for logistic sequence xn+1 = 4xn (1 − xn ).

Note that the above results have been proved only for a certain class of dynamicalsystems which generated deterministic chaos. As usual, the situation of mixture "chaoticrandomness" is a normal for the natural observed data (one of the main task is todetermine their correlation in the time series). It’s normal to expect that the quality of
the approximation of this mixture depends on the specified ratio in the proposed modelfbm

H = 0.2

H = 0.5

H = 0.6

Tent map

0.57660.0078

1.0569Mixture (α = 0.2) 0.9591

0.87212.6039Lorenz1.8544

1.63601.9381

Mixture (α = 0.5) 0.9244

1.06782.8903(approximation of a random process fbm and the quality is defined by the specified
statistics An , Bn , Dn ).
Key words: fractional Brownian motion, Lyapunov exponent, Lorenz system.

Influence of Chaotic Behavior in Complex Networks
by Changing Network Topology
T. Chikazawa, Y. Uwate, and Y. Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, Tokushima, 770-8506 Japan
Email: {chikazawa, uwate, nishio}@ee.tokushima-u.ac.jp
In our society, there are various type networks. We have lived our life by using
networks. Examples of networks are transportation network, ﬂight network and so on.
Recently, various networks around our life have became more complex and large scale.
Complex network have attracted grate deal of attention from various ﬁelds. Some researchers discover small-world network [1] and scale-free network [2]. These networkmodels have various types of feature quantities. Examples of feature quantities are path
length, degree distribution, clustering coeﬃcient and so on. Moreover, in the complex
network, there are various network with propagation. The pandemic outbreak of viral
infection and the traﬃc jam of the transportation network are mentioned as an example
of propagation in the real network. However, there are not many studies of large- scale
network of continuous-time real physical systems such as electrical circuits Therefore,
it is important to investigate the chaos propagationand the spread of chaotic behavior
10
under some diﬃcult situations for the circuits.
As previous studies, the chaos propagation and the spread of chaotic behavior have

infection and the traﬃc jam of the transportation network are mentioned as an example
of propagation in the real network. However, there are not many studies of large- scale
network of continuous-time real physical systems such as electrical circuits Therefore,
it is important to investigate the chaos propagation and the spread of chaotic behavior
under some diﬃcult situations for the circuits.
As previous studies, the chaos propagation and the spread of chaotic behavior havebeen investigated only in simple networks such as ladder and ring topology [3]. In thissimple network, the periodic attractors change to the chaotic attractors by increasing
the coupling strength.
The chaotic circuit is shown in Fig. 1. This circuit consists of a negative resistor, twoinductors, a capacitor and dual- directional diodes. This chaotic circuit is called NishioInaba circuit. We propose diﬀerent topology complex networks with coupled chaoticcircuit. Figure 2 shows the proposed two types networks. Proposed network models
consist of many nodes and edges. We set chaotic circuit in node, and resistor R in edge.
Each node is coupled by one edge. We use 25 coupled chaotic circuits in Fig. 2(A)and 49 coupled chaotic circuits in Fig. 2(B). Furthermore, one circuit is set to generatechaotic attractor and the other circuits are set to generate three-periodic attractors.
chaotic attractor and the other circuits are set to generate three-periodic attractors.
In this study, we investigate the influence of chaotic behavior in complex networks
In this study, we investigate the influence of chaotic behavior in complex networks
by changing network topology. First, we investigate ratio of spreading chaotic behavior
by changing network topology. First, we investigate ratio of spreading chaotic behavior
by changing network topology in small network. Second, we verify chaotic behavior in
by changing network topology in small network. Second, we verify chaotic behavior inlarge-scale complex network. Finally, we observe how to spread of chaotic behavior bylarge-scale complex network. Finally, we observe how to spread of chaotic behavior byincreasing the coupling strength.
increasing the coupling strength.
i2i2i1i1L1
L1-r-r

5C
C

L2L2vdvd

39

49

39

49

374840
48
46
46

44
44

41
42

43

42

43
47
47

45

32
20313214

41651516
19
15

713

7613

12
18

2119

1211

24

29

6

21122

1

102330 28

26
25

1023

28

25

2993
32

8
1

27 2624

30

4

98

18

21

22

31 27

41

1420
1733

38

40(A) 25 coupled chaotic circuits.
(A) 25 coupled chaotic circuits.

Figure 1: Chaotic circuit.
Figure 1: Chaotic circuit.

3337CnCn

34
1735

3638vv

3435

36(B) 49
45 coupled chaotic circuits.
(B) 49 coupled chaotic circuits.

Figure 2: Attractors of chaotic circuit.
Figure 2: Attractors of chaotic circuit.

References
References[1] D. J. Watts and S. H. Strogatz, “Collective dynamics of small-world,” Nature, vol.
[1] D. J. Watts and S. H. Strogatz, “Collective dynamics of small-world,” Nature, vol.
393, pp. 440-442, 1998.
393, pp. 440-442, 1998.
[2] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks,” Science,[2] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks,” Science,
vol. 286, pp. 509-512, 1999.
vol. 286, pp. 509-512, 1999.
11[3] Y. Uwate and Y. Nishio, “Chaos Propagation in a Ring of Coupled Circuits Gen[3] Y. Uwate and Y. Nishio, “Chaos Propagation in a Ring of Coupled Circuits Generating Chaotic and Three-Periodic Attractors,” Proceedings of IEEE Asia Pacificerating Chaotic and Three-Periodic Attractors,” Proceedings of IEEE Asia PacificConference on Circuits and Systems (APCCAS’12), pp. 643-646, Dec. 2012.

[2] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks,” Science,vol. 286, pp. 509-512, 1999.
[3] Y. Uwate and Y. Nishio, “Chaos Propagation in a Ring of Coupled Circuits Generating Chaotic and Three-Periodic Attractors,” Proceedings of IEEE Asia PacificConference on Circuits and Systems (APCCAS’12), pp. 643-646, Dec. 2012.

The Concept of Integrability for Multifunctions
and Dynamics of the Trace Map
Efremova L. S.
Nathional Research Nizhni Novgorod State University
lefunn@gmail.com
The concept of integrability of a continuous map in the plane introduced in [1] (seealso [2]), is generalized for an upper semicontinuous two-valued map defined in a convex
unbounded domain of the plane.
Criterion is proved for integrability of above multivalued maps. This criterion is basedon the reduction of a considered two-valued map to an upper semicontinuous two-valued
skew product of maps of an interval defined on an unbounded (with respect to secondvariable) rectangle of the plane.
Obtained results are applied to the investigation of the upper semicontinuous twovalued map connected with the trace map
F (x, y) = (xy, (x − 2)2 ).
This trace map arises in quasicrystal physics.
Considerations of this work are based on use of geometric results obtained in [1] forthe above trace map.
This is the joint work with S.S. Belmesova.
REFERENCES[1] Belmesova S. S., Efremova L. S. On the Concept of Integrability for Discrete Dynamical Systems. Investigation of Wandering Points of Some Trace Map// Nonlinear
Maps and their Applic. Springer Proc. in Math. and Statist., 112 (2015), 127–158.
[2] Efremova L. S. Dynamics of Skew Products of Maps of an Interval// Russian Math.
Surveys, 72:1(433) (2017), 107-192.

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