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]

f ( t ) = ∑ k s j 0 , k φ j 0 , k ( t ) + ∑ j ≥ j 0 ∑ k d j , k ψ j , k ( t ) , (1)

where

φ j , k ( t ) = 2 j / 2 φ ( 2 j t − k ) ,     ψ j , k ( t ) = 2 j / 2 ψ ( 2 j t − k ) . (2)

As introduced φ ( t ) and ψ ( t ) we will take Daubechies [3] scaling (scale) φ M ( t ) wavelet basis function ψ M ( t ) and they are defined by the equations:

φ ( t ) = 2 ⋅ ∑ k = 0 2 M − 1 h k φ ( 2 t − k ) ,             ψ ( t ) = 2 ⋅ ∑ k = 0 2 M − 1 g k φ ( 2 t − k ) , (3)

where M: positive integer.

Within the framework of multiresolution analysis [4] functions φ j , k ( t ) and ψ j , k ( t ) serve as a high-frequency and low-frequency filters, RHR, respectively h k . The general properties of the scaling functions and wavelet coefficients are uniquely determined. Example of calculation for Daubechies ( D 4 ) wavelet is given in [1] .

For Daubechies h k wavelets are real and

s j , k = 〈 f , φ j , k 〉 ,           d j , k = 〈 f , ψ j , k 〉 , (4)

where for any f 1 ( t ) and f 2 ( t ) ( L 2 ( R ) . Here its means that L 2 ( R ) space of quadratically summable functions on R

〈 f 1 , f 2 〉 = ∫ R f 1 ( t ) f 2 ( t ) d t . (5)

Using formulas fast wavelet transform

s j , n = ∑ k = 0 2 M − 1 h k ⋅ s j − 1 , 2 n + k ,               d j , n = ∑ k = 0 2 M − 1 g k ⋅ s j − 1 , 2 n + k . (6)

When time signal f ∈ L 2 ( R ) 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

s 0 , k : = 〈 f , φ o , k 〉 = ∫ R f ( t ) φ ( t − k ) d 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

s 0 , k : = f ( k ) ,         k ∈ Z . (8)

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

Denote the matrix elements of the operator in terms of T S S ( n ) ( j , k ; j , k ′ ) , T S D ( n ) ( j , k ; j , k ′ ) , T D S ( n ) ( j , k ; j , k ′ ) , T D D ( n ) ( j , k ; j , k ′ ) . Here

T S S ( n ) ( j , k ; j , k ′ ) = ∫ R φ j , k ( t ) ( T ( n ) φ j , k ′ ( t ) ) d t , (9)

replacement of the lower indices S → D ( D → S ) in the left side of (9) corresponds to the substitution φ → ψ     ( ψ → φ ) of the integral sign on the right side. We denote further r j , l ; j , l ′ ( n ) = T S S ( n ) ( j , l ; j , l ′ ) , r k ( n ) = T S S ( n ) ( 0 , 0 ; 0 , k ) . It’s obvious that r j , l ; j , l ′ ( k ) = r j , k / k = l − l ′ ( k ) , where r j , k ( k ) = r j , 0 ; j , k ( k ) .

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

r j , k ( n ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ h m ⋅ r j , 2 k − i + m ( n ) , ​ ​ ​ ​         ( L = 2 M − 1 ) , (10)

and the elements adjacent levels-ratio

r j + 1 , l ; j + 1 , l ′ ( n ) = 2 n ⋅ r j , l ; j , l ′ ( n ) . (11)

Conditions normalization coefficients r 0 , k ( n ) is determined as [1]

∑ k = 1 L k ⋅ r 0 , k ( n ) = n ! . (12)

If j = 0 (10) for the value of r 0 , k ( n ) (the r k ( n ) ) we obtain the system of equations

r k ( n ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ h m ⋅ r 2 k − i + m ( n ) , ​ ​ ​ ​           ​ ( k = 0 , 1 , ⋯ , L − 2 ) . (13)

In the domain of the L wavelet coefficients r k ( n ) in length, they have the symmetry property

r k ( n ) = ( − 1 ) k ⋅ r − k ( n )               ( k = 1 , 2 , ⋯ ) . (14)

Given a L ′ ≥ n ( L ′ = L − 1 ) system of linear algebraic Equations (12) - (14) has a unique exact solution u = ( r 0 ( n ) , r 1 ( n ) , ⋯ , r L − 2 ( n ) ) . The matrix elements of the differentiation operator recurrently expressed in terms of matrix elements of the op-

erator of differentiation T ( n − 1 ) = d n − 1 d t n − 1 [5] .

Solving the system of linear algebraic equations (12) - (14) n = 1 for all matrix elements r 0 , l ; 0 , l ′ ( k ) = r k ( 1 ) | k = l ′ − l can be recovered from the elements.

T S S ( 1 ) ( j , l ; j , l ′ ) = r j , l ; j , l ′ ( 1 ) ( j ≥ 1 ) 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

T S S ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ h m ⋅ r j , 2 l + i ; j , 2 l ′ + m (15)

replacement indexes S → D and respective substitute h → g , obtain for all j ≥ 0 presentation

T S D ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L h i ⋅ g m ⋅ r j , 2 l + i ; j , 2 l ′ + m ( 1 ) , T D S ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L g i ⋅ h m ⋅ r j , 2 l + i ; j , 2 l ′ + m ( 1 ) , T D D ( 1 ) ( j , l ; j , l ′ ) = 2 ∑ i = 0 L ∑ m = 0 L g i ⋅ g m ⋅ r j , 2 l + i ; j , 2 l ′ + m ( 1 ) . (16)

Let f ( t ) represented as g ( t ) = T ( n ) f ( t ) a series of wavelet coefficients ( s f j , k ;   d f j , k ) and ( s g j , k ;   d g j , k ) . Then

s g j , k = ∑ k ′ = 0 L ( T S S ( n ) ( j , k ; j , k ′ ) ⋅ s f j , k ′ + T S D ( n ) ( j , k ; j , k ′ ) ⋅ d f j , k ′ ) , d g j , k = ∑ k ′ = 0 L ( T D S ( n ) ( j , k ; j , k ′ ) ⋅ s f j , k ′ + T D D ( n ) ( j , k ; j , k ′ ) ⋅ d f j , k ′ ) . (17)

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

T S S ( 2 ) ( j , k ; j , k 2 ) = ∑ k 1 = 0 L ( T S S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S S ( 1 ) ( j , k ; j , k 2 ) + T S S ( 1 ) ( j , k ; j , k 1 ) ⋅ T D S ( 1 ) ( j , k ; j , k 2 ) ) , T S D ( 2 ) ( j , k ; j , k 2 ) = ∑ k ′ = 0 L ( T S S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S D ( 1 ) ( j , k ; j , k 2 ) + T S D ( 1 ) ( j , k ; j , k 1 ) ⋅ T D D ( 1 ) ( j , k ; j , k 2 ) ) , T D S ( 2 ) ( j , k ; j , k 2 ) = ∑ k ′ = 0 L ( T D S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S S ( 1 ) ( j , k ; j , k 2 ) + T D D ( 1 ) ( j , k ; j , k 1 ) ⋅ T D S ( 1 ) ( j , k ; j , k 2 ) ) , T D D ( 2 ) ( j , k ; j , k 2 ) = ∑ k ′ = 0 L ( T D S ( 1 ) ( j , k ; j , k 1 ) ⋅ T S D ( 1 ) ( j , k ; j , k 2 ) + T D D ( 1 ) ( j , k ; j , k 1 ) ⋅ T D D ( 1 ) ( j , k ; j , k 2 ) ) . (18)

Replacing in the equation oscillator Van der Pol x ( t ) , d x / d t and d 2 x / d t 2 expansions in the wavelet basis with the carrier. Applying the formula of differentiation (12) and taking t the points k = 0 , 1 , ⋯ , L , we get in the parameter λ system of linear algebraic equations type

A Λ = b (19)

where A = ( a 0 , a 1 , ⋯ , a L ) T - the vector of one element λ , b = ( b 0 , b 1 , ⋯ , b L ) T 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 A Λ and the vector), i.e. | A Λ − b | → min . Such a solution can be obtained by solving function MATHCAD system.