mark the area of convergence to z ∗ , gray tones mark Julia set. The darker color, thefaster escape to infinity.

Figure 2: The common structure of Julia and Fatou sets for a solitary map with
µ0 = 2.5i. As in Figure 1, the axes represent the real and imaginary parts of theinitial condition. Color indicates the number of iterations to convergence, gray tones
indicate the number of iterations before leaving orbit to infinity.

Synchronization Phenomena in Ringsof Coupled Three van der Pol Oscillators
Daiki Nariai, Tran Minh Hai, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,
Tokushima University,
2-1 Minami-Josanjima,{nariai, minhhai, uwate, nishio}@ee.tokushima-u.ac.jp
Synchronization phenomena have been studied in various ﬁelds since a long time ago,such as in electrical systems, in mechanical systems, in biological systems and basicallyeverywhere. Among them, synchronization phenomena of van der Pol oscillator are
similar to natural phenomena by changing
41 frequency. The coupled system of van der
Pol oscillators is simple and easy to handle. Many researchers have proposed various
coupled oscillatory networks of van der Pol oscillators [1] - [2]. We focus on the couplingSynchronization phenomena have been studied in various ﬁelds since a long time ago,such as in electrical systems, in mechanical systems, in biological systems and basicallyeverywhere. Among them, synchronization phenomena of van der Pol oscillator aresimilar to natural phenomena by changing frequency. The coupled system of van der
Pol oscillators is simple and easy to handle. Many researchers have proposed various
coupled oscillatory networks of van der Pol oscillators [1] - [2]. We focus on the couplingstrength of coupled oscillatory networks consisted of two kinds of oscillators including
van der Pol oscillator.
In this study, we propose a novel coupled oscillatory system. Figure 1 shows circuitsof van der Pol oscillators. Figure 2 shows the circuit model. We use two ring circuitswith six oscillators. Three VDP of the ﬁrst ring are connected by resistors, three NC ofthe second ring are connected by inductors and resistors. The ﬁrst and the second ringare connected by resistors (R1 , R2 , R3 ). We investigate how to change synchronization
phenomena of adjacent oscillators by changing the value of R1 , R2 and R3 by computer
simulations and circuit experiments. Furthermore, when we change only the value of R2
from 0 to 0.03 at intervals of 0.001, we investigate relationship between coupling strengthand phase diﬀerence. This research obtained an interesting results which synchronization
phenomena are observed by magnitude correlation between R1 , R2 and R3 .

ign

iLn
vCn

VDPLC

Figure 1: Circuit of van der Pol oscillators.
circuit 2i2bR
circuit 1

i1a

VDP

i2a
R

circuit 3i3bir2

VDP

VDP
i1bi3aR

ir3

ir1
R2

R3

R1
ir5igncircuit 52L

ir4
circuit 42L

i4a2L

NC

2L

i5ai5b
R'R'2L2L

ir6i6bNC

vCn

C

NC

circuit 6NC

i4b

R'

i6a

Figure 2: Circuit model.

References

42[1] Y. Uwate, Y. Nishio, “Two van der Pol Oscillators Coupled by Chaotically Varying

Figure 2: Circuit model.

References[1] Y. Uwate, Y. Nishio, “Two van der Pol Oscillators Coupled by Chaotically VaryingResistor", Proceedings of International Workshop on Nonlinear Dynamics of Electronic Systems (NDES’06), pp. 189-192, Jun. 2006.
[2] K. Matsumura, T. Nagai, Y. Uwate, Y. Nishio, “Analysis of Synchronization Phenomenon in Coupled Oscillator Chains", Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’12), pp. 620-623, May 2012.
[3] K. Oi, K. Ago, Y. Uwate, Y. Nishio, “Eﬀect of the Hub in Complex Networks ofCoupled Parametrically Excited Oscillators with Dispersion", Proceedings of IEEEWorkshop on Nonlinear Circuit Networks (NCN’15), pp. 11-14, Dec. 2015.

On the shadowing property and odometers
Li, J. and Oprocha, P.
Faculty of Applied Mathematics, AGH University of Science and Technology,
al. Mickiewicza 30, 30-059 Kraków, Poland– and –National Supercomputing Centre IT4Innovations, Division of the University of Ostrava,
Institute for Research and Applications of Fuzzy Modeling,
30. dubna 22, 70103 Ostrava, Czech Republic
oprocha@agh.edu.pl
When we investigate the space of invariant measures from ergodic theory point of view,
we are usually not that much interested in the topological structure of underlying space.
By famous Jewett-Krieger theorem, we can view invariant measures as supported onminimal systems and numerous further generalizations allow to add even more topological (dynamical) properties to the underlying system. On the other hand, there areexamples of systems with quite rich dynamical structure (e.g. topologically mixing) but
not that much interesting invariant measures (e.g. only trivial measure, only atomicmeasures, etc.). In other words, connections
43 between topology and ergodic theory (oncompact metric spaces) is not that tight.
In this talk we will provide some characterizations of invariant measures in the case