as Hopf Bifurcations in TimeDelay Systems with Bandlimited Feedback


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1 as Hopf Bifurcations in TimeDelay Systems with Bandlimited 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 2226, 2005
2 Motivation Chaos: Lowspeed highspeed Application: Signal Source for Ranging (Radar) Chaotic signals have broad spectrum Fast decaying correlations Application: Communications Bandwidth compatible with infrastructure privacy, power efficiency,... wwwchaos.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 RFcomponents are ACcoupled (high pass filtering) How does ACcoupling affect the dynamics? SIAM Snowbird 2005 p.2/24
4 LowSpeed DelaySystem Example: No ACcoupling Lowpass 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 Ikedatype 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 HighSpeed DelaySystem With ACcoupling Bandlimited 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 timedelay systems with bandlimited feedback Study consequences of ACcoupling Focus on instability of steady state (Hopf Bifurcation) SIAM Snowbird 2005 p.4/24
6 Consequences of ACcoupling? H 2 Periodic Dynamics H 2 Chaos Frequency Frequency LowPass Filter introduces distortions HighPass Filter irrelevant BandPass Filter Increases complexity of chaos 1 Changes route to chaos 2 Changes steadystate 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 TimeDelay 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 ACcoupling 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 Codimensionone bifurcations = 1D curves (Fold, Hopf) Codimensiontwo bifurcations = points (BogdanovTakens,FoldHopf,DoubleHopf) 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 DoubleHopf exist indicate quasiperiodicity, 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 highspeed chaos generators for applications At high speed: timedelays are present signals are bandpass filtered Exploit timedelay to generate complex dynamics Exploit bandlimited 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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