A State-Space Model for a Nonlinear Time-Delayed Feedback Loop

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1 A for a Nonlinear Time-Delayed Feedback Loop Karl Schmitt Advisors: Jim Yorke, Rajarshi Roy, Tom Murphy AMSC 663 October 14, / 21

2 Goal To implement an alternative, discrete time model for coupled nonlinear (chaotic) time-delayed feedback loops. 2 / 21

3 Introduction to Chaos Properities of a Chaotic System sensitive to initial conditions topologically mixing periodic orbits are dense Classic Example: The Lorenz System ẋ = σ(y x) σ = 10 ẏ = rx y 20xz r = 60 ż = 5xy bz b = / 21

4 Lorenz Synchronization Coupled Lorenz Equations: ẋ 1 = σ(y 1 x 1 ) + γ(x 2 x 1 ) ẋ 2 = σ(y 2 x 2 ) + γ(x 1 x 2 ) y 1 = rx 1 y 1 20x 1 z 1 ẏ 2 = rx 2 y 2 20x 2 z 2 ż 1 = 5x 1 y 1 bz 1 ż 2 = 5x 2 y 2 bz 2 σ = 10 r = 60 b = / 21

5 System Overview Laser Diode MZM Bias Photoreceiver Amplifier Delay (τ) V(t) Low-pass High-pass x(t) + τẋ(t) + 1 θ t t o x(s)ds = βcos 2 [x(t T ) + φ] x(t) is normalized RF voltage τ is the low pass filter time constant θ is the high pass filter time constant T is the time delay in the loop β is the feedback strength φ is the phase offset in nonlinearity [Kouomou, Thesis] 5 / 21

6 Modeling the system β = 1.15 β = 2.00 β = 2.45 Traditional numeric methods can be applied Generates a variety of dynamics β = 3.05 β = 4.30 Problems Step-Size Concerns Speed (3+ Min per run) Complicated filters difficult to model t (ns) / 21

7 Comparing Differentials vs. State-Space Differential Form x(t) + τẋ(t) + 1 t θ t o x(s)ds = y(t) Laser Diode MZM Bias Photoreceiver Amplifier Delay (τ) V(t) Low-pass High-pass Discrete State-Space Form u[n + 1] = Au[n] + By[n] x[n] = Cu[n] + Dy[n] 7 / 21

8 Some Details Discrete State-Space Form u[n + 1] = Au[n] + By[n] x[n] = Cu[n] + Dy[n] Choose canonical form from z-space transform of discrete filter Include nonlinear delayed feedback (y[n] f (x[n k]) This gives us for matrices: A= B=1 C= D=0 8 / 21

9 Simplified Model With all of the previous simplifications we are left with: u[n + 1] = Au[n] + Bf (x[n k]) x[n] = Cu[n] u[n + 1] = Au[n] + Bf (Cu[n k]) u[n + 1] = Au[n] + Bβcos 2 (Cu[n k] + φ) Where we have reintroduced the nonlinearity from the Mach-Zehnder. 9 / 21

10 Simplified Model With all of the previous simplifications we are left with: u[n + 1] = Au[n] + Bf (x[n k]) x[n] = Cu[n] u[n + 1] = Au[n] + Bf (Cu[n k]) u[n + 1] = Au[n] + Bβcos 2 (Cu[n k] + φ) Where we have reintroduced the nonlinearity from the Mach-Zehnder. 10 / 21

11 Simplified Model With all of the previous simplifications we are left with: u[n + 1] = Au[n] + Bf (x[n k]) x[n] = Cu[n] u[n + 1] = Au[n] + Bf (Cu[n k]) u[n + 1] = Au[n] + Bβcos 2 (Cu[n k] + φ) Where we have reintroduced the nonlinearity from the Mach-Zehnder. 11 / 21

12 Coupled State-Space Finally we build a coupled system similar to the coupled Lorenz System (with abstraction for easy reading) u 1 [n + 1] = Au 1 [n] + Bf (Cu 1 [n k]) +γb(f (Cu 2 [n k]) f (Cu 1 [n k])) u 2 [n + 1] = Au 2 [n] + Bf (Cu 2 [n k]) +γb(f (Cu 1 [n k]) f (Cu 2 [n k])) u 1 [n + 1] = Au 1 [n] + B((1 γ)f (Cu 1 [n k]) + γf (Cu 2 [n k])) u 2 [n + 1] = Au 2 [n] + B((1 γ)f (Cu 2 [n k]) + γf (Cu 1 [n k])) 12 / 21

13 Coupled State-Space Finally we build a coupled system similar to the coupled Lorenz System (with abstraction for easy reading) u 1 [n + 1] = Au 1 [n] + Bf (Cu 1 [n k]) +γb(f (Cu 2 [n k]) f (Cu 1 [n k])) u 2 [n + 1] = Au 2 [n] + Bf (Cu 2 [n k]) +γb(f (Cu 1 [n k]) f (Cu 2 [n k])) u 1 [n + 1] = Au 1 [n] + B((1 γ)f (Cu 1 [n k]) + γf (Cu 2 [n k])) u 2 [n + 1] = Au 2 [n] + B((1 γ)f (Cu 2 [n k]) + γf (Cu 1 [n k])) 13 / 21

14 Implementation Considerations Quick Graphical Output Vector Operations In-built filter functions Decision: Matlab (w/ C) Major Concern: Quantization 14 / 21

15 Validation Stage 1: Single Loop Characteristic Curves from Kouomuo [1] Stage 2: Coupled Lorenz Conditions by Anishchenko [4] Stage 3: Coupled Mach-Zehnders Open Loop: Argysis [5] Symmetric 50/50 coupling: Piel [6] 15 / 21

16 Use of Code Previously Explored 50/50 Coupling Synchronization for Coupling vs. Feedback Strength To be Explored Synchronization for: Coupling vs. Delay Coupling vs. Optical Bias Noise and Quantization Non-Symmetric Coupling Variations in Other parameters 16 / 21

17 Milestones Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion of Code November 1 st week December 1 st Feburary 1 st To be Determined 17 / 21

18 Milestones Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion of Code November 1 st week December 1 st Feburary 1 st To be Determined 18 / 21

19 Milestones Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion of Code November 1 st week December 1 st Feburary 1 st To be Determined 19 / 21

20 Milestones Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion of Code November 1 st week December 1 st Feburary 1 st To be Determined 20 / 21

21 References Kouomou, Yanne. Nonlinear Dynamics of Semiconductor Laser systems with feedback Doctoral Thesis Cohen, Adam, et. al. Using synchronization for prediction of high-dimensional chaotic dynamics Phys. Rev. Let., Publication Pending Murphy, Thomas. Personal Communications Anischchenko, V. S. et. al, Mutual synchronization and desynchronization of Lorenz systems Tech. Phys. Lett., Vol. 24, Nmb 4, Apr 1998, Amer. Inst. Phys. Argysis, Apostolos, et. al. Chaos Based Communications at high bit-rates using commercial fibre-optics Nature, 2005 Nature Publishing Piel, Michael, et. al, Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling Phys. Rev. E Vol 76, 2007 Amer. Phys. Soc. 21 / 21