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Global convergence of pulse-coupled oscillators on trees

Hanbaek Lyu

The Ohio State University

www.hanbaeklyu.com

Joint Mathematics Meeting 2018, San Diego

Jan. 13 2018

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1. Introduction

The 4-coupling

1 0

1

½ ¼

½

¼

½ 0 ½

½

0 1

1 1

1

, ,

∈ {0, ¼} ∈ (¼, ½) ∈ {½, ¾}

0

¾

½

¼

∩

∪ ∩

∖

Figure: PRC for the4-coupling

• A pulse-coupled oscillator evolves on unit circleS1 = R/Z with constant unit speed, fires pulseat phase 1, and adjusts its phase uponreceiving pulse from a neighbor.{

φ̇v (t) ≡ 1 not upon pulseφv (t

+) = f (φv (t)) upon pulse,

• The way an oscillator responses to pulse signalis given by the phase response curve (PRC).

• The 4-coupling is the pulse-coupling with thePRC to the left, which extends the 4-colorfirefly cellular automaton.

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1. Introduction

First result

Theorem (L. 2017)

Let T = (V ,E ) be a finite tree with diameter d .

(i) If T has maximum degree ≤ 3, arbitrary phase configuration on Tsynchronizes by time 51d .

(ii) If T has maximum degree ≥ 4, then there exists a non-synchronizingphase configuration on T .

u

u

u

u

𝑎 0 ¼

u

u

u

u

𝑏 u

u

u

u

𝑐 u

u

u

u

𝑑

u

u

u

u

𝑎 u

u

u

u

𝑏 u

u

u

u

𝑐 u

u

u

u

𝑑

𝛽 = 0 𝜇 = (0,0,0,1) 𝜎 = 0

𝛽 = 0 𝛽 = 0 𝜇 = (0,0,1,1) 𝜎 = 0

𝛽 = ¼ 𝜇 = (0,0,1,1) 𝜎 = 0

𝛽 = ¼ + 0 𝜇 = (0,1,1,1) 𝜎 = 0

u

u

u

u

𝑒

𝛽 = ½ 𝜇 = (0,1,1,1) 𝜎 = 0

u

u

u

u

f

𝛽 = ½ + 0 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

𝑔

𝛽 = ¾ 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

ℎ

𝛽 = ¾ + 0 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

𝑖

𝛽 = 1 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

𝑗

𝛽 = 0 𝜇 = (1,1,1,0) 𝜎 = 1

u u

u

𝑘

𝛽 = 3/8 𝜇 = (1,1,0,0) 𝜎 = 1

u

u u

u

𝑙

𝛽 = 3/8 + 0 𝜇 = (0,0,0,0) 𝜎 = 2

u

𝑣

𝑣 ¼

0

Figure: An example of 4-coupled phase dynamics on a star with center v = � andleaves = •. In every 1/4 second, one of the leaves blink and pulls the center by1/4 in phase, resulting in a non-synchronizing orbit.

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1. Introduction

First result

Theorem (L. 2017)

Let T = (V ,E ) be a finite tree with diameter d .

(i) If T has maximum degree ≤ 3, arbitrary phase configuration on Tsynchronizes by time 51d .

(ii) If T has maximum degree ≥ 4, then there exists a non-synchronizingphase configuration on T .

u

u

u

u

𝑎 0 ¼

u

u

u

u

𝑏 u

u

u

u

𝑐 u

u

u

u

𝑑

u

u

u

u

𝑎 u

u

u

u

𝑏 u

u

u

u

𝑐 u

u

u

u

𝑑

𝛽 = 0 𝜇 = (0,0,0,1) 𝜎 = 0

𝛽 = 0 𝛽 = 0 𝜇 = (0,0,1,1) 𝜎 = 0

𝛽 = ¼ 𝜇 = (0,0,1,1) 𝜎 = 0

𝛽 = ¼ + 0 𝜇 = (0,1,1,1) 𝜎 = 0

u

u

u

u

𝑒

𝛽 = ½ 𝜇 = (0,1,1,1) 𝜎 = 0

u

u

u

u

f

𝛽 = ½ + 0 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

𝑔

𝛽 = ¾ 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

ℎ

𝛽 = ¾ + 0 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

𝑖

𝛽 = 1 𝜇 = (1,1,1,1) 𝜎 = 1

u

u

u

u

𝑗

𝛽 = 0 𝜇 = (1,1,1,0) 𝜎 = 1

u u

u

𝑘

𝛽 = 3/8 𝜇 = (1,1,0,0) 𝜎 = 1

u

u u

u

𝑙

𝛽 = 3/8 + 0 𝜇 = (0,0,0,0) 𝜎 = 2

u

𝑣

𝑣 ¼

0

Figure: An example of 4-coupled phase dynamics on a star with center v = � andleaves = •. In every 1/4 second, one of the leaves blink and pulls the center by1/4 in phase, resulting in a non-synchronizing orbit.

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1. Introduction

The adaptive 4-coupling

• In order to overcome the degree constraint,introduce an auxiliary state variableσv (t) ∈ {0, 1, 2} for each node v ∈ V .

• Whenever σv = 0, v uses the 4-coupling PRC(top).

• Whenever σv ∈ {1, 2}, v ignores all input pulses(bottom)

• Dynamics of this auxiliary variable is carefullycoupled with the phase dynamics.

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1. Introduction

Main result

Theorem (L. 2017)

Let T = (V ,E ) be a finite tree with diameter d . Then arbitrary initialjoint configuration on T synchronizes under the adaptive 4-coupling bytime 83d .

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1. Introduction

Simulations: on lattices and its spanning tree

Figure: A4C on square lattice (withMoore neighborhood, deg 8)

Figure: A4C on a uniform spanning treeof the lattice on the left

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1. Introduction

Thank You!

Reference.Hanbaek Lyu, “Global synchronization of pulse-coupled oscillators ontrees” SIAM Journal on Applied Dynamical Systems (to appear)arXiv:1604.08381

1. Introduction