Numerical analysis of proximity oscillator networks · Numerical analysis of proximity oscillator networks Giovanni Pugliese Carratelli - M58/30 DIETI - Univesity Federico II of Naples

Complex Networks overview Proximity oscillator networks Results Bibliography

Numerical analysis of proximity oscillator networks

Giovanni Pugliese Carratelli - M58/30

DIETI - Univesity Federico II of Naples

26 September 2014

To follow knowledge like a sinking star,

Beyond the utmost bound of human thought.

–Ulysses, Lord Tennyson

Supervisor Prof. Franco Garofalo Co-Supervisors Dr. Piero De LellisEng. Francesco Lo Iudice

Giovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks


Complex Networksoverview

Proximity oscillatornetworks

Results





Results



Introduction

Network: ensemble of interacting dynamical entities over a web of interconnectionsComplex: behaviour that cannot be explained in terms of the behaviour of each agent

Complex Networks model

xi (t) = fi (xi ) + gN∑i=1

aij (h(xi )− h(xj )), ∀i = 1, . . . ,N (1)

where

fi (xi ) is the independent dynamics of the i-th node

g∑N

i=1 aij (h(xi )− h(xj )) is the interaction term between nodes

g is the coupling gain

aij are the terms of the adjacency matrix A: defines the network topology

h is the output function



Topology

The structure of the network influences the dynamics!

Figure: All-to-all topology

Figure: Star topology

Figure: Ring topology

(c)

(a) (b)

Figure: In Fig.(a) an Erdos Renyi network, in Fig.(b) a Small worldnetwork, in Fig.(c) a Scale Free network



State dependent topologies

We aim to study the dynamics of a

Proximity Networks

Nodes are connected if their distance lies below a given threshold




We aim to study the dynamics of a

Proximity Networks

Nodes are connected if their distance lies below a given threshold


xi (t) = fi (xi ) + gN∑i=1

aij (x(t))(h(xi )− h(xj )), ∀i = 1, . . . ,N (2)





Results



The Kuramoto model

The Kuramoto model

θi = ωi + gN∑j=1

sin(θi − θj ), ∀i = 1, . . . ,N (3)

Synchronization

A Kuramoto network is completely synchronized [1]⇐⇒

Frequency synchronization,limt→∞ θi (t) = ω, ∀i = 1, . . . ,N

Phase-locking,limt→∞ ||θi (t)− θj (t)|| = 0, ∀i , j = 1, . . . ,N



The restricted visibility Kuramoto model

We consider N heterogeneous Kuramoto oscillators moving on a circular route

The proposed model

θi = ωi + gN∑j=1

aij (t)sin(θij ), ∀i = 1, . . . ,N (4)

where, θij = θi − θj is the relative angular position

aij (t) =

{1, if min

{mod (θij ), mod (−θij )

}≤ θvis ≤ π

2

0, otherwise(5)

θvis

Figure: In red an oscillator for which aij = 0, in green an oscillator for which aij = 1



Equilibria topologies

Results in [2] hold, i.e. entrainment frequency

ω =

∑Ni=1 ωi

N(6)

is reached, phase-locking is achieved and thus the topology is steady with respect totime.Hence by imposing θi = ω, ∀i = 1, . . . ,N, in Eq. (4) we obtain

ω = ωi + gN∑j=1

aij sin (θji ), ∀i = 1, . . . ,N (7)

that can be recast to

ω − ωi

g=

N∑j=1

aij sin (θji ), ∀i = 1, . . . ,N (8)

a necessary condition for the exitance of a solution is that

g > gmin =maxi |ω − ωi |

N − 1(9)

Topology bifurcation

Multiple equilibria may exist, depending on g and the initial conditions of network (4)



Experimental plan

We are interested to

Verify that also for network (4) entrainment frequency is reached for sufficientvalues of g

Evaluate as many possible equilibria topologies for values by varying g and initialconditions and thus build a topology bifurcation diagram of a N = 5 node network

Show that for high values of g the reached steady-state topology is the all-to-alltopology

with these aims we

build a grid for g ranging from g = 0.1 to g = 12, with a pace ∆g = 0.1

use Montecarlo techniques to generate 20 initial conditions for each topology

Note that with N = 5, 2N2−N

2 = 210 permutations could be possible; although somematrixes are not valid topologies for the system (4).Thus we account for 20 conditions for 687 topologies leading to 12740 simulations tobe performed for all the values of the gain grid g , which leads to 1528800 simulations!





Results



A simple example: the chain topology

Figure: Qualitative diagram of a chain topology for a 5 node network

0 5 10 15 20 25 30 35 40 45 50

1

2

3

Time[s]

ωi[

rad s

]

Figure: Synchronisation is not achieved for g = 1.0, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π





0

1

2

3

4

ωi[

rad s

]

π10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

π2

π

32π

2π

Time[s]

θ 1−θ i

[rad

]

Figure: All-to-all topology is achieved for g = 3.0, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π





1.5

2

2.5

ωi[

rad s

]

π10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

Time[s]

θ 1−θ i

[rad

]

Figure: Not All-to-all topology is achieved for g = 6.6, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π





0

2

4

6

8

ωi[

rad s

]

π10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

Time[s]

θ 1−θ i

[rad

]

Figure: All-to-all topology is achieved for g = 11.0, N = 5, θvis = π10 ,∇1,i [ωi ] = 0.1 · 2π





0 2 4 6 8 10 12 14 16

0

5

10

15

20

g [ 1t

]

Narcs

Figure: Narcs diagram, with respect to g , for a fixed initial condition: the chainGiovanni Pugliese Carratelli - M58/30 Numerical analysis of proximity oscillator networks


Topology bifurcation diagram

0 2 4 6 8 10 12 14

0

5

10

15

20

g [ 1t

]

Narcs

Figure: N = 5 network topology bifurcation diagram




0 2 4 6 8 10 12 14

0

5

10

15

20

g [ 1t

]

Narcs

Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization




0 2 4 6 8 10 12 14

0

5

10

15

20

g [ 1t

]

Narcs

Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization. The blue area denotes values of g that no matter the initialcondition lead to the all-to-all topology.




0 2 4 6 8 10 12 14

0

5

10

15

20

g [ 1t

]

Narcs

Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization. The blue area denotes values of g that no matter the initialcondition lead to the all-to-all topology. The area in green denotes values of g for which by varyingthe initial conditions more and more equilibria topologies appear.




0 2 4 6 8 10 12 14

0

5

10

15

20

g [ 1t

]

Narcs

Figure: N = 5 network topology bifurcation diagram. The red area denotes values of g notsufficient for synchronization. The blue area denotes values of g that independently for initialcondition lead to the all-to-all topology. The area in green denotes values of g for which by varyingthe initial conditions more and more equilibria topologies appear. The red area on the rightdenotes values of g that independently from the initial condition lead to the all-to-all topology.



Conclusions and future work

We have introduced a proximity rule in a network of heterogenous Kuramotooscillators, developing a restricted visibility model

We have illustrated how multiple equilibria topologies may exist

We have numerically shown the emergence of an interesting phenomenon that wecalled topological bifurcation topology bifurcation

The number of times the not all-to-all equilibrium topologies are seen isapproximately 5% of the total number of simulations

Statistical analysis have shown a dependance between the equilibria topologiesand the initial conditions

Future works will be devoted to investigate the possible emergence of thesephenomenon for different individual dynamics and coupling rules

We envision that control strategies may be developed to control both theindividual dynamics and the emerging topologies



A. Arenas, A. Diaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou,“Synchronization in complex networks,” pp. 1–80, May 2008. [Online]. Available:http://arxiv.org/abs/0805.2976

F. Radicchi and H. Meyer-Ortmanns, “Reentrant synchronization and patternformation in pacemaker-entrained Kuramoto oscillators,” Physical Review E,vol. 74, no. 2, p. 026203, Aug. 2006. [Online]. Available:http://link.aps.org/doi/10.1103/PhysRevE.74.026203


http://arxiv.org/abs/0805.2976

http://link.aps.org/doi/10.1103/PhysRevE.74.026203

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Numerical analysis of proximity oscillator networks · Numerical analysis of proximity oscillator networks Giovanni Pugliese Carratelli - M58/30 DIETI - Univesity Federico II of Naples