## 1. Introduction

What does it mean that the cosmic radio wave flux density varies with the passage of time is an interesting question; the radio wave is of the quasar, a system of galaxy, which is distributed in our universe from a few billions of light years to the distanse close to the big bang age and has been radiating immense electromagnetic energy from it by the synchrotron radiation that we may able to make a measurement of the flux density at micro wave bands with a radio interferometer[3,4]. A group of radio observers and astronomers has been monitoring daily so far over several years extragalactic radio sources (radio galaxies, quasars, etc.) and the monitored data were kindly shared with us who were interested in using for analysis[5]. In a few recent decades, the chaos and fractal theory has been intensively studied and developed in the fields of mathematics, computer numerical analysis, natural sciences and technologies[1], and in same decades, the nonlinear time series analysis methods have been developed intensely based on the newly understood ideas of the theory for analyzing the nonlinear phenonena[2].

The study in this chapter is motivated by the three factors mentioned above to analyze the time series of the radio wave flux density from the cosmological object, primarily, with one of the nonlinear methods, for finding the dynamics related to the cosmic object, including its information in the flux density variations. We hoped that if we could infer the dynamics and if the result would be found to have any rule changing with the magnitude of the red shift of the object we might have some knowledge concerning to the evolution of our universe. The hope has been prompted us to continue consistently to analyze the time series data. The period of monitor over several years is extremely short compared with the cosmic age, however, the analysis result of the time series data in newly developing methods may give us a new sight viewed from the nonlinear dynamics in the short time scale for the cosmic dynamical system.

## 2. Linear and nonlinear systems

### 2.1. Linear system

The result of a linear data computed with Fourier spectral analysis gives a pure periodicity, which means predictability of event(s) for unlimited future. No event of evolution can be expected by a linear data, and a linear system is not a way of our natural world. On the other hand, the result of a pure random data computed with it gives no structure of the periodicity, a continuous spectrum, and means no eventof predictability.

### 2.2. Nonlinear system

One of discrete nonlinear dynamical systems is given in this section for explanation. The logistic map function is given in Eq.(1) with a difference equation. Given that the time series data is generated from thissystem by iterating at

The result of the Fourier analysisfor the time series data at parameter

The time series data measured from natural and social systems is not generated in such a mathematical way and so complicated that more intricated process is required to deal with data. This problem will be discussed in later sections. Before we do this, some methods of quantifying fractal data and some characteristics of chaos dynamics (the initial value dependence and the parametric dependence) are exemplified by the logistic system.

Time series data were generated from Eq.(1) at parameter

The parametric dependence is characteristic to a chaos system. The vertical axis on the right diagram in Fig. 3 shows the values of number computed by Eq.(1) at parameter

## 3. Quantitative propertiesof nonlinearity

Time series data of a natural system may often be of a nonlinear dynamics, about which we know little for the system and need to analyze in an appropreate method in assumed state space dimensions. In this section we discuss on the way how to quantify such data assumed to be nonlinear by exemplifying the analysis for the logistic system.

### 3.1. Lyapunov exponent and information entropy

The initial value dependence of a system is evaluated by the Lyapunov exponent given by

for the discrete system, and

The information entropy of a system, as its manifold is given in a state space, is defined as

where

Figure 4 shows the bifurcation diagram for the logistic system (left diagram), the Lyapunov exponent

### 3.2. Fractal dimension

In this chapter we aim to infer the cosmic system’s evolution by the flux density data radiated from cosmic object and measured by the interferometer (of the radio wave). The fractal dimension is useful to study the system with which the data is related. We have a variety of fractal dimensions; the box counting dimension

The fractal dimensions in Eq.(4) are derived from the generalized dimension

## 4. Dynamical system and time series data

The purpose of this section is to discuss on the problem: if we measure the time series data from a system of nature how we access to a function of the system with which the data is generated. If the nature was constructed by the mathematics it would have been going well to solve the problem. Unfortunately, the nature, I believe, do not go so easy. We need to solve it by devising the data reconstruction and by applying above nonlinear methods to it.

### 4.1. Time series data

The originally measured data is defined as follows: A measurement starts at time

Vector

### 4.2. Embedding theory

The attractor is the manifold of a dynamical system,from which a physical quantity is continuausly released to be observed and the time series is, as a result, accessed to be analized. The time series is reconstructed in the form of a vector

The data set *-*th dimension. Takens’ theory claims that the attractorreconstructed on the embedding space inFig. 6 is generic embedding under condition

Takens‘ theorem is summarized in Eq. (7)

It is same to say that Galileo Galilei could find experimentally the gravity on the earth, but could not express it in a deterministic expression as the Newton’s equation. The map (

It is useful to give attention to noise inevitaby coming into the dynamical system and the observation system. The noise comes into the two systems[2],

where

### 4.3. Correlation dimension

The attractoris reconstructed in the embedded state space from observed time series. We are interested in the dynamical system of the quasar, the galactic object distributed at cosmological distances (up to ten billions of light years), which releases vast energy by synchrotron radiation and enables us to research some cosmic information by the fluctuations of the microwave flux density. We study the dynamics involving in the time series to know the structure of the dynamics for the system at different cosmological distances.We may be to have somedynamical knowledge of the systems‘s evolution in the experimental method. The system’s information is, in our context, is the manifold for the dynamics of the quasar system at different cosmological distances. The data is the time series of the flux density of the microwaves, 2.7GHz and 8.1GHz, for more than twenty quasars, daily monitored over thousand days. The fractal dimension of the manifold is to be analyzed. The dimensions introduced in the section 3 are difficult to compute withthe reconstructed data. Fortunately we have an useful method to compute the correlation dimension developed by Grassberger and Procaccia (GPA) as the substitute of

Given each point

in which

The optimal correlation dimension

### 4.4. Other methods of analysis

We introduce briefly three methods of analyzing the characteristics of the flux density variation, in which the same data were also computed for reference. The results will cover different aspects of the variation. The first is the spectral index

The detailed introductions for the methods be referred in [9-11][1] -. The result analyzed in these methods will be shown later. The results computed in above principles are useful for cross-checking the knowledge of the result of source dynamics.

## 5. Time series data

The extragaractic radio sources generate the time series data of the radio wave flux density for us to observe and to analyze their system‘s dynamics to see a mechanism how the cosmic object has been evolved in the cosmological age from a dynamic aspect of view.[3]

### 5.1. Monitored cosmic objects

Compact extragaracticradio sourceshad been monitored daily by Waltmann et al. at GBI radio wave observatory over 3000 days from 1979 [5,6]. Waltmann et al were kind to sendus the data of 46 extragalactic objects, from which 21 QSOs and 7 BL Lacs were selected for analysis. At the beginning we analyzed the data in the methods of the spectral index, of the Higuchi‘s fractal dimension and of the Hurst exponent. The methods will be explained briefly, and the result was published in [7].

The monitoredmicrowave frequencies wereat 2.7and 8.1GHz; and the red shift (the indicator of cosmological distance) of the monitored objects ranged from 0.15 to 2.22 ( from one billion to ten billions of light years). The name of the objects are shown in Table 1 as well as the red shifts. In Figure 7, the diagrams of the flux density variation over nine years are shown for several quasars.

0133+476* | 0202+319 | 0224+671 | 0235+164* | 0237-234 | 0333+321 | 0336-019 |

0.860 | 1.466 | 0.524 | 0.851 | 2.224 | 1.253 | 0.852 |

0420-014 | 0552+398 | 0828+493 | 0851+202* | 0923+392 | 0954+658 | 1245-197 |

0.915 | 2.365 | 0.548 | 0.306 | 0.699 | 0.368 | 1.275 |

1328+254 | 1328+307 | 1502+106 | 1555+001 | 1611+343 | 1641+399 | 1741-038 |

1.055 | 0.849 | 1.833 | 1.770 | 1.404 | 0.595 | 1.054 |

1749+096* | 1749+701* | 1821+107 | 2134+004 | 2200+420* | 2234+282 | 2251+158* |

0.322 | 0.760 | 1.036 | 1.936 | 0.070 | 0.795 | 0.859 |

Symbol * | BL Lac (7) | ; QSO (21) | ; |

### 5.2. Process of analysis

#### 5.2.1. Correlation dimension

The method of estimating the correlation dimension of the flux density time series of the quasars listed in Table 1 is plainly described here. The correlation sum for the time series data is calculated in the method given in Eq.(12). The time series for the quasar 0224+671, for example, is reconstracted in the way given in subsection 4.2 and embedded in the reconstructed state space (at

## Other. indices

A process of analyzing the data withother methods introduced in subsection 4.4 is shown in Fig.9. The diagrams gives us an insight of the way how each index is derived. We will show all of the result analyzed in these methods for a cross reference with the correlation dimension.

## 6. Result of analysis

### 6.1. Correlation dimension

The correlation dimension reflects the dimension of a dynamical function

It may be taken care to seethe diagram in Fig.10 that the radio wave frequency (2.7 or 8.1 GHz) from which the correlation dimension was derived is the value on the earth; the frequency at the radio wave source must be modified by the red shift (See Table 2); the second is that the sampling rate (one day on the earth) must be also modified on the quasarby the theory of relativity (See Table 2).[7]

### 6.2. Other indices

Figure 12 shows the result of the indices analyzed in the methods introduced in subsections 4.4 and subsubsection 5.2.2: the diagrams of the spectral index

### 6.3. Incident angles of the radio wave to the solar system and to our galaxy

It is natural to have a question that the radio wave flux may be strongly scattered by the matters whose density may be high around the earth (our solar system) and our galaxy; if it is true, there may be a possibility that the indices may affected by the insident angles to our solar system and to our galaxy. The distributions the index

## 7. Conclusion

The correlation dimension

For readers from different fields the literature on the radio galaxies and quasars may be referredin reference[3].

As discussed in section 2., the nonlinear dynamics is sensitive to the initial condition and the parameter. Our hope is to infer a definite dynamical function

We may need to take in mind that our accessed data might have added by noise of Eq.(15) and the analyzed attractor (a reflection of

### Acknowledgement

I would like to thank many Japanese principal radio astrophysicists for giving their knowledges on this field, comments and frequent encouragements in meetings, Dr. E.B. Waltmann and her group for sending their data to the computer center at Osaka Prefecture University, Mr. M. Takano and Dr. M.R. Khan for their computing works as graduate students at OPU.

## Notes

- It is true in the second dimension of coordinates. If the time series is generated by the function of higher degree, the correlation may be true in the graph at a higher dimension of coordinates.
- 1/𝑓 noise characterized as the power law events in the electronic circuit is in reference [11].