1

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

13

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− =

= +

14

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== − − + −