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 Contents 1.Exact solution to a modulation model 2.Appearance of “quasi-intermittency” 3.Hurst’s index 4.Signal complexity 5.Correlation between “quasi-intermittency” and complexity 6.Conclusions

3 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 , ] A correlation between the Hurst’s index of the transformed electromagnetic signal and its complexity is shown Abstract

4 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

5 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

6 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.

7 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"

8 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

9 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

10 T is the duration of the cycle of the parameters change is the duration of the disturbance interval The process of the modualtion

11 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. Further, all time variables are normalized to a frequency of the initial wave:

12 On the first cycle of the medium modualtion is the new, transformed, normalized frequency takes into account the medium dissipation

13 The field on the other cycles consists of two waves: on the inactivity intervals 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 , 1999 ]

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

15 Parameters of Transformation on the disturbance intervals on the inactivity intervals The ratios of the forward and the backward wave amplitudes: The controlling sequence The generalized parameter

16 Behaviour of the ratios in the modulation process: for the disturbance intervalsfor the inactivity intervals monotone behaviour

17 near irregular behaviour

18 irregular behaviour

19 Lamerey’s diagram for the controlling sequence irregular behaviourmonotone behaviour

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

21 Otherwise, when, the sequence as well as the field have irregular behaviour The field decreases as the medium possesses the dissipation.

22 QUASI-INTERMITTENCY AND THE HURST’S INDEX The presence of the quasi-intermittency is confirmed by the Hurst’s method The Hurst’s index

23 The sequence has almost regular behaviour and the Hurst's index has a corresponding value. [Nerukh A.G., J. of Physics D: Applied Physics, 32, pp , 1999 ] long-range correlation when the time series exhibits persistence (antipersistence) corresponds to

24 For the white noise (a completely uncorrelated signal) The sequence has irregular behaviour and the Hurst's index has a corresponding value.

25 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]

26 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

27 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 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:

28 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 :  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

29 The statistical complexity is defined as the Shannon entropy of the causal states: 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, and D. Nerukh, G. Karvounis, and R. C. Glen, J. Chem. Phys., 117(21), (2002)

30 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

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

32 Similar behaviour for darkening of the medium

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

34 The detailed behaviour of Hurst's index

35 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