Post on 08-Aug-2020
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Slow-fast asymptotics for delay differential equations
Thomas Erneux and Lionel WeickerUniversité Libre de Bruxelles
Laurent LargerUniversité de Franche-Comté
2
1. Delay equation?
2. 2τ − periodic square-wave oscillations
3. τ − periodic square-wave oscillations
PLAN
3
( )( ) exp( )
dy tky t y A kt
dt= − → = −
t
y
1. Delay Equation?
4
( )( ) exp( )
dy tky t y A kt
dt= − → = −
( )( ) y sin( t)
if 2k
dy tky t A
dtτ ω
πτ
= − − → =
>> =
t
y
t
y
1. Delay Equation?
5
2009 – 2011
1. T. Erneux, “Applied Delay Differential Equations”, Springer (2009)2. A. Balachandran, T. Kamár-Nagy, D.E. Gilsinn, Eds. “Delay Differential
Equations, Recent Advances and New Directions”, Springer (2009)3. F.M. Atay, Ed. “Complex Time-Delay Systems” , Springer (2010)4. Theme Issue “Delay effects in brain dynamics” compiled by G. Stepan,
Phil. Trans. Roy. Soc A 367, 1059 (2009)5. Theme Issue “Delayed complex systems” compiled by W. Just, A. Pelster,
M. Schanz and E. Schöll, Phil. Trans. Roy. Soc A 368, 303 (2010)6. Special Issue on Time Delay Systems, T. Kalmár-Nagy, N. Olgac, and G.
Stépán, Eds., J. Vibration & Control, June/July 16 (2010)7. Hal Smith, An Introduction to Delay Differential Equations with
Applications to the Life Sciences, Springer (2010)8. M. Lakshmanan and D.V. Senthilkumar, Dynamics of Nonlinear Time-
Delay Systems , Springer Series in Synergetics (2011)9. T. Insperger and G. Stépán, Semi-discretization for time-delay systems –
Engineering applications, Springer (2011)10. Complex Systems, Fractionality, Time-delay and Sync hronization,
A.C.J. Luo and J.-Q. Sun (Eds.) Springer (2012)
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2. 2τ – periodic square-waves
1
3
' ( ( -1))
1.2, 0.02
x x f x t
f ax x
a
εε τ
ε
−
= − +== − += =
t
-2 -1 0 1 2 3 4
x
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
2+2εr
7
2. 2τ – periodic square-waves
1
3
' ( ( -1))
1.2, 0.02
x x f x t
f ax x
a
εε τ
ε
−
= − +== − += =
t
-2 -1 0 1 2 3 4
x
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
2+2εr
1
0
( ( 1)) 0
( )
n n
x f x t
x f x
ε
+
→− + − =
=
Period 2 fixed points = square-wave plateaus
1983 - 1996: S.N. Chow, J. Mallet-Paret, R.D. Nussbaum, J.K. Hale and W. Huang
8
' - ( ( -1))
11 cos( ( 1))
2
x x f x t
f x t
ε
β
= +
= + −
Larger et al. JOSA B 18, 1063 (2001)
Experiments using an optoelectronic oscillator
9
1
' - ( ( -1))
1( ) 1 cos( )
2
( ( 1)) 0
( )n n
x x f x t
f x x
x f x t
x f x
ε
β
+
= +
= +
− + − ==
Hopf1 Hopf2 Chaos
map 2.08 5.04 6.59
experiment 2.07 5.30 6.69
Larger et al. JOSA B 18, 1063 (2001)
Experiments using an optoelectronic oscillator
β1 2 3 4 5 6 7 8
xn
0
2
4
6
8
10
12
1010
3. τ – periodic square – wavesExperiments using an optoelectronic oscillator
Feedback gain
1111
2 2
1 -3 -1 -3
'
' cos ( ( 1) ) cos ( )
ε 10 δ τθ 8 10
y x
x x y x sε δ β
µτ −
=
= − − + − + Φ − Φ
≡ = ≡ = ×
1 2 2
0
1
0
' ( ) ' cos ( ( ) ) cos ( )
/ , y ( ') '
t
t
x x x t dt x t
s t x t dt
µ θ β τ
τ τ
−
−
= − − + − + Φ − Φ
≡ ≡
∫
∫
Model equations Peil et al, Phys. Rev. E 79, 026208 (2009)
12
2 2
'
' cos ( ( 1) ) cos ( )
y x
x x y x sε δ β=
= − − + − + Φ − Φ
s10006 10007 10008 10009 10010
y
-2.5
-2.4
-2.3
-2.2
-2.1
x
-0.8
0.0
0.8
s0x02
x011 − s0
-3 -3ε 10 δ 8.43 10
/ 4 0.1
1.2
1 ( 1 1/ 3)
1 ( 1/ 3 0)
(0) 0
x s
x s
y
πβ
= = ×Φ = − +
== − − ≤ < −= − ≤ <
=
Numerical Simulations
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Asymptotics
2 2
'
' cos ( ( 1) ) cos ( )
y x
x x y x sε δ β=
= − − + − + Φ − Φ
Part 1. Seek a -periodic solution:
( ) ( )
where 1 ( )
T
x s T x s
T εα ε− =
= +
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Asymptotics
2 2
'
' cos ( ( 1) ) cos ( )
y x
x x y x sε δ β=
= − − + − + Φ − Φ
2 2
Part 1. Seek a -periodic solution
( ) ( )
where 1 ( )
leading 0
'
0 cos ( ) cos ( )
0 ignore fast transitions layers
T
x s T x s
T
y x
x y x
εα εε
δ β
ε
− == +=
=
= − − + + Φ − Φ
= →
րx
-0.5 0.0 0.5
δy
-0.06
-0.04
-0.02
0.00
0.02
0.04 1β >
15
2 2
10 1
0 1
0 0
'
0 cos ( ) cos ( )
Part 2. Try
( ) ( ) ...
( ) ( ) ...
1 (0 ), 2 ( 1)j j
y x
x y x
y y s y s
x x s x s
j s s j s s
δ β
δδ
−
=
= − − + + Φ − Φ
= + += + +
= < < = < <
x
-0.8
0.0
0.8
s0
x02
x01
0 1
16
2 2
10 1
0 1
0 0
'
0 cos ( ) cos ( )
Part 2. Try
( ) ( ) ...
( ) ( ) ...
1 (0 ), 2 ( 1)j j
y x
x y x
y y s y s
x x s x s
j s s j s s
δ β
δδ
−
=
= − − + + Φ − Φ
= + += + +
= < < = < <
0
2 20 0 0
0 0
leading 0
' 0
0 cos ( ) cos ( )
' 0 ?
j j
y
x y x
y y cst
δ
β
==
= − − + + Φ − Φ
= → = x
-0.5 0.0 0.5
δy
-0.06
-0.04
-0.02
0.00
0.02
0.04
x
-0.8
0.0
0.8
s0
x02
x01
0 1
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1 0
1 1 0 1
0
O( )
'
0 2 sin(2 2 )
Solve for 1,2
Periodicity conditions
continuity condition at
j
j j j
y x
x y x x
j
s s
δ
β=
= − − − + Φ
=
=
x
-0.8
0.0
0.8
s0
x02
x01
0 1
1818
10007 10008 10009 s
x
-0.8
0.0
0.8
s0
0.5 1.0 1.5 2.0β β0.5 1.0 1.5 2.0
s0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
1 - s0(a) (b) (c)
Numerical solution Asymptotic analysis
β = 1.2
Bifurcation diagrams
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x
-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05
δy
-0.0020
-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
0.0020
s
10006 10007 10008 10009 10010
x
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
1.01β =
- Not branching from Hopf bifurcations
- Branching from a SN of limit-cycles?
Bifurcation point near β = 1
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Conclusions
1. Stable τ − periodic asymmetric square-waves not possible for first order scalar DDEs
2. Possible for second order scalar DDEs
Note: 1. Other periodic solutions coexist with the asymmetric
square waves2. No connection with the Hopf bifurcations from the zero
solution3. Isolated branch of periodic solutions4. We have ignored the fast transition
' ( ( -1))x x f x tε = − +
'
' ( 1)
y x
x x y f sε δ== − − + −