This document summarizes research on the synchronous and oscillatory behavior of a network of heterogeneous Van der Pol oscillators under strong coupling. The key points are: 1) The network can achieve approximate synchronization if the "emergent Van der Pol oscillator" derived from blended dynamics has a stable limit cycle. 2) The blended dynamics reduce the high-dimensional network to a lower dimensional system that characterizes the network behavior. 3) If conditions for a stable limit cycle in the emergent oscillator are met, then the network oscillators will approximately synchronize their behavior to this limit cycle for sufficiently strong coupling.

1. Heterogeneous Van der Pol oscillators
under strong coupling
synchronous and oscillatory behavior of the network
Jin Gyu Lee and Hyungbo Shim
Control & Dynamic Systems Lab. Seoul National University
December 18, 2018

2. Biological systems
Characteristics:
Non-identical cells
Collective behavior, e.g., synchronization and oscillation
Robustness: Many good oscillators with few bad agents
Single channel communication
2 / 22

3. Heterogeneous Van der Pol oscillators
Internal model principle (Wieland, Wu, and Allgöwer, IEEE TAC, 2013):
Heterogeneous Van der Pol oscillators cannot achieve
synchronization unless they have a common internal model.
Previous works
Homogeneous Van der Pol oscillators
Rand and Holmes, Int. J. Nonlinear Mechanics, 1980
Low, Reinhall, Storti, and Goldman, Struct. Control Health Monit., 2006
Omelchenko, Zakharova, Hövel, Siebert, and Schöll, Chaos, 2015
Heterogeneous frequency, but a common internal model exists
Banning, Ph.D. dissertation, San Diego State University, 2011
3 / 22

4. Problem setting
Parameters
Van der Pol oscillator:
¨x − µω(1 − rx2
) ˙x + ω2
x = κu
µ determines the shape of the limit cycle
ω determines the period of the limit cycle
r determines the size of the limit cycle
Let ¯x:=
√
rx and ¯κ:=
√
rκ, then we get
¨¯x − µω(1 − ¯x2
) ˙¯x + ω2
¯x = ¯κu.
4 / 22

5. Problem setting
Parameters
Van der Pol oscillator:
¨x − µω(1 − rx2
) ˙x + ω2
x = κu
µ determines the shape of the limit cycle
ω determines the period of the limit cycle
r determines the size of the limit cycle
Let ¯x:=
√
rx and ¯κ:=
√
rκ, then we get
¨¯x − µω(1 − ¯x2
) ˙¯x + ω2
¯x = ¯κu.
4 / 22

6. Problem setting
Parameters
Van der Pol oscillator:
¨x − µω(1 − rx2
) ˙x + ω2
x = κu
µ determines the shape of the limit cycle
ω determines the period of the limit cycle
r determines the size of the limit cycle
Let ¯x:=
√
rx and ¯κ:=
√
rκ, then we get
¨¯x − µω(1 − ¯x2
) ˙¯x + ω2
¯x = ¯κu.
4 / 22

7. Problem setting
Parameters
Van der Pol oscillator:
¨x − µω(1 − rx2
) ˙x + ω2
x = κu
µ determines the shape of the limit cycle
ω determines the period of the limit cycle
r determines the size of the limit cycle
Let ¯x:=
√
rx and ¯κ:=
√
rκ, then we get
¨¯x − µω(1 − ¯x2
) ˙¯x + ω2
¯x = ¯κu.
4 / 22

8. Problem setting
Heterogeneity and Diffusive coupling
Network of heterogeneous Van der Pol oscillators:
¨xi − µiωi(1 − rixi
2
) ˙xi + ωi
2
xi = κiui, i ∈ N := {1, . . . , N},
with diffusive coupling:
ui = k
j∈Ni
αij(δjyj − δiyi), yi = axi + b ˙xi,
where the underlying graph is directed (Ni := {j ∈ N : αij > 0}).
5 / 22

9. State space representation
Individual dynamics:
˙xi = vi
˙vi = µiωi(1 − rix2
i )vi − ω2
i xi + κiui, i ∈ N
Diffusive coupling:
ui = k
j∈Ni
αij(δjyj − δiyi), yi = axi + bvi
Approximate synchronization:
lim sup
t→∞
xi(t)
vi(t)
−
xj(t)
vj(t)
≤ , ∀i, j ∈ N
6 / 22

10. State space representation
Individual dynamics:
˙xi = vi
˙vi = µiωi(1 − rix2
i )vi − ω2
i xi + κiui, i ∈ N
Diffusive coupling:
ui = k
j∈Ni
αij(δjyj − δiyi), yi = axi + bvi
Approximate synchronization:
lim sup
t→∞
xi(t)
vi(t)
−
xj(t)
vj(t)
≤ , ∀i, j ∈ N
6 / 22

11. Simulation results
Homogeneous Van der Pol oscillators
7 / 22

12. Simulation results
Heterogeneous Van der Pol oscillators
8 / 22

13. Summary of the contents
Q) When does the network approximately synchronize?
A)
Q) What would be the synchronized behavior?
A)
9 / 22

14. State space representation
Individual dynamics:
˙xi = vi
˙vi = µiωi(1 − rix2
i )vi − ω2
i xi + κiui
Diffusive coupling:
ui = k
j∈Ni
αij(δjyj − δiyi), yi = axi + bvi,
where the underlying graph is directed.
10 / 22

15. State space representation
Individual dynamics:
˙xi = vi
˙vi = µiωi(1 − rix2
i )vi − ω2
i xi + κiui
Diffusive coupling:
ui = k
j∈Ni
αij(δjyj − δiyi), yi = axi + bvi,
where the underlying graph is undirected.
10 / 22

16. Blended dynamics
Brief introduction
Analysis and synthesis of a multi-agent system,
˙xi = fi(t, xi) + kBi
j∈Ni
αij(xj − xi) ∈ Rn
, i ∈ N,
or
˙xi = fi(t, xi) + k
j∈Ni
αij(Cjxj − Cixi) ∈ Rn
, i ∈ N,
by the reduced dimensional system called blended dynamics.
The behavior of the original network is characterized by the
behavior of the blended dynamics.
J. G. Lee and H. Shim, “A tool for analysis and synthesis of heterogeneous multi-agent systems under
rank-deficient coupling,” under review for Automatica, available at arXiv:1804.00638.
11 / 22

17. Blended dynamics
A special case
Original network, 2N dimension:
˙yi = gi(t, yi, zi) ∈ R, i ∈ N,
˙zi = hi(t, yi, zi) + kb
j∈Ni
αij(zj − zi) ∈ R, i ∈ N,
where b is a positive constant.
Blended dynamics, N + 1 dimension:
˙ˆyi = gi(t, ˆyi, ˆz) ∈ R, i ∈ N,
˙ˆz =
1
N
N
i=1
hi(t, ˆyi, ˆz) ∈ R.
J. G. Lee and H. Shim, “A tool for analysis and synthesis of heterogeneous multi-agent systems under
rank-deficient coupling,” under review for Automatica, available at arXiv:1804.00638.
12 / 22

18. Blended dynamics
Network of heterogeneous Van der Pol oscillators
Consider ¯xi := xi and ¯vi := (a/b)xi + vi. Then, we obtain
˙¯xi = −
a
b
¯xi + ¯vi, i ∈ N,
˙¯vi = a
b + µiωi(1 − ri¯x2
i ) −a
b ¯xi + ¯vi − ω2
i ¯xi
+ kb
j∈Ni
αij(¯vj − ¯vi), i ∈ N.
13 / 22

19. Blended dynamics
Network of heterogeneous Van der Pol oscillators
Blended dynamics, N + 1 dimension:
˙ˆxi = −
a
b
ˆxi + ˆv, i ∈ N,
˙ˆv =
1
N
N
i=1
a
b + µiωi(1 − riˆx2
i ) −a
b ˆxi + ˆv − ω2
i ˆxi
13 / 22

20. Blended dynamics
Asymptotic behavior
Blended dynamics, N + 1 dimension:
˙ˆxi = −
a
b
ˆxi + ˆv, ⇒ lim
t→∞
|ˆxi − ˆxj| = 0, ∀i, j ∈ N,
˙ˆv =
1
N
N
i=1
a
b + µiωi(1 − riˆx2
i ) −a
b ˆxi + ˆv − ω2
i ˆxi
14 / 22

21. Blended dynamics
Asymptotic behavior
Blended dynamics, ˆxi → ˆx, 2 dimension:
˙ˆx = −
a
b
ˆx + ˆv,
˙ˆv =
1
N
N
i=1
a
b + µiωi(1 − riˆx2) −a
b ˆx + ˆv − ω2
i ˆx
= a
b + 1
N
N
i=1 µiωi(1 − riˆx2) −a
b ˆx + ˆv − 1
N
N
i=1 ω2
i ˆx
14 / 22

22. Blended dynamics
Asymptotic behavior
Let x := ˆx and v := −(a/b)ˆx + ˆv. Then, we obtain
˙x = v,
˙v = 1
N
N
i=1 µiωi v − 1
N
N
i=1 µiωiri x2
v − 1
N
N
i=1 ω2
i x.
=: ˆµˆωv − ˆµˆωˆrx2
v − ˆω2
x.
Assume
1
N
N
i=1 µiωi > 0, 1
N
N
i=1 ω2
i > 0, 1
N
N
i=1 µiωiri > 0.
=: ˆµˆω =: ˆω2
=: ˆµˆωˆr
15 / 22

23. Blended dynamics
Asymptotic behavior
Let x := ˆx and v := −(a/b)ˆx + ˆv. Then, we obtain
˙x = v,
˙v = 1
N
N
i=1 µiωi v − 1
N
N
i=1 µiωiri x2
v − 1
N
N
i=1 ω2
i x,
=: ˆµˆωv − ˆµˆωˆrx2
v − ˆω2
x.
Assume
1
N
N
i=1 µiωi > 0, 1
N
N
i=1 ω2
i > 0, 1
N
N
i=1 µiωiri > 0.
=: ˆµˆω =: ˆω2
=: ˆµˆωˆr
15 / 22

24. Blended dynamics
Asymptotic behavior
Let x := ˆx and v := −(a/b)ˆx + ˆv. Then, we obtain
˙x = v,
˙v = 1
N
N
i=1 µiωi v − 1
N
N
i=1 µiωiri x2
v − 1
N
N
i=1 ω2
i x,
=: ˆµˆω(1 − ˆrx2
)v − ˆω2
x.
Assume
1
N
N
i=1 µiωi > 0, 1
N
N
i=1 ω2
i > 0, 1
N
N
i=1 µiωiri > 0.
=: ˆµˆω =: ˆω2
=: ˆµˆωˆr
15 / 22

25. Emergent Van der Pol oscillator
Now, we obtain
˙x = v,
˙v = ˆµˆω(1 − ˆrx2
)v − ˆω2
x,
which we call the emergent Van der Pol oscillator.
Limit cycle of the blended dynamics:
¯Γ :=








x
...
x
(a/b)x + v





∈ RN+1
:
x
v
∈ γ ⊂ R2



γ: Limit cycle of the emergent Van der Pol oscillator
16 / 22

26. Blended dynamics has a limit cycle
Theorem (Blended dynamics)
Blended dynamics has a stable limit cycle if and only if the
emergent Van der Pol oscillator has a stable limit cycle, i.e.,
1
N
N
i=1
µiωi > 0,
1
N
N
i=1
µiωiri > 0.
Moreover, the limit cycle is unique and is explicitly given as
¯Γ :=








x
...
x
(a/b)x + v





∈ RN+1
:
x
v
∈ γ ⊂ R2



.
17 / 22

27. Network of Van der Pol oscillators approximately synchronize
Theorem (Network of Van der Pol oscillators)
If the emergent Van der Pol oscillator has a stable limit cycle,
i.e.,
1
N
N
i=1
µiωi > 0,
1
N
N
i=1
µiωiri > 0,
then for sufficiently large k, we have
lim sup
t→∞







x1
v1
...
xN
vN







Γ
≤ , where Γ:=










x
v
...
x
v







∈ R2N
:
x
v
∈γ ⊂R2



,
from which we obtain approximate synchronization.
18 / 22

28. Oscillation near the limit cycle
In fact, ¯Γ :=








x
...
x
(a/b)x + v





∈ RN+1 :
x
v
∈ γ ⊂ R2



is a
periodic orbit of the blended dynamics.
There is a periodic orbit near Γ:=










x
v
...
x
v







∈ R2N :
x
v
∈γ ⊂R2



.
Each individual
xi
vi
oscillates near the limit cycle γ.
19 / 22

29. Oscillation near the limit cycle
In fact, ¯Γ :=








x
...
x
(a/b)x + v





∈ RN+1 :
x
v
∈ γ ⊂ R2



is a
periodic orbit of the blended dynamics.
There is a periodic orbit near Γ:=










x
v
...
x
v







∈ R2N :
x
v
∈γ ⊂R2



.
Each individual
xi
vi
oscillates near the limit cycle γ.
19 / 22

30. Oscillation near the limit cycle
In fact, ¯Γ :=








x
...
x
(a/b)x + v





∈ RN+1 :
x
v
∈ γ ⊂ R2



is a
periodic orbit of the blended dynamics.
There is a periodic orbit near Γ:=










x
v
...
x
v







∈ R2N :
x
v
∈γ ⊂R2



.
Each individual
xi
vi
oscillates near the limit cycle γ.
19 / 22

31. Conclusions
Answering the questions
Q) When does the network approximately synchronize?
A) When the emergent Van der Pol oscillator has a stable limit
cycle.
Q) What would be the synchronized behavior?
A) Oscillation near the limit cycle of the emergent Van der Pol
oscillator.
20 / 22

32. Conclusions
Answering the questions
Q) When does the network approximately synchronize?
A) When the emergent Van der Pol oscillator has a stable limit
cycle.
Q) What would be the synchronized behavior?
A) Oscillation near the limit cycle of the emergent Van der Pol
oscillator.
20 / 22

33. Conclusions
Summary
Non-identical agents
˙xi = vi, ˙vi = µiωivi − µiωirix2
i vi − ω2
i xi + ui
Single channel communication
ui = k
j∈Ni
αij(yj − yi), yi = axi + bvi
Collective behavior: Emergent Van der Pol oscillator
˙x = v, ˙v = ˆµˆωv − ˆµˆωˆrx2
v − ˆw2
x
Robustness: Many good oscillators with few bad agents
ˆµˆω = 1
N
N
i=1 µiωi > 0, ˆµˆωˆr = 1
N
N
i=1 µiωiri > 0.
21 / 22

34. Conclusions
Summary
Non-identical agents
˙xi = vi, ˙vi = µiωivi − µiωirix2
i vi − ω2
i xi + ui
Single channel communication
ui = k
j∈Ni
αij(yj − yi), yi = axi + bvi
Collective behavior: Emergent Van der Pol oscillator
˙x = v, ˙v = ˆµˆωv − ˆµˆωˆrx2
v − ˆw2
x
Robustness: Many good oscillators with few bad agents
ˆµˆω = 1
N
N
i=1 µiωi > 0, ˆµˆωˆr = 1
N
N
i=1 µiωiri > 0.
21 / 22

35. Conclusions
Summary
Non-identical agents
˙xi = vi, ˙vi = µiωivi − µiωirix2
i vi − ω2
i xi + ui
Single channel communication
ui = k
j∈Ni
αij(yj − yi), yi = axi + bvi
Collective behavior: Emergent Van der Pol oscillator
˙x = v, ˙v = ˆµˆωv − ˆµˆωˆrx2
v − ˆw2
x
Robustness: Many good oscillators with few bad agents
ˆµˆω = 1
N
N
i=1 µiωi > 0, ˆµˆωˆr = 1
N
N
i=1 µiωiri > 0.
21 / 22

36. Conclusions
Summary
Non-identical agents
˙xi = vi, ˙vi = µiωivi − µiωirix2
i vi − ω2
i xi + ui
Single channel communication
ui = k
j∈Ni
αij(yj − yi), yi = axi + bvi
Collective behavior: Emergent Van der Pol oscillator
˙x = v, ˙v = ˆµˆωv − ˆµˆωˆrx2
v − ˆw2
x
Robustness: Many good oscillators with few bad agents
ˆµˆω = 1
N
N
i=1 µiωi > 0, ˆµˆωˆr = 1
N
N
i=1 µiωiri > 0.
21 / 22

37. Conclusions
Summary
Non-identical agents
˙xi = vi, ˙vi = µiωivi − µiωirix2
i vi − ω2
i xi + ui
Single channel communication
ui = k
j∈Ni
αij(yj − yi), yi = axi + bvi
Collective behavior: Emergent Van der Pol oscillator
˙x = v, ˙v = ˆµˆωv − ˆµˆωˆrx2
v − ˆw2
x
Robustness: Many good oscillators with few bad agents
ˆµˆω = 1
N
N
i=1 µiωi > 0, ˆµˆωˆr = 1
N
N
i=1 µiωiri > 0.
21 / 22

38. Conclusions
22 / 22