathematics and Computer Science,
University of Latvia,
Rainis boulevard 29, Riga,
LATVIA
Abstract: We apply Sabatini’s transformation for the study of a class of nonlinear oscillators, dependent on
quadratic terms. As a result, an initial equation is reduced to Newtonian form, for which in a standard way the
period-amplitude relation can be established.
Key-Words: Differential equations, Oscillation, Period-Amplitude relation.
Received: August 26, 2022. Revised: March 24, 2023. Accepted: April 19, 2023. Published: May 8, 2023.
1 Introduction
The problem of vibrations in mechanical systems
is important from a practical point of view and
interesting as a descent object of investigation for
theoreticians. The mathematical models often are
formulated in terms of the second-order ordinary
differential equations. The second-order oscillators
are of great importance in mechanics, engineering,
and other practical areas. The oscillatory behavior
of solutions was and continues to be an object of
intensive studies (books, [19], [7], [15]). The main
characteristics of these oscillators are the
amplitude and periods of solutions, as well as the
amplitude-period relations, [2], [3], [4], [5], [6],
[8], [9], [10], [11], [12], [14], [16], [17], [21]. This
theory arose from elementary harmonic
oscillations represented by equation
(1)
An interesting question arises immediately. What
happens if the coefficient is not constant? If it
depends on the independent variable t, then the
equation is still linear, and the linear theory
applies. But if is dependent on or (or
both), the equation can become nonlinear. The
remarkable feature of nonlinear equations is that the
oscillation amplitude is dependent on the period of
a solution, and vice versa. In a series of papers, [8],
[9], [10], [11], [12], [14], this problem was treated
using the “ancient Chines” technique.
For instance, in the work, [11], the nonlinear
oscillator
(2)
was investigated. The trial solution in the form
was used, where is the
frequency to be determined. For equation (2) the
frequency-amplitude relation was found in the form
(3)
Another approximate relation was found using “the
ancient Chinese inequality called Chengtian’s
inequality”, [13]. The resulting formula is
(4)
Let us look at this problem from a different
point of view. Rewrite equation (2) as
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(5)
It resembles the classical Liénard equation
(6)
and the generalized one
(7)
The second term in (5), however, is quadratic.
In the work, [20], a special transformation was
invented which can be used for the reduction of the
equation
(8)
to the Newtonian form
(9)
We will apply this transformation to the study of
our selected cases. We are interested in establishing
the relation period versus amplitude.
2 The Scheme of the Study
Consider the equation
(10)
Introduce the new variable u by the relation
(11)
where
. Then
.
Therefore is a monotone function and, as a
consequence, the inverse function exists. If
is an arbitrary solution of (10), then the
corresponding function
(12)
satisfies the second-order conservative equation
(13)
where . Equation (13) has an
integral
(14)
where
The purpose of this article is to use the
described approach to equations of the form (8).
Several cases will be considered.
Since we are interested in periodic solutions of
differential equations, the following assertion is
important.
Proposition 2.1. If is a periodic solution
of the equation (15), then the corresponding
function , obtained by the formula (11), is the
periodic solution of (13).
Proof. If is a periodic solution of (15),
then , for some
. The respective trajectory in the -
phase plane is closed. The respective solution ,
defined by (11) is also periodic, because
. Due to
the autonomy of both equations (15) and (13), these
solutions on a phase plane are represented by closed
trajectories.
3 Equation
First, notice that this equation can be written in the
form
(15)
This is a Liénard type equation with quadratic
dependence on . The variable change, described
above, is applicable.
Equation (15), written as
, can be considered as a perturbation of the
harmonic equation, which is known to have
periodic solutions. Does equation (15) have a
periodic solution? Let us write equation (15) in the
form (13). We get ,
,
The latter can be
written as the differential equation, given the initial
condition,
(16)
Evidently, is strictly monotonically
increasing function with the graph, symmetrical
with respect to the origin and passing through the
origin. It is known as the function
,
[22]. The inverse function exists and has
similar properties. Both functions are depicted in
Figure 1.
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Fig. 1: Blue: , red: .
The function is the solution of the Cauchy
problem
(17)
The function in equation (13) is
(18)
The functions and are depicted in
Figure 2.
Fig. 2: Blue: , red: .
Equation (13) has the only equilibrium
and, therefore, the periodic solution ought to
oscillate around it.
Let us apply the above transformation to
equation (2) written in the form (15).
(19)
The graphs of and the inverse function
are depicted in the Figure 3 below (blue – , red
- ).
Fig. 3: Left: u(x), right: x(u).
The inverse function exists and the
graph is symmetrical with respect to the bisectrix.
Equation (13) in this case is
(20)
Periodic solutions of the equation (2) are in
one-to-one correspondence to periodic solutions of
(20).
3.1 Period-amplitude Relation
Our goal in this subsection is to state the
frequency-amplitude, or, which is almost the same,
period-amplitude relation for the equation (15). For
this, we have the function , defined in (19), or
as a solution to the Cauchy problem (16). We have
the inverse function , which can be obtained
explicitly or numerically as in (17). The solution
of the initial value problem
(21)
has a counterpart , which solves the initial
value problem
(22)
since
.
It follows that if is a periodic solution of
the problem (21), the same is a solution of the
problem (22), and their periods are equal. As an
illustration, and are depicted in Figure
4.
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Fig. 4: Red: , blue: .
It follows that
where is the primitive for
. One has, in a standard manner, that
and, at the same time,
, where M is the maximal va+lue of
. Hence formula for the amplitude of a solution
of the initial value problem (19) is
(23)
or, equivalently,
(24)
Periods of solutions of the problem (21) and
of the problem (22) are the same. The period
T of a solution to the problem (22) can be found in
the relation
(25)
The amplitudes of solutions to the problem (21) and
(22) may differ. The relation between the amplitude
of and of is
(26)
Table 1. The amplitudes A of x(t) versus M of u(t).
Period
Amplitude
Amplitude
0.1
6.32
0.10
0.10
0.5
6.11
0.47
0.49
1.0
5.72
0.82
0.92
1.5
5.40
1.06
1.30
2.0
5.16
1.26
1.69
4 Equation
The equation under investigation is
(27)
Suppose that
(28)
We get, using the transformation
(29)
that the equation (27) takes the Newtonian form
(30)
where is the inverse
function of .
Both functions and are odd
functions, in view of (28). Their graphs are
symmetrical with respect to the origin. Then
Therefore, if is a solution of the equation
(30), then also, is an arbitrary
constant. The function is a solution also,
since
The positive and negative amplitudes of
have the same absolute value, denote it M again.
One has for M, using the integral relation
(31)
that for a solution with the initial conditions
(32)
holds
(33)
where T is the period.
Let be a solution of (27) with the initial
conditions
(34)
Both solutions and have the same
period . Let A be the amplitude of . In
view of (29), the relation between T and A is
(35)
for given. The relations (33) and (35) fully
describe the period-amplitude relation.
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5 Generalizations
Using the same approach, more general oscillatory
equations can be investigated.
More general equations of the form
(36)
can be studied by using Sabatini’s transformation.
The function f in (36) can be interpreted as the
stiffness coefficient, dependent generally on
. The equation with a “generalized” stiffness
coefficient may be in the form
(37)
It can be represented as
(38)
In this form, it is convenient to study using
Sabatini’s transformation.
There are many articles on the subject, for
instance, [16], [18], [11]. The main problem solved
is the relation between oscillation frequency and
amplitude. Equations of the form (38) can be shown
to have multiple embedded period annuli.
Investigation of oscillation of solutions, in that
case, deserves special attention.
Multiple cases are possible, depending on
whether the function has an asymptote for
or , or two asymptotes.
The equation in the next section can be studied
in the generalized form
In particular, it was shown in the work, [1],
that a period annulus may appear in this equation,
while in the shortened equation without the
middle-term period, annuli are absent. The function
g(x) was chosen as a polynomial of the 5th degree.
In the work, [23], the above equation was
studied together with the Dirichlet boundary
conditions. The number of solutions was estimated
provided that g(x) is a cubic polynomial.
6 Equation
Consider equation
(39)
This equation and its generalization were studied in
the paper, [1]. The effectiveness of Sabatini’s
transformation was tested and confirmed. This
equation exhibits an especially simple relation
between the original one and its counterpart in
terms of the variable u.
Proposition 6.1. Equation (39) by Sabatini’s
transformation turns into an equation
(40)
Proof. Indeed,
,
Then
(41)
Since
,
, the
integral of (41) is
(42)
for a solution of (41) with the initial conditions
(43)
One has, as before, that
(44)
and the relation between amplitude A of ad
the period T is
(45)
for given.
7 Conclusion
A relatively broad class of nonlinear oscillators can
be treated using Sabatini’s transformation.
Relations between period/frequency and the
amplitudes of oscillation can be established with
the accuracy, allowed by used computational
instruments. In further studies of nonlinear
oscillators represented by the equations of the form
focus can be made on
the coefficient f(x). The new variable u can be
defined on a bounded interval, in contrast with the
variable x. This is the case if the integral in (11) is
convergent. A great variety of variants are possible
if f(x) is a somewhat arbitrary polynomial with
multiple zeros. If g(x) is a polynomial of
sufficiently high degree, period annuli can appear
in a related equation The
interrelation between period annuli in this
shortened equation and the above one with the
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middle term is interesting for practical purposes and
challenging for theoreticians’ problems. An
interesting problem is to compare the equation in
question with its dissipative counterpart
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