Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
1/7
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
2/7
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.
3/7
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.
3/7
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.
4/7
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.
5/7
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 .
6/7
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
7/7
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