On locally linearizable billiard systems
Treschev D. V.
Steklov Mathematical Institute of RAS
treschev@mi.ras.ru

On positive metric entropy conjecture
Berger P., Turaev D.
Imperial College London
dturaev@imperial.ac.uk
We prove the folowing conjecture by Herman:
Arbitrarily close, in the C ∞ -topology, to the identity map of a two-dimensional disc there
exists an area-preserving diffeomorphism with positive metric entropy.

Investigation of Ring - Star Network of van der Pol Oscillators
K. Ueta, Y. Uwate, Y. Nishio
Tokushima University{kazuki, uwate, nishio}@ee.tokushima-u.ac.jp
There are a lot of synchronization phenomena in this world. This is one of thenonlinear phenomena that we can often observe by natural animate beings which docollective actions. For example, firefly luminescence, cry of birds and frogs, applauseof many people, and so on. Synchronization phenomena have a feature that the set ofsmall power can produce very big power by synchronizing at a time. Therefore, studyof synchronization phenomena has been widely reported not only in the engineering
but also the physical and the biological fields. Investigation of coupled oscillators isfocused on many researchers, because coupled oscillatory network produces interesting
synchronization phenomena, such as the phase propagation wave, clustering, and complex patterns. In addition, it has the advantage
of being able to manufacture for circuit
55
on the board[1, 2, 3].
In this study, we investigate synchronization phenomena observed in the systembut also the physical and the biological fields. Investigation of coupled oscillators isfocused on many researchers, because coupled oscillatory network produces interesting
synchronization phenomena, such as the phase propagation wave, clustering, and complex patterns. In addition, it has the advantage of being able to manufacture for circuiton the board[1, 2, 3].
In this study, we investigate synchronization phenomena observed in the systemmodel containing a ring and a star of van der Pol oscillators by circuit experimentand computer simulation. We observe several types of synchronization phenomena byincreasing the coupling strength of the ring. Then, we observe the synchronization phenomena with computer simulation. van der Pol oscillator is shown in Fig. 1.
Figure 2 shows a system model constituted van der Pol oscillators (VDP-A andFigure 2 shows a system model constituted van der Pol oscillators (VDP-A andVDP-B). We couple each VDP-B via inductor L and ground by coupling resistor R. InVDP-B). We couple each VDP-B via inductor L and ground by coupling resistor R. Inaddition, We couple VDP-A via resistor r. VDP-A is the only one central circuit which
addition, We couple VDP-A via resistor r. VDP-A is the only one central circuit whichis connected to all VDP-B in this system by resistor r.
is connected to all VDP-B in this system by resistor r.

CiAg
iAg

vAvA

CC

VDP A
VDP A

L

iA
iA
L
L

ikg
ikg

VDP B
VDP B

vkvk

CC

Figure 1: van der pol oscillator.

rr

R
R

VDP B
VDP B

VDP A
VDP A

rr

2L
2L
2L
2L

R
R

rr

r
r
2L
2L

2L
2L
VDP B
VDP B

2L
2L

R
R

2L
2L

R
R
VDP B
VDP B

Figure 2: System model.
Figure 2: System model.

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
56 of scaling in random networks", Science,[2] A. L. Barabasi and R. Albert, “Emergence[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

[1] D. J. Watts and S. H. Strogatz, “Collective dynamics of small-world", Nature, vol.
393, pp. 440-442, 1998
[2] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks", Science,vol. 286, pp. 509-512, 1999
[3] S. H. Strogatz, “Exploring complex networks", Nature, vol. 410, pp. 268-276, 2001

Dynamics
maps
aa one-dimensional
Dynamics of monotone
maps ofona one-dimensional
connected locallyDynamics
of monotone
monotonemaps ononlocally
one-dimensional
locallf monotone maps on a one-dimensional continuum
locally connected continuum
continuum
continuum

Vaniukova K. S.

Vaniukova
Vaniukova K.
K. S.
S.

Vaniukova
K. S.of information
TheInstituteinformation
technology,
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The Institutemechanics,Thetechnology,
Institute ofofmathematics
informationandtechnology,
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mechaInstitute of informationtechnology,
mathematics
and mechanics,
Lobachevsky
StateUniversity
ofofNovgorod,
NizhniNovgorod,
Lobachevsky
State Universityof Nizhni
Novgorod,
Nizhni
RussianNizhni
FederationLobachevsky
StateUniversity
Nizhni Novgorod,
Novgorod,
NizhniNovgorod, Russia
Russi

State University of Nizhni Novgorod, Nizhni Novgorod,
Russian Federationvaniukovaks@gmail.com
vaniukovaks@gmail.com
vaniukovaks@gmail.com
vaniukovaks@gmail.com
continuum
weaacompact
By continuum we mean aBy
compact
connected
metricspace. connected
By
continuum
wemeanmean
compact
connectedmetric
metricspace.
space.

uum we mean a compact
metric space.
LetXXbeaaone-dimensional
locally
ff : :XX →Let Xconnected
be a one-dimensional
locallycontinuum,
f : connectedX→ X becontinuum,
a continuousLetbeconnected
one-dimensional
locallyconnectedcontinuum,→XXbebea one-dimensionalmap.
locally
continuum,:X→ fX
becalled
a continuous
map.
AA
map
toifif for
A connected
map f is called
tobe fmonotone,
for
every
connected
subsetCevery⊂ X,connectedf −1 (C) subset
map.
map
f isifiscalled
to be
be monotone,monotone,
for
every
connected
subset CC ⊂f is called to beismonotone,
subset C ⊂ X, f −1 (C)isisconnected.
connected.if for every connectedconnected.

One-dimensional
continuahave
One-dimensional locally connected
continualocallyhave aconnectedcomplicatedtopological
One-dimensional
locallyconnected
continua
haveastructure.
acomplicatedcomplicatedtopologi
topolognsional locally connected
continua
have a complicatedtopologicalstructure.
Therefore,
evenmapsondynamics
Therefore,
even monotone
maps on themhave
nontrivial(see,e.g.,[1] - [5]).
Therefore,
evenmonotone
monotone
mapsdynamics
onthemthemhave
havenontrivial
nontrivial
dynamics(see,(see,e.even monotone maps on
have nontrivial(see,mapse.g.,[1]In
thisdynamics
ofof monotoneon
one-dimensional
In themthis report
dynamics dynamics
ofon -a [5]).
one-dimensional
locally
Inmonotone
this reportreportdynamics
monotone maps
maps
on aaconnected
one-dimensional local
loca

port dynamics ofcontinuum
monotone ismaps
on a one-dimensional
locally connectedcontinuumstudied.
continuumisisstudied.
studied.
studied.

ces

References

References
References

[1]E.
The
dynamics
ofofmonotone[1] L. S. Efremova, E. N. Makhrova,
The dynamics
ofmonotone
maps
of dendrites,
Mat. maps[1] L.
L.S.S.Efremova,
Efremova,
E.N.
N.Makhrova,
Makhrova,
The
dynamics
monotone
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Sb.
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[2]
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graph,dendriteand[2] I. Naghmouchi. Dynamic
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maps,
Interna[2] I.ofI. Naghmouchi.
Naghmouchi.
Dynamicmonotone
graph,dendriteand dendroid
dendroid mmouchi. Dynamic of monotone
graph,ofdendrite
anddendroid
International
Journalofof
Bifurcation
andtional Journal
Bifurcation
andChaos.maps,- 2011.
- V. 21,
pp.Chaos.
1-11. --2011.
tionalJournalBifurcation
andChaos.
2011.--V.
V.21,
21,pp.
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properties of monotone
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Topology dendrite
andits maps,[3] I.Naghmouchi.
I.Naghmouchi.
Dynamicaldendrite
properties
monotonedendritemaps, TopTop

ouchi. Dynamical propertiesof monotone
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Topology
its159,Applications.
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2012.-and-Vol.
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144-149.

ons. - 2012. - Vol. 159, pp. 144-149.
H.
setsmonotone
local
dendritemaps,[4] Abdelli H. ω-limit sets[4]forAbdellimonotone
local
dendritemaps,Chaos,Solitons& Fractals.
[4]
Abdelli
H.ω-limit
ω-limitsetsforformonotone
local
dendritemaps,Chaos,Chaos,Soliton
Solito. ω-limit sets for monotone
maps,
Chaos,Solitons& 66-72.
Fractals.
--2015.
- 2015. local- V. dendrite71, pp. 66-72.
2015.--V.
V.71,71,pp.
pp.
66-72.

V. 71, pp. 66-72.

E.
K.
S.S.non-wandering
Vaniukova
ofofmonotone
non-wandering[5] E. N. Makhrova, K. S.[5]Vaniukova
On57the setpoints[5]E. N.
N. Makhrova,Makhrova,
K.of
Vaniukova On
On thethe set
setof
non-wandering points
pointskhrova, K. S. Vaniukova
of non-wanderingof monotone
maps
local
dendrites,
Journalofof Physics:
mapsOnon thelocalsetdendrites,
JournalPhysics:
Conference- 2016.Conference- Vol. 692,Series.
maps ononofpoints
local
dendrites,JournalSeries.
Physics:
Conference
Series. -- 2016
201

local dendrites, Journalof Physics: Conference
Series. - 2016. - Vol. 692,
012012.
012012.
012012.