as
Hopf Bifurcations in Time-Delay Systems with
Band-limited Feedback
Lucas Illing and Daniel J. Gauthier
Department of Physics
Center for Nonlinear and Complex Systems
Duke University, North Carolina
Siam Conference on Applications of Dynamical Systems
Snowbird, UT, May 22-26, 2005
Motivation
Chaos: Low-speed → high-speed
Application: Signal Source for
Ranging (Radar)
Chaotic signals have broad
spectrum
Fast decaying correlations
Application: Communications
Bandwidth compatible with
infrastructure
privacy, power efficiency, ...
www-chaos.umd.edu
SIAM Snowbird 2005 – p.1/24
High Speed Circuits (RF)
Delay always present
Microwaves : f = 0.3-30 GHz
Propagation with speed of light
Wavelength λ = 100-1 cm
Transfer Function
|H|
2
ωh ωb
ωl
ω
Many RF-components
are AC-coupled (high
pass filtering)
How does AC-coupling affect the dynamics ?
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Low-Speed Delay-System
Example:
Low−Pass Filter
No AC-coupling
|H|
2
ωl
Low-pass feedback
1
ωl ẋ(t)
= −x(t) + γ f [x(t − τ )]
ω
T
Nonlinearity
xout = f(x in )
γ
Ikeda-type systems (scalar DDE) studied intensively
K. Ikeda, J. K. Hale, W. Huang, T. Erneux, L. Larger, J. P. Goedgebuer, P. Mandel,
R. Kapral, J. Othsubo, P. L. Buono, J. Belair, A. Longtin, F. Giannakopoulos, S.
Yanchuk, ...
[Reference:] K. Ikeda, Opt. Commun. 30 (1979) 257
SIAM Snowbird 2005 – p.3/24
High-Speed Delay-System
Band−Pass Filter
With AC-coupling
Band-limited feedback
1
ωl ẋ(t) = −x(t) + γ f [y(t − τ )]
1
1
ẏ(t)
=
−y(t)
+
ωh
ωh ẋ(t)
|H|
2
ωh ωb
ωl
ω
T
Nonlinearity
xout = f(x in )
γ
Little is known about time-delay systems with
band-limited feedback
Study consequences of AC-coupling
Focus on instability of steady state (Hopf
Bifurcation)
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Consequences of AC-coupling?
Periodic Dynamics
2
Chaos
|H| 2
|H|
Frequency
Frequency
Low-Pass Filter
introduces distortions
High-Pass Filter
irrelevant
Band-Pass Filter
Increases complexity of
chaos1
Changes route to chaos2
Changes steady-state
bifurcations2
[1] V. S. Udaltsov, et al., IEEE Trans. Circuits Syst. I 49 (2002) 1006
[2] J. N. Blakely, et al., IEEE J. Quantum Electron. 40 (2004) 299
SIAM Snowbird 2005 – p.5/24
2.) Experimental Results
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80
60
60
40
40
20
0
20
0
-20
-20
-40
-40
-60
0
10
Time (ns)
Steady State
20
400
Output (mV)
80
Output (mV)
Output (mV)
Route to Chaos
-60
0
200
0
10
Time (ns)
20
Periodic Quasi−Periodic
-200
0
10
20
Time (ns)
Chaos
Increasing Feedback Strength
Andronov−Hopf Bifurcation
SIAM Snowbird 2005 – p.7/24
Hopf Bifurcation in Experiment
7
Interferometer Output
Amplitude (mW)
6
Frequency stays
roughly constant
as γ is increased
5
4
3
Amplitude
smoothly
2
1
0
4
5
6
7
Feedback Gain γ (mV/mW)
grows
8
SIAM Snowbird 2005 – p.8/24
Hopf Bifurcation in Experiment
Positive Feedback
100
80
6/(2 τ)
50
4/(2 τ)
2/(2 τ)
0
0
20
40
Feedback Delay τ (ns)
60
Frequency (MHz)
Frequency (MHz)
100
Negative Feedback
5/(2 τ)
60
3/(2 τ)
40
20
0
0
1/(2 τ)
10
20
30
Feedback Delay τ (ns)
40
Even modes reach instability Odd modes reach instability
threshold first
threshold first
Only the 1/(2τ ) mode exists in the Ikeda system.
SIAM Snowbird 2005 – p.9/24
Heuristic explanation
Even
0
Odd
τ
0
One Round−trip
τ
2τ
2τ
2τ
One Round−trip
τ
Positive Feedback
τ
τ
2τ
3τ
Negative Feedback
τ
2τ
3τ
SIAM Snowbird 2005 – p.10/24
Heuristic explanation
Increase Time-Delay τ
1/(2 τ)
3/(2 τ)
Frequency
5/(2 τ)
1/(2 τ) 3/(2 τ) 5/(2 τ) 7/(2 τ)
Frequency
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2.) Theory
Clearly AC-coupling can change the dynamics.
Can we quantitatively predict the observed behavior?
How general is the observed behavior?
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Model Equations
Band−Pass Filter
|H|
1
1
ẋ
=
−x
+
ωh
ωh ẏ
1
ωl ẏ = −y + γ f [xτ ]
2
ωh ωb
ωl
ω
T
Nonlinearity
xout = f(x in )
γ
ẋ(t) = −x(t) + y(t) + γf [x(t − τ )]
ẏ(t) = −ωb 2 x(t)
Parameters: γ , τ , ωb
Maximal transmission at ωb
f (0) = 0 → steady state solution is x = y = 0
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Linear Stability Analysis
Investigate how system evolves after small perturbation
Nonlinear DDE
Linearized DDE
U
U
W
E
S
W
S
E
Sufficient to determine stability of Linearized DDE
Ansatz : cλ eλt → Characteristic Equation
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Re(λ)
nU
n
Im(λ)
n
S
C
Effective Slope: b = γf 0 (0)
λ2 + λ + ωb 2 − [γf 0 (0)]λe−λτ = 0.
Characteristic Equation
Bifurcations: Re λ = 0 ( Im λ = ΩC )
Plot in τ − b−space locations where Re λ(τ, b) = 0
Codimension-one bifurcations = 1-D curves
(Fold, Hopf)
Codimension-two bifurcations = points
(Bogdanov-Takens,Fold-Hopf,Double-Hopf)
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Result - Critical Gain
Effective Slope b = γ f’(0)
1.01
Effectiv Slope b = γ f’(0)
1.12
1.08
4
8
6
4
2
8
6
Unstable
6
4
2
4
2
1.00
0
Unstable
50
1.04
100
2
4
2
Stable
Delay τ
150
200
1.00
Hopf Bifurcation
Stable
-1.00
Unstable
-1.02
0
100
200
Delay τ
300
400
500
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Result - Double Hopf
Effective Slope b = γ f’(0)
1.01
4
8
6
4
2
8
6
Unstable
6
4
2
4
2
1.00
0
50
2
4
2
Stable
100
Delay τ
150
200
Double−Hopf Bifurcation
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Result- Frequency
Imaginary Part of Eigenvalue Ω
n
ΩC(τ) numerical solution
n
ΩC(τ)=2π n/(2τ) (n=1,3,5,...)
0.2
0.1
0
0
50
Delay τ
100
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Result- Linear Stability Analysis
For general nonlinear f
Generically Hopf bifurcations
Can determine quantitatively critical gain and
frequency at onset
Double-Hopf exist
→ indicate quasi-periodicity, chaos
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Hopf bifurcation type
Is the Hopf bifurcation supercritical or subcritical ?
Supercritical
Subcritical
|x|
|x|
0
0
µ
Found in our experiments
µ
Is it possible ?
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Result - bifurcation type
ẋ(t) = −x(t) + y(t) + γf [x(t − τ )]
ẏ(t) = −ωb 2 x(t)
Derived for general nonlinearity f condition for Hopf
bifurcation type1
Both supercritical and subcritical bifurcation possible
[1] L. Illing and D. J. Gauthier, submitted
SIAM Snowbird 2005 – p.21/24
Examples - bifurcation type
Effective Slope b
Example: f (x) = (x +
1 2
2x
2
3
−x
+ x )e
Supercirtical
Subcritical
Unstable
Unstable
Stable
Delay τ
Example: f (x) = sin(x) → Always supercritical
SIAM Snowbird 2005 – p.22/24
Summary
Want simple high-speed chaos generators for
applications
At high speed:
time-delays are present
signals are bandpass filtered
Exploit time-delay to generate complex dynamics
Exploit band-limited feedback, e.g. tailor signal to fit
communication band
Many open question concerning the dynamics
For steady state instability:
Quantitative theory for general nonlinear f
Agreement of experiment and theory
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Thank you for your attention !
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