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1 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

2 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

3 High Speed Circuits (RF) Delay always present Microwaves : f = GHz Propagation with speed of light Wavelength λ = cm H 2 Transfer Function ω h ω b ω l ω Many RF-components are AC-coupled (high pass filtering) How does AC-coupling affect the dynamics? SIAM Snowbird 2005 p.2/24

4 Low-Speed Delay-System Example: No AC-coupling Low-pass feedback 1 ω l ẋ(t) = x(t) + γ f[x(t τ)] H 2 Low Pass Filter Nonlinearity x out = f(x in) T ω l ω 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

5 High-Speed Delay-System With AC-coupling Band-limited feedback H 2 Band Pass Filter ω h ω b ω l ω 1 ω l ẋ(t) = x(t) + γ f[y(t τ)] 1 ω h ẏ(t) = y(t) + ω 1 h ẋ(t) Nonlinearity x out = f(x in) T 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) SIAM Snowbird 2005 p.4/24

6 Consequences of AC-coupling? H 2 Periodic Dynamics H 2 Chaos Frequency Frequency Low-Pass Filter introduces distortions High-Pass Filter irrelevant Band-Pass Filter Increases complexity of chaos 1 Changes route to chaos 2 Changes steady-state bifurcations 2 [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

7 2.) Experimental Results SIAM Snowbird 2005 p.6/24

8 Route to Chaos Output (mv) Output (mv) Output (mv) Time (ns) Time (ns) Time (ns) Steady State Periodic Quasi Periodic Chaos Increasing Feedback Strength Andronov Hopf Bifurcation SIAM Snowbird 2005 p.7/24

9 Hopf Bifurcation in Experiment 7 Interferometer Output Amplitude (mw) Frequency stays roughly constant as γ is increased Amplitude smoothly grows Feedback Gain γ (mv/mw) SIAM Snowbird 2005 p.8/24

10 Hopf Bifurcation in Experiment Positive Feedback 100 Negative Feedback Frequency (MHz) 50 6/(2 τ) 4/(2 τ) 2/(2 τ) Frequency (MHz) /(2 τ) 3/(2 τ) 1/(2 τ) Feedback Delay τ (ns) Even modes reach instability threshold first Feedback Delay τ (ns) Odd modes reach instability threshold first Only the 1/(2τ) mode exists in the Ikeda system. SIAM Snowbird 2005 p.9/24

11 Heuristic explanation Even Odd 0 τ One Round trip 0 τ 2τ One Round trip τ 2τ τ 2τ 3τ Positive Feedback Negative Feedback τ 2τ τ 2τ 3τ SIAM Snowbird 2005 p.10/24

12 Heuristic explanation Increase Time-Delay τ 1/(2 τ) 3/(2 τ) 5/(2 τ) Frequency 1/(2 τ) 3/(2 τ) 5/(2 τ) 7/(2 τ) Frequency SIAM Snowbird 2005 p.11/24

13 2.) Theory Clearly AC-coupling can change the dynamics. Can we quantitatively predict the observed behavior? How general is the observed behavior? SIAM Snowbird 2005 p.12/24

14 Model Equations 1 ω h ẋ = x + 1 ω h ẏ 1 ω l ẏ = y + γ f [x τ ] H 2 Band Pass Filter ω h ω b Nonlinearity x out = f(x in) T ω l ω γ ẋ(t) ẏ(t) = x(t) + y(t) + γf[x(t τ)] = ω 2 b x(t) Parameters: γ, τ, ω b Maximal transmission at ω b f(0) = 0 steady state solution is x = y = 0 SIAM Snowbird 2005 p.13/24

15 Linear Stability Analysis Investigate how system evolves after small perturbation Nonlinear DDE Linearized DDE W U E U W S E S Sufficient to determine stability of Linearized DDE Ansatz : c λ e λt Characteristic Equation SIAM Snowbird 2005 p.14/24

16 Characteristic Equation λ 2 + λ + ωb 2 [γf (0)]λe λτ = 0. Effective Slope: b = γf (0) Im(λ) Re(λ) n S n U n C 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) SIAM Snowbird 2005 p.15/24

17 Result - Critical Gain Effectiv Slope b = γ f (0) Effective Slope b = γ f (0) Unstable Delay τ Stable Unstable Delay τ 0 4 Unstable Stable Hopf Bifurcation SIAM Snowbird 2005 p.16/24

18 Result - Double Hopf Effective Slope b = γ f (0) Unstable Delay τ 6 Stable Double Hopf Bifurcation SIAM Snowbird 2005 p.17/24

19 Result- Frequency Imaginary Part of Eigenvalue Ω Ω C n (τ) numerical solution Ω C n (τ)=2π n/(2τ) (n=1,3,5,...) Delay τ SIAM Snowbird 2005 p.18/24

20 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 SIAM Snowbird 2005 p.19/24

21 Hopf bifurcation type Is the Hopf bifurcation supercritical or subcritical? Supercritical Subcritical x x 0 µ 0 µ Found in our experiments Is it possible? SIAM Snowbird 2005 p.20/24

22 Result - bifurcation type ẋ(t) = x(t) + y(t) + γf[x(t τ)] ẏ(t) = ω b 2 x(t) Derived for general nonlinearity f condition for Hopf bifurcation type 1 Both supercritical and subcritical bifurcation possible [1] L. Illing and D. J. Gauthier, submitted SIAM Snowbird 2005 p.21/24

23 Examples - bifurcation type Example: f(x) = (x x2 + x 3 ) e x2 Effective Slope b Unstable Stable Supercirtical Subcritical Unstable Delay τ Example: f(x) = sin(x) Always supercritical SIAM Snowbird 2005 p.22/24

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 SIAM Snowbird 2005 p.23/24

25 Thank you for your attention! SIAM Snowbird 2005 p.24/24

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