Complexity and “Quasi-Intermittency” of Electromagnetic Waves in Regular Time-Varying Medium

KHARKIV NATIONAL UNIVERSITY OF RADIO ELECTRONICS

Complexity and “Quasi-Intermittency” of

Electromagnetic Waves in Regular Time-Varying Medium

Alexander Nerukh, Nataliya Ruzhytska,Dmitry Nerukh

Kharkov National University of RadioElectronics Kharkov, Ukraine

Department of Chemistry, Cambridge University, Cambridge, UK

2 KHARKIV NATIONAL UNIVERSITY OF RADIO ELECTRONICS

Contents

1. Exact solution to a modulation model2. Appearance of “quasi-intermittency”3. Hurst’s index4. Signal complexity5. Correlation between “quasi-intermittency”

and complexity6. Conclusions


• The electromagnetic wave transformation in a medium with parameters that change in time as a finite packet of periodic rectangular pulses is considered

• Regularity of the transformation is estimated by two characteristics, the Hurst's index [Hurst H E, Black R P and Simaika Y M, 1965 Long-Term Storage: An Experimental Study (London: Constable) ]

and the complexity [Crutchfield J. P. and K. Young, Phys. Rev. Lett., 63, pp. 105-108, 1989. ]

• A correlation between the Hurst’s index of the transformed electromagnetic signal and its complexity is shown

Abstract


Introduction

• The problem of complexity is being increasingly studied in physical, biological, and chemical sciences. There are few, if any, applications of complexity approaches to electromagnetic problems

• This is surprising, as an electromagnetic signal possesses some complexity and it can changes significantly during signal interaction with media in devices and in an environment


• In many practical problems we deal with complexity at all levels: in the observed signals and responses, the models and the solution algorithms. It appears reasonable to assert that all these complexities must be commensurate with each other

• There is however a lack of generally accepted quantitative measures of complexity covering signals, models and algorithms which can be used to design the optimum solution algorithms for particular problems

Three kinds of complexity


• We shall consider time evolving phenomena when the complexity of this formulation can be different, depending on the nature of the phenomena considered. This complexity has become the subject of active research, as it opens a fundamentally new viewpoint on the information capability of physical systems, the way they store and transform information. Once the problem is formulated a solution to it is needed. The method of solution also has its own complexity. Once the problem is solved, the behaviour of the output obtained can also be characterised by different levels of complexity. This intrinsic complexity can be estimated and is directly connected to the information content of the signal itself.


• Even though the possibility of quantifying the three complexities (that intrinsic to the signal itself, that of the mathematical description and that of the solution method employed) has been recognised, to the best of our knowledge there is no attempts to compare them on the same ground

• We consider one of these aspects of complexity, namely, the intrinsic complexity of the signal under its transformation in a modulated medium having dissipation

• Modulation is forced by external sources and it means that such a model represents a process in an open system

• The latter can lead to irregular behaviour of the process or to "quasi-intermittency"


• The transformation is described by equations with changing in time parameters and here the Volterra integral equations in time domain method are used

• In the systems with distributed parameters the main features of the wave transformation by the medium nonstationarity can be revealed when a simple law changes the medium parameters and an exact solution to the problem can be constructed

Wave transformation under medium modulation


• It allows to use an exactly solvable model and to investigate the process of modulation of the electromagnetic field in a time-varying medium

• The modulation of an unbounded dielectric dissipative medium change of the permittivity and the conductivity, which are modulated according to the law of the finite packet of rectangular periodic pulses is given by

1 11

1 11

( ) ( ) ( ( 1) ) ( ( 1) )

( ) ( ( 1) ) ( ( 1) )

N

k

N

k

t t k T t T k T

t t k T t T k T


T is the duration of the cycle of the parameters change is the duration of the disturbance interval

F 0 F 1 2F

T 2 T

E = F + R F 1 0 0E = F 1 1E = F + R F 1 0 0E = F 1 1 E = F + R F 2 1 1 2 2E = F E = F + R F 3 2 2

E = F 3 3

T 2 T

T

+ T1

1

1 + 2 T1

+ 2 T1+ T1

T TT

TT

0 t

1T

The process of the modualtion


The initial field exists before zero moment of time when the modulation begins and it is given by the function

• Each time jump of the medium properties changes the electromagnetic field, so there are:

• is the field on the disturbance intervals

• is the field on the inactivity intervals.

0 ( , ) exp[ ( )] exp[ ( )]E t x i t kx i t kx

t t

Further, all time variables are normalized to a frequency

of the initial wave:

nE

nF


On the first cycle of the medium modualtion

1 1 1exp exp( ) exp( )E st ikx C iq t D iq t

1 1 1exp( ) exp( ) exp( )F ik A i t B i t

12 2 2( )q a s

21/a

1 0/s

is the new, transformed, normalized frequency

takes into account the medium dissipation


The field on the other cycles consists of two waves:

on the inactivity intervals

exp( ) exp( ) exp( )n n nE st ikx C iq t D iq t

exp( ) exp( ) exp( )n n nF ikx A i t B i t

on the disturbance intervals

The expressions for the direct and the inverse secondary wave amplitudes are given in [Nerukh A.G., J. of Physics D: Applied Physics, 32, pp. 2006-2013, 1999 ]


The transformed field at any moment t of the N-th modulation period

1 1

1 1

( , ) ( ( 1) )

( ( 1)( ))

( ( 1)( )) ( )

N

nn

N

nn

n n

E t x E t n T

E t nT n T T

F t nT n T T F t nT


Parameters of Transformation

on the disturbance intervals 2( 1) 2 1 2 1 21 1 2 22

2 1 2 1 11 1 2 12

{ ( ) }

{ ( ) }i N qTN N

NN N

D p p p r p pw e

C p p p r p p

on the inactivity intervals 2 1 2

2 1 2( )i NTN

NN N

B p pp e

A p p p r

2 2

1 4 /(4 )N Nr u u r

2

1 1 1 1

1cos( )cos( ) sin( )sin( )

2

au qT T T qT T T

q

The ratios of the forward and the backward wave amplitudes:

The controlling sequence

The generalized parameter


Behaviour of the ratios in the modulation process:

npnw

1 0 1 1 2 2/ 1.5, /( ) 0.01, 3.68, 0.52, 1,0002a s T T T T u

0 5 10 15 20 25 300,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

pN w

N

Rel

atio

ns

N

for the disturbance intervals for the inactivity intervals

monotone behaviour


0 5 10 15 20 25 300,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

pN

wN

Re

latio

ns

N

1 21.5, 0.01, 3.65, 0.52, 0.98988a s T T u

near irregular behaviour


0 5 10 15 20 25 300,00

0,05

0,10

0,15

0,20

0,25

0,30

wn

pn

R

elat

ions

N

1 21.5, 0.01, 0.55, 0.52, 0.18978a s T T u

irregular behaviour


Lamerey’s diagram for the controlling sequence

irregular behaviourmonotone behaviour


If then the sequence has a regular character and the transformed field undergoes a parametric amplification with time

1u

0 100 200 300 400 500

-15

-10

-5

0

5

10

15

500t/(TN)

a=1.3; T=6; T1=1.15; N=10; u=1.032312;

E


Otherwise, when , the sequence as well as the field have irregular behaviour

1u

The field decreases as the medium possesses the dissipation.

0 100 200 300 400 500-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

a=1.3; T=6; T1=3; N=20; u=0.819154;

E

500t/(TN)


QUASI-INTERMITTENCY AND THE HURST’S INDEX

The presence of the quasi-intermittency is confirmed by the Hurst’s method

~ ln( / ) / lnn nH R S n

11

max ( , ) min ( , )nk nk n

R X k n X k n

2

1

1 n

n i ni

S r rn

1

( , )k

i ni

X k n r r

1

1 n

n ii

r rn

The Hurst’s index


20 40 60 80 1000.40

0.45

0.50

0.55

0.60

0.65

0.70

u =0.98988

2)

a =1.5, s =0.01T1 =3.65, T = 4.17

ln(R

N /S

N ) / ln(N)

N

0 20 40 60 80 100

0

10

20

30 1)r N

The sequence has almost regular behaviour and the Hurst's index has a corresponding value.

Nr

[Nerukh A.G., J. of Physics D: Applied Physics, 32, pp. 2006-2013, 1999 ]

long-range correlation when the time series exhibits persistence (antipersistence) corresponds to

0.5 ( 0.5)H H


Nr

0.5H

20 40 60 80 1000.30

0.35

0.40

0.45

0.50

0.55

2)

ln(R

N /S

N ) / ln(N)

N

0 20 40 60 80 100

-15

-10

-5

0

5

10

15

1)u=0.18978

a =1.5, s =0.01, T1=0.55, T =1.02

r N

For the white noise (a completely uncorrelated signal)

NrThe sequence has irregular behaviour and the Hurst's index has a corresponding value.


THE COMPLEXITY OF THE SIGNALS

In order to estimate how complex the signals are we calculated the ‘finite statistical complexity’ measure of the signals

This approach of estimating the complexity of dynamical process rests on such well-known theories as Kolmogorov-Chaitin algorithmic complexity and Shannon entropy

The formalism is called ‘computational mechanics’ and was originated in the works by Crutchfield and others [Crutchfield J. P., Physica D, 75, 11, 1994 ]

The algorithm of computing the finite statistical complexity follows the method described in [Cover T. M. and J.A. Thomas, Elements of information theory, John Wiley & Sons, Inc., 1991]


In the computational mechanics framework symbolic dynamics is considered, i.e. the signal is described by discrete symbols assigned to discrete time steps for that a continuum signal is converted into a sequence of symbols from predefined alphabet

Outline of the theory and specific characteristics used to quantify dynamic

complexity


A set of symbols corresponding to each time step form a sequence S. To calculate the statistical complexity, S is decomposed into a set of left (past) of length l and right (future) of length r halves joined together at time points

lis

it

ris

lks

Pr |r lks s

it

Consider all equivalent left subsequences . Collect a set of all right subsequences following this unique left subsequence

Each right subsequence has its probability conditioned on the particular left one:


The equivalence relation between any two left subsequences is defined as follows: two unique left subsequences and are equivalent if their right distributions are the same up to some tolerance value :

lis l

js

Pr | Pr |r l r li js s s s

Pr iA

Pr |r lis s

The equivalence classes represent the states of the system that define the

dynamics at future moments – the “causal states”

A set of all equivalent left subsequences forms an “equivalence class”. The

equivalence classes have their own probabilities calculated from the

probabilities of the constituent left subsequences

The time evolution of the system can be viewed as traversing from one causal

state to the other with a transition probability equal to


The statistical complexity is defined as the Shannon entropy of the causal states:

2Pr log Pri

i iA

C A A

where are causal states.

The algorithm of computing the finite statistical complexity follows the method described in Perry N. and P.-M. Binder, Phys. Rev. E, 60, 459, 1999.andD. Nerukh, G. Karvounis, and R. C. Glen, J. Chem. Phys., 117(21), 9611-9617 (2002)

iA


• This measure of complexity shows how much information is stored in the signal

• It also indicates how much information is needed to predict the next value of the signal if we know all the values up to some moment in time

• In two limiting cases, when a signal has constant value at all times and when the signal is completely random, a complexity is equal to zero in this framework because of no information about the previous evolution needed to predict the signal in both cases

• All intermediate cases have a finite, non-zero value of a complexity


• The dependence of the complexity measure on the duration of the modulation period shows a correlation between the complexity and the generalized parameter u

0 1 2 3 4 5 6 7

-1

0

1

2

3

4

5

6

7

a=1.5; s=o.o1; T1=1

Co

mp

lexi

ty

T2

pn

rn

wn

u

Clearing of the medium


Similar behaviour for darkening of the medium

0 1 2 3 4 5 6 7

-1

0

1

2

3

4

5

6

7

a=0.75; s=o.o1; T1=1

Co

mp

lexi

ty

pn

rn

wn

u

T2


A correlation between the Hurst's index and the complexity of the signal

0 2 4 6 8 10-2

-1

0

1

2

3

4

5

6

7

8

Com

plex

ity a

nd H

urst

's in

dex

a=0.5T=10

T1

Comp rN

HrN

u


The detailed behaviour of Hurst's index

0 2 4 6 8 100,40

0,45

0,50

0,55

0,60

0,65

0 2 4 6 8 10

-1

0

1

a=0.5T=10

Hur

st's

inde

x H

T1

u


CONCLUSIONS

• The quasi-intermittency that occurs during the wave transformation under the time changes of the medium properties can be described by two characteristics, the Hurst's index and the complexity measure

• These two characteristics correlate• They also correlate with the generalized

parameter that controls the process of the wave transformation

Documents

Complexity and “Quasi-Intermittency” of Electromagnetic Waves in Regular Time-Varying Medium