On the Homotopy-First Integral method

for non-conservative oscillators

ANDRÉS GARCÍA

GESE-Departamento de Ingeniería Eléctrica

Universidad Tecnológica Nacional-Facultad Regional Bahía Blanca

11 de abril 461, Bahía Blanca, Buenos Aires

ARGENTINA

Abstract: - This paper presents a ready-to-use formula for determining the number and approximate location of

periodic orbits in second-order Lienard systems. As a result of the exact closed-form derived in [16], in which

an ordinary differential equation (ODE) must be solved to determine the existence and location of periodic

orbits for general non-conservative oscillators, a homotopy functional is defined for Lienard-type systems. This

provides a closed-form and ready-to-use polynomial formula with roots as an approximation of the periodic

orbit's amplitude.

In addition, some examples are analyzed, along with conclusions and future plans.

Key-Words: -Periodic orbits, Homotopy method, First-integral, Series expansion

Received: May 6, 2023. Revised: May 3, 2024. Accepted: June 15, 2024. Published: July 16, 2024.

1 Introduction

It is well known that dynamical systems

written as a single or a collection of ordinary

differential equations (ODEs) need to be evaluated

to determine their parametric properties (see for

instance [1], [2] and [3]).

Oscillations and periodic orbits are key behaviors of

general ODEs ([4]). Determining periodic orbits can

be complex and sometimes impossible using closed-

form or accurate approximate formulas or reduced-

order models ([5], [6]).

Even second-order ODEs in the plane can exhibit

intricate periodic patterns that defy closed-form

analysis (see [7]).

Simplified versions of these systems have been

proposed as toy models for such complexity ([8]).

Two key studies in this field are discussed in [9]:

 Determine all possible periodic orbits

with a given initial amplitude

 Time parameterization in time

The first case involves determining a single number,

the period, while the second case is more complex

and intricate. See references [4] and [10] for further

details.

It is well-known that a second-order oscillator has a

non-differential relation between its period and

amplitude known as a first- integral (see for instance

[11] and [12]). However, computing this first-

integral can be challenging due to complex

integrals.

In [13], a reduced dynamics approach was

introduced to calculate a subset of periodic orbits by

solving an equivalent first-order ODE. Applying

this method to Mickens’ oscillator (referenced in

[14]-[15]) yields a closed-form formula for their

amplitude-period relation ([16]).

Solving the reduced order ODE presented in [16]

provides a complete solution to upper bound the

number of limit cycles for Lienard systems ([6]).

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DOI: 10.37394/232021.2024.4.3

Andrés García

E-ISSN: 2732-9976

Volume 4, 2024

However, estimating the location and amplitudes of

periodic orbits remains challenging.

This paper presents a formula for identifying the

number and location of periodic orbits in second -

order Lienard systems. It introduces a homotopy and

a reduced order ODE from a previous study,

offering a polynomial formula with roots to

approximate the amplitude of the periodic orbits.

This paper is organized as follows: Section

2 outlines the method with an appropriate

homotopy, Section 3 provides application examples,

Section 4 discusses precision and the potential

Artificial Computational Intelligence extensions,

and Section 5 offers some conclusions and future

research directions.

2 The Homotopy-First Integral

method

According to [13] and [16], given a second

order ODE:

󰇘󰇛󰇜 󰇛󰇛󰇜󰇗󰇛󰇜󰇜

Where 󰇗 means time derivatives, a periodic orbit of

amplitude A exists if and only if the following first

order ODE possess solution for   󰇟󰇠:

󰇛󰇜

 󰇛󰇜

󰇛󰇜  󰇛󰇜 

If we restrict ourselves to the Lienard’s case:

󰇛󰇗󰇜   󰇛󰇜

  󰇗

Then, it is possible to define an homotopy (see for

instance [17]):

 

 

()

1 ( )

( ) ( )

( ) ( ) 0, 0,1

dF x

dF x d x

p x x x p

dx dx







   







       





(1)

Assuming the classical series representation for



(x)

([17]):

󰇛󰇜 󰇛󰇜



 (2)

Replacing (2) into (1), the homotopy equation to

solve leads:

󰇛󰇜

  󰇛󰇜



    󰇡   󰇛󰇜



 

󰇛󰇜





 󰇢     󰇟󰇠 (3)

To maintain the equality with zero, every all

coefficients in this series must be zero:



󰇛󰇜

       󰇛󰇜  

󰇛󰇜󰆒

Where 󰇛󰇜󰆒󰇛󰇜

 .



󰇛󰇜

     󰇛󰇜 



󰇛󰇜

  󰇛󰇜 󰇛󰇜 󰇛󰇜󰆒  

󰇛󰇜 󰇛󰇜 󰇛󰇜

󰇛󰇜

Where 󰇛󰇜  󰇛󰇜

 . The sequence can be

continued, by adding more terms, but (2) can be also

truncated at i=3:

󰇛󰇜 󰇛󰇜 󰇛󰇜  

󰇛󰇜󰆓󰇛󰇜󰇛󰇜󰆒

󰇛󰇜󰆓

󰇡󰇡 

󰇛󰇜󰆓󰇢󰇛󰇜󰆒󰇢

󰇛󰇜󰆒 , 󰇛󰇜  (4)

Equation (4) can be further developed:

( )' ( )''

( )' ( )'

( ) , ( ) 0

( )'

x F x x F x

xF x F x









  















(5)

Then, replacing x=A in (5):

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DOI: 10.37394/232021.2024.4.3

Andrés García

E-ISSN: 2732-9976

Volume 4, 2024

( )' ( )''

( )' ( )' ( )' ( )''

( )' ( )' ( )'

A F A A F A

AF A F A A F A A F A

F A F A F A







  



 







     











(6)

Where 󰇛󰇜󰆒 󰇛󰇜 and 󰇛󰇜󰆒󰆒 

󰇛󰇜. It is important to note that:

If ( )' 0 (3) does not depend on ( )F A F x

Finally, from (6) with 󰇛󰇜  :

󰇛󰇜 󰇛󰇜󰆒   󰇛󰇜 (7)

3 Application examples

Using (7), we can predict the presence of limit

cycles and estimate their amplitudes.

Example 1

Considering the well-known Van der Pol’s oscillator

([18]):

󰇛󰇗󰇜   󰇛 󰇜 󰇗

Equation (7) leads:

󰇛󰇜 󰇛󰇜󰆒   󰇛󰇜󰆒󰆒 󰇛   󰇜

 󰇛   󰇜     

That is   . In numerical simulations, this

oscillator’s the amplitude is approximately A=1.16.

Example 2

In [19], the following Lienard system was

considered:

󰇛󰇗󰇜   󰇛           󰇜 󰇗

In other words:

( ) 0.8 0.32

'( ) 0.8 4 0.32 5

"( ) 8 0.32 20

F x x x x

F x x x

     



     



     





Then, equation (7) yields:

 

12 10 8

( ) ( )' ( )"

4.096 30.72 82.944 94.72

36.672 3.68 1.312 0

'F A F A A F A

A A A A

    



      



    

Finding the real roots:

1.72721

0.803071



Based on the values provided in [19]:

1.8, 1AA

, the accuracy obtained is around

80% in the worst case.

Example 3

For instance, in [19], the paper introduces the

following parameterized system:

󰇛󰇗󰇜   󰇛       󰇜 󰇗

Two limit cycles are present for µ>2.5, while no

limit cycles are observed for µ<2. Applying

equation (7):

 

2 4 2

()

'( ) 1 3 5 Replacing

"( ) 6 20

1 3 5 15 3 1

F x x x x

F x x x A y

F x x x

y y y y





   

        



     



           

Obtaining Fig. 1 and Fig. 2:

Figure 1: Parametric curve for Example 2

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DOI: 10.37394/232021.2024.4.3

Andrés García

E-ISSN: 2732-9976

Volume 4, 2024

Figure 2: Zoom in the parametric curve

Clearly, for µ>0.55 two limit cycles exist.

Example 4

Building on the previous discussion in paper [19],

we can now consider a final example:

6 4 2

( , )

7 ( 11.1143 29.6914 13.1657)

x f x x

x x x x x



         

Again, equation (7) yields:

     

 

2 2 2 2

6 4 2

1 2 3

( ) 1.6 4 9

'( ) 7 11.1143 29.6914 13.1657

"( ) 42 7.40952 9.89714

0.7701, 1.8380, 2.7133

F x x x x x

F x x x x

F x x x

A A A

      



       



    





   

Figures 3 and 4 in Matlab illustrate a potential limit

cycle with an approximate amplitude of 3.1 in x(t).

In this way, the homotopy-first integral serves as

both a numerical method for approximating periodic

orbit amplitudes and an upper bound for the number

of limit cycles.

Based on the conjecture from reference [18] of a

single limit cycle without specifying its amplitude,

the method in this paper approximates this

amplitude value with A3 above, suggesting a

possible maximum of three limit cycles.

Figure 3: Limit cycle in x(t)

Figure 4: Limit cycle’s phase portrait

Example 5

Practical applications utilizing oscillation circuits in

power electronics, including DC-DC, DC-AC, and

LED drivers, can be achieved through the use of

relaxation oscillator circuits [21]:

󰇛󰇗󰇜     󰇛 󰇜 󰇗

An harmonic oscillator is obtained when λ=0 in

equation (7):   

The polynomial approximation is inaccurate for

pure harmonic oscillators due to the third-order

truncation in equation (7). However, for λ≠0, the

condition for periodic orbits can be determined:





The amplitude obtained wit the border value λ=1/27

using equation (7) is A=5.3805. This value is higher

than the conjectured value in [20] which falls

between 2 and 2.0235, indicating an error of more

than double. The following section will provide

insights into the accuracy issue.

4 Discussion

4.1 Approximation accuracy

To assess the accuracy of our approximation

formula (7), we can analyze the classical Van der

Pol equation in more depth:

󰇛󰇗󰇜     󰇛 󰇜 󰇗

According to [20] pp.7, the unique limit cycle has

an amplitude between 2 and 2.0235 for all µ>0.

When formula (7) is applied:

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Andrés García

E-ISSN: 2732-9976

Volume 4, 2024

   

3 2 2 2

1 1 2A A A

  

        

In other words:

 







(8)

Plotting the curve µ-A as shown in Fig. 5:

Figure 5: A-µ curve using (7)

Figure 6: Figure 5 zoom

The conjectured interval   󰇟󰇠 is

approximated by the curve in Fig. 5 with  

󰇟󰇠.

On the other hand, for A<10, an interval  

󰇟󰇜 is covered. Furthermore, by calculating

the estimated amplitude using equation (8) and

comparing it to the minimum amplitude suggested

in [20], we observe that near-zero or zero error is

achieved when µ values are close to unity (refer to

Figures 7 and 8). It is also beneficial

to compare the small values approximation in [20]:

(      ) with equation (8), as illustrated

in Figure 9.

Figure 7: Relative error

Figure 8: Figure 7 zoom

Figure 9: Comparison with reference [20] for small

values of µ

4.2 Artificial/Computational Intelligence

extensions

Possible extensions to a broader scope include

utilizing Deep Neural Networks (DNN) for

simulations based on collected data from the non-

linear oscillator ([22] and [23]). This approach can

help focus on central points for more accurate

results.

5 Conclusion

This paper presents a method for determining

periodic orbits of general nonlinear oscillators by

using closed-form reduced order ODEs derived in a

previous study. An homotopy approach is developed

to combine with the resulting polynomial from

applying the reduced order ODE to Lienard systems.

By defining a classical infinite sum as a solution for

the homotopy and truncating it around its third term,

formula (7) offers a quick and efficient way to

determine both:

 Approximate number of periodic orbits

 Approximate amplitude of periodic orbits

The real roots of the polynomial indicate the

locations and quantities of periodic orbits in a

Lienard system.

In future work, a higher truncation order will be

used in the homotopy defined in this paper. The

extensions discussed here will be further

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DOI: 10.37394/232021.2024.4.3

Andrés García

E-ISSN: 2732-9976

Volume 4, 2024

investigated for general non-linear oscillators,

beyond just Lienard oscillators, as the scope of the

paper [16] is broad.

As suggested exploring the potential combination of

artificial intelligence or computational intelligence

is an important future research direction.

The focus will be on extending the formulas and

homotopy from this study to develop

a general method for approximating and validating

oscillations with diverse behaviors, particularly in

neuroscience and power electronics, where non-

linear oscillators play a vital role.

Acknowledgement:

This work is supported by Universidad Tecnológica

Nacional-Facultad Regional Bahía Blanca,

Departamento de Ingeniería Eléctrica and GESE.

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Volume 4, 2024

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The author contributed in the present research at all

stages from the formulation of the problem to the

final findings and solution.

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Scientific Article or Scientific Article Itself

The author would like to acknowledge

Departamento de Ingeniería Eléctrica at Universidad

Tecnológica Nacional and GESE.

Conflict of Interest

The author have no conflicts of interest to declare

that are relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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EQUATIONS

DOI: 10.37394/232021.2024.4.3

Andrés García

E-ISSN: 2732-9976

Volume 4, 2024