weakly non-linear time delay systems with modified parameters

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International Journal of Scientific Research and Engineering Studies (IJSRES)

Volume 1 Issue 4, October 2014

ISSN: 2349-8862

www.ijsres.com Page 1

Weakly Non-Linear Time Delay Systems With Modified Parameters

Abstract: Delay in a gi ven system makes the dimension

of the phase space high. I n th is paper, it i s shown that the

modif icati on of parameters method can be appli ed to weakly

non- li near systems with delay. For systems with order one

delay, the method leads to reduced systems without delay.

For systems with l arger delay, the method is extended and

the appli cation of the extended method l eads to reduced

systems with delay. The extended method can also be applied

to systems with order one delay; the resultant reduced

equations are dif ferent f rom those obtained by the use of the

conventional r enormalization method. A time delay term of a

given system makes the dimension of the phase space hi gh.

Even in such a case, this method can lead to reduced

equations.

Keywords: Parameters; Weakly non-Linear Systems;

Pertu rbation; Renormalization.

I. INTRODUCTION

Many non-linear dynamical systems in various scientificdisciplines are influenced by the finite propagation time ofsignals in feedback loops. A typical physical system is

provided by a laser system where the output light is reflectedand fed back to the cavity. Time delays also occur in othersituations; for example, in traffic flow model including adrivers reaction time, in Biology due to physiological controlmechanisms, or in economy where the finite velocity ofinformation processing has to be taken into account.Moreover, realistic models in population dynamics or inecology include the duration for the replacement of resources.

In some situations, such as lasers and electromechanicalsystems, systems with large-delay appear. It is with this reasonthat there is need for a Mathematical tool to study them,especially for weakly non-linear systems.

If a perturbation term were added to a given system, thesystem will not be guaranteed to be structurally stable. So the

perturbation result is plagued with singularities. It has beenrecognised that these singularities in the result of the

perturbation method can be renormalized away by themodification of parameters associated with the unperturbedsystem. The modified parameters are governed by therenormalization equations that turn out to be reducedequations. To obtain a more useful and sophisticated tool to

study weakly non-linear systems, reformulated versions of theoriginal method have been used. There are a variety ofapplications of renormalization method to physical systems,

such as plasma physics, general relativity and quantum optics,in addition to studies in standard non-linear dynamicalsystems.

II. CONVENTIONAL RENORMALIZATION METHOD

A.

LINEAR SYSTEMS

Consider the following system,2

2

2

( )( ) ( ) 0

d x tx t x t

dt (2.1)

where is a parameter, represents the time

delay, and is a small parameter where 1 . In this

system there is an oscillatory solution that is can be expressedanalytically without any approximation. The exact solution isgiven by;

2

( ) exp( ) .x t A it c c (2.2)under the condition that;

2.

(2.3)

Here c.c represents the complex conjugate terms of the

proceeding expression, A is the integration constant. Inthe perturbation analysis, the exact solution (2.2) is not used.As in the case of a differential equation without delay, the

perturbation solution,(0) (1) 2 (2) 3( ) ( ) ( ) ( ) ( )x t x t x t x t is found.

This perturbation solution is obtained by solving the

equations;(0) ( ) 0;Lx t ( ) ( 1)( ) ( ),j jLx t x t

( 1,2,....)j , where2

2

2( ) ( )

dLx t x t

dt

We obtain;

(0) .i tx t Ae c c , (1) ( ) .2

i tiAx t te c c

,

(2) 2 ( 2 )2

2 . .,8

i tA ix t t t t e c c

where A

is the integration constant of the solutionof the unperturbed system,

0 ( )x t . The solutions

Okwoyo, M. JamesSchool of Mathematics, University of Nairobi, Nairobi,

Kenya


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( ) ( ),jx t 1j contain the terms const.exp ( )i t and

const.exp ( )i t . These terms are included in

exp( )A i t and its complex conjugate term in (0) ( )x t .

Apparently the validity of the perturbation is invalid in theregime

1( )t

due to the terms 2 2( , )t t etc

Define the normalized variable as;

2 2 2

2 2

2 8

i iiA A iA A te t t t e

(2.4)

This definition is a form of near identity transformation atthe constant A and the perturbation solution is expressed interms of the normalized variable;

3( ) ( ) exp( ) . ( )x t A t i t c c (2.5)

The equation which ( )A t should satisfy perturbatively is

then constructed and such an equation is the renormalizedequation. From equation (2.4), the following two relations can

be obtained

2 2 2 3

2( ) 2 2 ( )

2 8

i iiA A iA t A e t e

(2.6)

and

2( )( ) ( )2

iiA tA A t te

(2.7)

where is a parameter. Substituting equation (2.7)into equation (2.6), we have the approximate closed relation;

2 2 3

2 3

( ) ( ) ( )( ) ( , )

2 4 8

i iA t A t iA t ie A t e

The renormalization equation can then be obtained as the

limit 0 ;

2 2

2 3

( ) ( )( )( )

2 4 8

i idA t iA t ie A t edt

(2.8)

The solution of equation (2.8) is given by;( )( ) (0) tA t A e , where

2 2

2 3( ) ( )2 4 8

i ii i

t t e e

(2.9)

To compare the solution of the renormalization methodwith the exact solution (2.2), it is restricted to the approximatesolution imposed on the condition (2.3). Using equation (2.3),equation (2.9) can be written as;

2 3

3( ) (0) exp( ( )

2 8

it it A t A

(2.10)

In terms ofx (t),we obtain the approximate solution usingequation (2.5) and (2.10) as;

23 3

3( ) (0)exp ( ( )) . ( )

2 8x t A it c c

(2.11)

In fact, due to the

relation 2

2

32 8

3( ) , equation

(2.11) same as equation (2.2) up to the second term,2( )

B. NON-LINEAR SINGLE OSCILLATOR

Consider the non-linear equation;

31 3( )

( ) ( ) ( ) ( )dx t

x t x t x t x tdt

(2.12)

where 1, , and 3 are parameters. The

value of represents the time delay, and is thesmall parameter. The unperturbed system has analyticallyexpressible oscillatory solution when the following conditionis satisfied;

1

2 2

cos ( )

;2 2 (2.13)

The oscillatory solution of the unperturbed system is

given by;(0) ( ) .i tx t Ae c c

where, 2 2 ,A is the integration

constant, and the relation exp( )i i issatisfied. The exact solution can be expressed analytically

though it is difficult in the case where. 0 Therenormalization equation is constructed to investigate theeffect of the perturbation term. The perturbation solution

(0) (1) 2( ) ( ) ( ) ( )x t x t x t is obtained by solving thefollowing equations;

(0) ( ) 0L x t (1) (0) (0)3

1 3( ) ( ) ( )L x t x t x t , where

( )( ) ( ) ( )

dx tL x t x t x t

dt

The solutions are given by;

(0) ( ) .i tx t Ae c c .,2

1 3(1) 3

( ) .1 ( )

i tA A A

x t t e c ci

.,

By defining the renormalization variable ( )A t by;2

1 33

( )1 ( )

A A AA t A t

i

,the renormalization

equation is thus obtained as;2

1 3( ) 3 ( ) ( )( )

1 ( )

A t A t A tdA t

dt i


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This system can be split into two parts; dynamicsdescribed by its amplitude and phase;

22 2 1

3

3

3 ( )3

dRQR R

dt

(2.14)

23 1

2 2

3

3

(1 ) ( ) 3

dR

dt

(2.15)

Here;( )( ) ( ) i tA t R t e and

2 2

2 (1 )

(1 ) ( )Q

. From

Equations (2.14) and (2.15), R R ,1

33

R

is a

stable fixed point when 1

3

0

and 3 0Q . Thus the limit-

cycle oscillation appears when the conditions are satisfied.

C. LANG-KOBAYASHI PHASE EQUATION

Consider the equation;

1''' '' ' cos ( ) ( ) 0S S (2.16)where 1, , are parameters, is the small

parameter, and the prime denotes differentiation with respectto S. A reduced equation is derived from equation (2.16) andthe result is compared with that obtained using the multiple-scale method [12] First the perturbation solution

(0) (1) 2( ) ( ) ( ) ( )S S S is obtained bysolving the following equations;

(0) ( )L S (2.17)

(1) (0) (0) (0)

1( ) ( ) cos ( ) ( )L S S S S (2.18)

where;3

3( ) ( )

d dL S S

dS dS

The solution to the unperturbed system (2.17)(0) ( )S is

obtained as;

(0) ( )( ) . ,2

i S vAS e c c S B (2.19)

where ,A v and B are integration constants.Substituting equation (2.19) into equation (2.18), thefollowing equation is obtained;

(1)

1( ) cos( ) ( ) cos( )oL S A S v j D

( )2( )

2

2,4,... 1,3,...

cos( ) ( ) sin( ) ( ) .

inS vin

S vn n

n n

eJ D e J D c c

i

(2.20)

Here D is defined by 2 sin( )2

D A

, nJ denotes the

nthorder Bessel functions, and the following relation has been

used in deriving equation (2.20);

sin

0

1,3,... 2,4,...

( ) 2 ( )sin( ) 2 ( ) cos( )iz

n n

n n

e J z i J z n J z n

The solution of equation (18) is given by;

(1)

1 0( ) cos( ) ( ) cos( )

2

ASS S v j D S

1 1( )sin( ) sin( )

2S j D S v

( )2

1 22,4,...

cos( ) ( )(1 )

inS ve

n

n

ej Din n

( )2

23,5,...

sin( ) ( ) .(1 )

inS v

n

n

ej D c c

n n

(2.21)

This perturbation solution includes the terms ( , , )S .

Next these terms are removed using the renormalization

method. Define the renormalization variables ( )C S and

( )B S as follows;

21 1

( ) 2 ( ( )) sin( ) ,2

iSC S A A i j D A e

1 0( ) ( ( )) cos( )B S B S j D A

The set of the renormalization equations up to ( ) is

obtained as;

21 1

( )( ) ( ( ( ) ))sin( ) ,

2

iC S

C S i j D C S edS

(2.22)

1 0

( )( ( ( ) )) cos( ),

B Sj D C S

dS

(2.23)

Now, comparison of the renormalization equations (2.22)and (2.23) with the reduced ones obtained by the multi-scalemethod is necessary. Using the decomposition

( )( ) ( ) iv SC S A S e where, ( ( ) , ( ) )A S iv S , youobtain;

1 1

( )( ) sin( ) ( ( ( )))sin( )

2 2

dA SA S j D A S

dS

(2.24)

11

( )sin( ) ( ( ( ))) cos( )

2( )

dv Sj D A S

dS A S

(2.25)


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When the reduced variable is introduced, the renormalizationequations (2.23), (2.24), and (2.25) become the reducedequations derived in [12].

D. WEAKLY NON-LINEAR SYSTEM

Here, the method leads to a discrete complex Ginzburg-Landauequation from a weakly non-linear lattice with delay.In this lattice system, the finite propagation time of motionfrom the nearest oscillators is taken into account. The stabilityof the trivial solution is predicted by studying the derivedreduced equation.The weakly non-linear oscillator is given by;

( )( )

j

j

dx tp t

dt (2.26)

2 31 1( )

( ) ( ) ( ) 2 ( ) ( ) ,j

j j j j j

dp tx t x t x t x t x t

dt

(2.27)

where , are parameters, is the small

parameter, and represents the time delay. Thevariables ( )jx t and ( )jp t denote the displacement

and momentum of a single oscillator located at lattice site

j respectively. This system becomes Hamiltonian

system where 0 .Firstly the perturbation solutions

(0) (1) 2( ) ( ) ( ) ( )j j jx t x t x t are obtained as;

2

1 1( ) ( ) 2 3 . .2

i ti t i

j j j j j j j

te

x t A e e A A A A A c ci

(2.28)

Here jA are the integration constants, and the higher

harmonic terms in (1) ( )

jx t are omitted.

Secondly from equation (2.28), the renormalized variables aredefined as;

2

1 1( ) ( ) 2 32

i

j j j j j j j

tA t A e A A A A A

i

From the definitions, we have the relation

( ) . .,i tj jx t A e c c and the renormalization equations as;

2

1 1

( )( ( ) ( )) 2 ( ) 3 ( ) ( )

2

j i r

j j j j j

dA te A t A t A t A t A t

dt i

(2.29)

The system (2.29) becomes the discrete Schrdinger

equation and Hamiltonian system, in the case 0 .When 0 , equation (2.29) becomes the discrete Ginzburg-

Landauequation.Using the renormalization equations, the behaviour of

motion of the original system is predicted and confirmed

numerically, on restriction to the conditions ( ) ( )j N jx t x t ,

withNbeing the number of oscillators, this conditions lead to

( ) ( )j N jA t A t . There is the trivial solution ( ) 0jA t inequation (2.29). The uniform solution, expressed as

( ) ( )j

A t A t (for any j), can be viewed as one of the local

stable manifolds of the fixed point 0j

A when a certain

condition is satisfied and this has to be shown. To do this, we

study the linear stability for 0jA . Substituting the equation

( ) ( )j jA t a t where ( ( ) 1)ja t into equation (2.29) we

have the linearized equation of motion in Fourier space;

( ) 21 cos( ) ( ),k

k

db t i ke i b t

dt N

(2.30)

where;

21

( )

0

1( ) j

i kjN

Na t

k

j

b t eN

and2

1

( )

0

1( ) ki kj

N

Nb tj

k

a t eN

with (0,......, 1)k N . From equation (2.30) we canpredict which modes increase or decrease in time. The

absolute value of ( )k

b t decreases to zero as time evolves

when;

2sin( ) cos( ) 0

k

N

(2.31)

and this condition with 0k , gives that the uniformsolution can be viewed as the local stable manifold of

0jA .

E.

SPATIALLY EXTENDED SYSTEM

In the case that a spatially extended system modelled byincluding unperturbed terms with delayed arguments, thefollowing system is considered;

23

2

( , ) ( , )( , ) ( , )

2

u t x u t xu t x u t x

t x

(2.32)

where , are parameters, is the small

parameter, and r represents the time delay. Although thesystem is described by a delay partial differential equation, themethod is not changed.

First the perturbation solution(0) (1) 2( , ) ( , ) ( , ) ( )u t x u t x u t x is obtained by

solving the following equations;(0) ( , ) 0L u t x ,

2 ( 0)(1) (0)3

2

( , )( , ) ( , )

u t xL u t x u t x

x

where

( , ) ( , )( , )

2

u t x u t xL u t x

t

The solutions are given by;


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(0 ) (2 )( , ) ( ) . .,

i t

u t x A x e c c

and2

2

2(1) (2 )

( )3 ( ) ( )

( , )1

2

i tA x

A x A xx

u t x tei

where ( , )A t x is an arbitrary differentiable functionof x. Next with the introduction of the renormalized variable

( , )A t x we have;

22

2

( )( , ) ( ) 3 ( ) ( )

12

t A xA t x A x A x A x

xi

Finally, the renormalized equation which should besatisfied is derived as;

22

2

( , ) ( )3 ( , ) ( , )

12

A t x A xA t x A t x

t xi

which is the well known complex Ginzburg-Landauequation

III. EXTENDED RENORMALIZATION METHOD

An extended renormalization method can lead to areduced equation with delay from a given system with large ororder-one delay. Large delay means that the delayed

arguments are of order1

( 0)

with being the small

parameter associated with the given weakly non-linear system.

A. LINEAR SYSTEM

The model which is studied here is given by the equation;2

2

2

( )

( ) 0

d x t

x t x tdt

(3.1)

where

represents large delay with being a

small parameter and 0 is a parameter. When 1 and1 the system was analysed using the multiple-scale

method as in Ref 23. This system will be analysed using theextended renormalization method and the result comparedwith that of the previous work.

Firstly the perturbation solution,(0) (0) 2 (0) 3( ) ( ) ( ) ( ) ( )x t x t x t x t is obtained

by solving the following equations;

(0) ( ) 0Lx t , (3.2)

(1) (0)( )Lx t x t

(3.3)

(2) (1)( )Lx t x t

, where;

22

2( ) ( ).

dLx t x t

dt

When deriving these equations, the magnitude of1

in

the delayed argument is treated as a large value, this treatmentcorresponds to the non-standard expansion in the previousstudy23

The solution of equation (3.2) is given by;(0) ( ) (0) exp( ) . .,x t A i t c c (3.4)

where (0)A represents the contribution to thesolution

(0) ( )x t except for the first motion exp( )i t . We

assume that the solution(0)

x att

is expressed as;

(0) exp( ) . ..

i

x t A e i t c c

Here,A

represents the contribution to

(0)

x t

, except for exp( )

i

e i t

.This implies that

the argument of ( )A t is only affected by a large time shift. At

this stage the functional form of ( )A t is not known. The

existence of ( )A t is the most fundamental assumption in this

extended method. It is noted that the equation which

( )A t should perturbatively satisfy is the extended

renormalization equation. This extended reduced equation isconstructed by removing the secular behaviour coming fromthe resonance between the frequency in the operator L and

exp( )i t in the forcing terms. When the delay becomes

zero, the extended method corresponds to the conventionalmethod. Substituting this solution

(0) ( )x t into equation (3.3),

the following equation is obtained;

(1) ( ) exp( ) . ..

i

Lx t A e i t c c

Whose solution is given by;

(1) ( ) exp( ) . .2

iit

x t A e i t c c

At the next order in , we obtain;


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22(2 )

3 3

2( ) exp( ) . .

8 8 4

rt i r r

x t t A e i t c c

At this stage, the secular behaviour, as already seen in thecase with delay, is not large.Define the extended renormalized variable as;

( ) (0)2

itA t A A e

22

2

3 3

2

8 8 4

t it A e

(3.5)

This definition is a form of a near-identity transformation

at the function (0)A , instead of the constantA in the

conventional renormalization method. From this definition of

( )A t , we obtain;

( ) ( )

2

A t A iA e

2

2 3

3 3

2 2( ) ( , )

8 8 4

t iA e

(3.6)

where, is a parameter whose value is smaller than

. The inverse of equation (3.5) is given by;

2(0) ( ) ( , )2

itA A t A t e

Using the above expression, the following relation can beobtained

2

( )

( , )2

im

i t

A m A t m A t m e

(3.7)

with m . Substituting equation (3.7) into equation(3.6) and taking the limit as 0, the extended

renormalization method which ( )A t should perturbativelysatisfy is obtained, that is;

2

2

3

( )2

2 8

i idA t i i

e A t e A t dt

(3.8)

There are delay terms in equation (3.8), and this

renormalization equation in the case of 1 and 1 isequivalent to reduced equations derived in [13], wherenumerical simulation and some analysis have shown that thereduced system reproduces the behaviour of slow motion in

the original system. In the case that 0 and is given byequation (2.3), it can be shown that one of the solutions to

equation (3.8) up to2( ) is given by equation (2.10).

By comparing this extended renormalization method withthe conventional method discussed in section two above, whenusing the conventional method one cannot obtain a reducedequation.

The perturbation solutions are;

(0) ( ) .i tx t Ae c c , (1)( ) . .2

i

i titx t Ae e c c

22(2 )

3 3( ) . .

8 8 4

i

i tt ix t t Ae e c c

The renormalized variable is defined as;

222

3 3( ) .

2 8 8 4

i iit t i

A t A Ae t Ae

The renormalization equation up to ( ) becomes;

( )( ).

2

idA t i

e A tdt

(3.9)

and up to2( ) becomes;

2

2

3

( )( ) ( ).

2 4 8

i iA t i i

e A t e A t dt

(3.10)

Since the magnitudes of the terms calculated as higher-

order correction in equation (3.10) are2( ) and 2( ) ,

this approximation is in contradiction with equation (3.9)

except for the case of 0 . When 0 , equation (3.10)becomes equation (2.8), and there is no contradiction only inthe case. For systems with large-delay, the extendedrenormalization method can easily be used.

B. NON-LINEAR OSCILLAOR

Here, a weakly nonlinear system with large delay, whichusually appears in Optics, is considered. In [14] a system withoptoelectronic feedback is analyzed, and the system isdescribed as;

2 2

2

( ) 2( ) ( ) 1 ( ) 1 ( )

1 2

dx s P y s x s y s C y s

ds P

( ) 1 ( ) ( ),dy s y s x sds

where s is the scaled time, , ,C P are parameters

and is the small parameter. The solution of great focusis the small amplitude regime, described by the followingassumptions;

(1) 2 (2) 3 (3) 4

( ) ( ) ( ) ( ) ( )x s x s x s x s


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(1) 2 (2) 3 (3) 4( ) ( ) ( ) ( ) ( )y s y s y s y s The reduced equation is constructed using the extended

method, and the result is compared with that reported in [14].

First the perturbation equations are;

(1)(1)( )

( ),dx s

y sds

(1)(1)( )

( ),dy s

x sds

(2 )(2 )( )

( ) ,dx s

y s Cds

(2 )(2) (1) (1)( )

( ) ( ) ( ),dy s

x s x s y sds

(3)

(3) (1) (1)( ) ( ) ( ) ( ),dx s y s x s Cy sds

(3)(3) (2) (1) (1) (2)( ) ( ) ( ) ( ) ( ) ( ),

dy sx s y s x s y s x s

ds

where2

. The solutions (1) ( )x s , (2 ) ( )x s and

(3) ( )x s are given by;(1)( )x s (0) . .,ise c c

(2 ) ( )x s

3

i(0)

2

2 . .,ise c c

(3) ( )x s 2

s ( iC(0)

3

i | (0)|

2

(0) (0) iC ( ) ) . .ise c c higher harmonics.The definition of the renormalized variable is;

(t) = (0) 22

s ( iC(0)

3

i

|(0)|2(0) (0) iC(-)). (3.11)

The renormalization equation is given by;

d(s) ds =2

2

s( iC(s)

3

i| (s) |

2(s) (s)

iC(s-)). (3.12)

Equation (3.12) is the reduced equation derived in [14].The extended method gives the same results given by the useof multiple-scale method.

C. DISCUSSION

For both the renormalization methods, terms in thereduced equation arise from the terms appearing in the

perturbation analysis. This implies that higher harmonics inthe perturbation analysis does not contribute to the reducedequation in the first order approximation. In this sense, thereduced equation can be obtained from a wide class includingthe original system.

In this paper, it has been shown that the renormalizationmethod can be extended to study systems with delay, and themethod gives reduced systems successfully. The application ofthe renormalization method can help elucidate the behaviourof time delayed systems in a non-chaotic regime.

REFERENCES

[1] Lang, R and Kobayashi, K., IEEE Journal. QuantumElectron. 16 (1980), 347

[2] Pieroux, D. and Mandel, P., Physics Revision Journal. 67(2003), 056213

[3] Uchida, A. et al Physics Revision Journal. 69 (2004),056201

[4] Mackey, M. C., Science 197 (1977), 287[5] Gopalsamy, K., Stability and Oscillations in Delay

Differential Equations of Population Dynamics, KluwerAcademic, 1992

[6] Johnson, M. A., and Moon, F. C., Int. J. Bifurcation andChaos 9 (1999), 49

[7] Nambu, Y. et al., Phys. Rev. D 60 (1999), 104011, M.,Phys. Rev. A 56 (1997), 1548

[8] Alsing, P. M., et alPhys. Rev. A 53 (1996), 4429, S. L.,and Chatterjee, A., Non-Linear Dynamics 39 (2005), 375

[9] Tzenov, S., Contemporary Accelerator Physics, WorldScientific, 2004

[10] Goto, S. and Nozaki, K., Prog. Theor. Phys. 105 (2001),99

[11] Erneux, A., et alSIAM J. Appl. Math. 61 (2000), 966[12] Mirollo, R.E., and Strogatz, S.H., synchronization of

Pulse Coupled Biological[13] Oscillators. SIAM J. Appl. Math., 50:1645-1662, 1990

weakly non-linear time delay systems with modified parameters

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