Identification of the Oscillator Parameters of the Van der Pol Oscillator Using Wavelet Analysis

ISSN Online: 2333-9721 ISSN Print: 2333-9705

DOI: 10.4236/oalib.1103718 Sep. 22, 2017 1 Open Access Library Journal

Identification of the Oscillator Parameters of

the Van der Pol Oscillator Using Wavelet

Analysis

N. T. Abdullayev, O. A. Dyshin, D. A. Dadashova

Azerbaijan Technical University (Baku), Baku, Azerbaijan

Abstract

A procedure for determining the current parameters of the oscillator is based on the calculation of the wavelet coefficients of the signal system using fast discrete wavelet transform and application of the differentiation formulas wavelet expansions.

Subject Areas

Computational Physics

Keywords

Oscillator, Range of Power, Wavelet-Transform, Area of Synchronization

1. Decomposition of a Function into a Wavelet Series

1.1. The Definition of the Scale Function and the Basis Wavelet

Any function f t

( )

quadratically summable R= −∞ ∞

(

,

)

_{in the space can be}

expanded at a given level of resolution j j= 0 in the wavelet series [1] [2]

( )

_{0 0}

( )

( )

0

, , , ,

j k j k j k j k

k j j k

f t s

ϕ

t d

ψ

t

≥

=

∑

+

∑ ∑

_{, (1)}

where

( )

2

(

)

( )

2

(

)

, 2j 2j , , 2j 2j

j k t t k j k t t k

ϕ = ϕ − ψ = ψ − . (2)

As introduced ϕ

( )

t and ψ

( )

t we will take Daubechies [3] scaling (scale)

( )

M t

ϕ _{wavelet basis function} ψ_M

( )

t _{and they are defined by the equations:}

( )

2 1

(

)

( )

2 1

(

)

0 0

2 M k 2 , 2 M k 2

k k

t h t k t g t k

ϕ − ϕ ψ − ϕ

= =

= ⋅

∑

− = ⋅

∑

− _{, (3)}

How to cite this paper: Abdullayev, N.T., Dyshin, O.A. and Dadashova, D.A. (2017) Identification of the Oscillator Parameters of the Van der Pol Oscillator Using Wavelet

Analysis. Open Access Library Journal, 4:

e3718.

https://doi.org/10.4236/oalib.1103718

Received: June 7, 2017 Accepted: September 19, 2017 Published: September 22, 2017

Copyright © 2017 by authors and Open Access Library Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

DOI: 10.4236/oalib.1103718 2 Open Access Library Journal

where M: positive integer.

Within the framework of multiresolution analysis [4] functions ϕj k,

( )

t and

( )

,

j k t

ψ _{serve as a high-frequency and low-frequency filters, RHR, respectively}

k

h . The general properties of the scaling functions and wavelet coefficients are uniquely determined. Example of calculation for Daubechies

( )

_D4 _{wavelet is}

given in [1].

For Daubechies hk _{wavelets are real and}

, , , , , , ,

j k j k j k j k

s = f ϕ d = f ψ _,

₍₄₎

where for any f t1

( )

and f t2

( )

(L R2

( )

. Here its means that L R2

( )

space of

quadratically summable functions on R

( ) ( )

1, 2 1 2 d .

R

f f =

∫

f t f t t (5)

Using formulas fast wavelet transform

2 1 2 1

, 1,2 , 1,2

0 , 0

M M

j n k j n k j n k j n k

k k

s −h s− + d − g s− +

= =

=

∑

⋅ =

∑

⋅ _{. (6)}

1.2. Dependence of Signal Scale on Its Resolution Level

When time signal f L R∈ 2

( )

analysis should choose, the thinnest scale to the

subsequent signal synthesis receives it in its original form. We can assume that such a scale is associated with the level of permissions j=0_{. Therefore, the}

analysis begins with the calculation

( ) (

)

0,k: , o k, d

R

s = f ϕ =

∫

f t ϕ t k t− .

(7)

These values can be calculated, for example by means of numerical integration. In the case where f the initially set is a discrete array

{

f k

( )

}

, k Z∈ (Z: set of integers), just suppose

( )

0,k: ,

s = f k k Z∈ _{. (8)}

Applying wavelet transform (7), we can now calculate all the factors s dj n, , j n, .

Denote the matrix elements of the operator in terms of ( )n

(

, ; ,

)

SS

T j k j k′ _,

( )n

(

_{, ; ,}

)

SD

T j k j k′ _, ( )n

(

, ; ,

)

DS

T j k j k′ _, ( )n

(

, ; ,

)

DD

T j k j k′ _{. Here}

( )

(

)

( )

(

( )

( )

)

, ,

, ; , d ,

n n

SS j k j k

R

T j k j k′ =

∫

ϕ t T ϕ ′ t t (9)

replacement of the lower indices S→D D

(

→S

)

_{in the left side of (9)}

corres-ponds to the substitution ϕ→ψ ψ

(

→ϕ

)

_{of the integral sign on the right side.}

We denote further rj l j l( ), ; ,n ′=TSS( )n

(

j l j l, ; , ′

)

, rk( )n =TSS( )n

(

0,0;0,k

)

. It’s obvious that

( ) ( )

, ; ,k , /k

j l j l j k k l l

r ′=r = −′, where rj k( ),k =rj( ),0; ,k j k.

Elements rj k( ),n of the same level are related

( ) ( )

(

)

, ,2

0 0

2L L , 2 1

n n

j k i m j k i m i m

r h h r − + L M

= =

=

∑ ∑

⋅ ⋅ = − _{, (10)}

DOI: 10.4236/oalib.1103718 3 Open Access Library Journal

( ) ( )

1, ; 1, 2 , ; ,

n n n j l j l j l j l

r+ + ′= ⋅r ′. (11)

Conditions normalization coefficients r0,( )nk is determined as [1]

( )

0,

1 !

L n k k

k r n

=

⋅ =

∑

.

(12)

If j=0 _{(10) for the value of 0,}( )n k

r (the rk( )n ) we obtain the system of

equa-tions

( ) ( )

(

)

2

0 0

2 L L , 0,1, , 2

n n

k i m k i m

i m

r h h r − + k L

= =

=

∑ ∑

⋅ ⋅ = − .

(13)

In the domain of the L wavelet coefficients rk( )n in length, they have the

symmetry property

( )n

( )

₁k ( )n

(

_1,2,

)

k k

r = − ⋅r− k= . (14)

2. Solution of a System of Linear Algebraic Equations

2.1. Operator Matrix Elements of Differentiation the Signal

Given a L n′ ≥

(

L L′ = −1

)

system of linear algebraic Equations (12) - (14) has a unique exact solution u=

(

r0( ) ( )n,r1n, , rL( )−n2

)

. The matrix elements of the

diffe-rentiation operator recurrently expressed in terms of matrix elements of the op- erator of differentiation ( )1 1

1

d d

n n

n

T

t

− −

− = _[5].

Solving the system of linear algebraic equations (12) - (14) n=1 _{for all}

ma-trix elements ( ) ( )1 0, ;0,kl l k _{k l l}

r ′=r _{= −′} can be recovered from the elements.

( )1

(

)

( )1

(

)

, ; ,

, ; , 1

SS j l j l

T j l j l′ =r ′ j≥ using the recurrence relation (11). The remaining

matrix elements are n=1 _{defined as follows. By implementing the equivalent}

of (10) in the n=1 expression

( )1

(

)

,2 ; ,2

0 0

, ; , 2L L

SS i m j l i j l m

i m

T j l j l h h r + ′+ = =

′ =

∑ ∑

⋅ ⋅

(15) replacement indexes S→D and respective substitute h→g, obtain for all

0

j≥ _presentation

( )

(

)

( )

( )

(

)

( )

( )

(

)

( )

1 1

,2 ; ,2 0 0

1 1

,2 ; ,2 0 0

1 1

,2 ; ,2 0 0

, ; , 2 ,

, ; , 2 ,

, ; , 2 .

L L

SD i m j l i j l m i m

L L

DS i m j l i j l m i m

L L

DD i m j l i j l m i m

T j l j l h g r

T j l j l g h r

T j l j l g g r

′

+ +

= =

′

+ +

= =

′

+ +

= =

′ = ⋅ ⋅

′ = ⋅ ⋅

′ = ⋅ ⋅

∑ ∑

∑ ∑

∑ ∑

(16)

2.2. Representation of Time Functions in the Form of Series with

Wavelet Coefficients

DOI: 10.4236/oalib.1103718 4 Open Access Library Journal

( )

(

)

( )

(

)

(

)

( )

(

)

( )

(

)

(

)

, , ,

0

, , ,

0

, ; , , ; , ,

, ; , , ; , .

L

n n

g j k SS f j k SD f j k

k L

n n

g j k DS f j k DD f j k

k

s T j k j k s T j k j k d

d T j k j k s T j k j k d

′ ′

′=

′ ′

′=

′ ′

= ⋅ + ⋅

′ ′

= ⋅ + ⋅

∑

∑

(17)

The matrix elements are n=2 connected to the matrix elements n=1 by relations with

( )

(

)

(

( )

(

)

( )

(

)

( )

(

)

( )

(

)

)

( )

(

)

(

( )

(

)

( )

(

)

( )

(

)

( )

(

)

)

( )

(

)

( )

(

)

( )

(

)

( )

(

)

( )

1

2 1 1 1 1

2 1 2 1 2

0

2 1 1 1 1

2 1 2 1 2

0

2 1 1 1 1

2 1 2 1

, ; , , ; , , ; , , ; , , ; , ,

, ; , , ; , , ; , , ; , , ; , ,

, ; , , ; , , ; , , ; , ,

L

SS SS SS SS DS

k L

SD SS SD SD DD

k

DS DS SS DD DS

T j k j k T j k j k T j k j k T j k j k T j k j k

T j k j k T j k j k T j k j k T j k j k T j k j k

T j k j k T j k j k T j k j k T j k j k T j

=

′=

= ⋅ + ⋅

= ⋅ + ⋅

= ⋅ + ⋅

∑

∑

(

)

(

)

( )

(

)

(

( )

(

)

( )

(

)

( )

(

)

( )

(

)

)

2 0

2 1 1 1 1

2 1 2 1 2

0

; , ,

, ; , , ; , , ; , , ; , , ; , .

L

k L

DD DS SD DD DD

k

k j k

T j k j k T j k j k T j k j k T j k j k T j k j k

′=

′=

= ⋅ + ⋅

∑

∑

(18)

Replacing in the equation oscillator Van der Pol x t

( )

,d dx t and _d2_{x td}2

expansions in the wavelet basis with the carrier. Applying the formula of diffe-rentiation (12) and taking t the pointsk=0,1, , L, we get in the parameter

λ system of linear algebraic equations type

AΛ =b

₍₁₉₎

where

(

)

T

0, , ,1 L

A= a a a - the vector of one element λ,

(

)

T

0, , ,1 L

b= b b b

T: _{a sign of transposition.}

Overdetermined system (19) does not have an exact solution, so instead of an exact solution is organized search of the vector Λ that will best meet all the equations of the system (19), i.e., minimize the residual (the difference between the vector AΛ and the vector), i.e. AΛ − →b min_{. Such a solution can be}

obtained by solving function MATHCAD system.

3. Advantages of Determining the Parameters of an

Oscillator by Solving a System of Equations in the Space of

Wavelet Coefficients

Thus, the calculation of the original signal wavelet coefficients using fast wavelet transform and application of the formulas of differentiation of the discrete wavelet decomposition reduces the problem of determining the oscillator para-meters directly to solving a system of linear algebraic equations directly in the space of wavelet coefficients, which significantly increases the speed of calcula-tions in comparison with method of direct and inverse wavelet transforms. This approach has undoubted advantages compared with the numerical x t

( )

_signal

DOI: 10.4236/oalib.1103718 5 Open Access Library Journal

suitable choice of wavelet basis. The undoubted advantage of wavelet transform is the possibility of eliminating the incorrectness of the operation of numerical differentiation of noisy time series by transitioning into the space of wavelet coefficients.

References

[1] Dremin, I.M., Ivanov, O.V. and Nechitailo, V.A. (2001) Wavelets and Their Uses.

Successes of Physical Sciences, 171, 465-501.

[2] Blatter, K. (2006) Wavelet Analysis. Basic Theory. Translated from English, Tech-nosphere, 272 p.

[3] Daubechies, I. (2004) Wavelet Ten Lectures. Moscow Izhevsk, IKI, 163 p.

[4] Mallat, S. (1989) Multiresolution Approximation and Wavelets. Transactions of the American Mathematical Society, 315, 69-88.

[5] Dyshin, O.A. (2008) The Method of Calculating the Matrix of Sustainable Perfor-mance in Wavelet Basis Functions of Differential Operators. Proceedings of the Azerbaijan, 27, 76-82.

[6] Koronovsky, A.A., Kurovskaya, M.C. and Khramov, A.E. (2009) Distribution of the Laminar Phases for the Intermittency of Type I in the Presence of Noise. Applied Nonlinear Dynamics, 17, 43-59.

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