Chaos: Zero lag synchronisation

1. Chaos: Zero lag synchronisation Mathijs Vermeulen & Bart van Lith

2. Contents • Introduction • Lang-Kobayashi equations • Zero lag synchronization • Bernoulli map • Perturbed Bernoulli map • Conclusions Don’t panic

3. Introduction • Paper: Zero Lag Synchronization of Chaotic Systems with Time Delayed Couplings. Englert et al. • Two chaotic semi-conductor lasers are synchronized. • Bernoulli map is investigated analytically. Experimental setup used in the paper. Don’t panic

4. Lang Kobayashi equations Set of coupled differential equations for that describe the dynamics of a chaotic diode laser. Don’t panic

5. Experimental data Don’t panic

6. Bernoulli map • Nonlinear because of the modulo. • Chaotic for α > 1. • Solution • Lyapunov exponent α = 1.5 Don’t panic

7. Zero lag synchronization • Two signals xt and yt; • Two time delayed couplings with delay time τ1 and τ2; • τ1, τ2 << internal time scales; • Synchronized signals is trivial solution; • Linear stability inspected by adding small perturbation; • Only works if the same mapping is used; • Only works if are prime numbers. Don’t panic

8. Zero Lag synchronization Don’t panic

9. Perturbed Bernoulli map • Small non-linear perturbation on Bernoulli map. • ‘Rounds’ the saw-tooth edges. • Works very well for small ε. • Here ε = 0.13, α = 1.5 Don’t panic

10. Perturbed Bernoulli map α = 1.5, ε = -1.7 Don’t panic

11. Conclusions • Small perturbations in ZLS are damped if: • Coupling time constants are smaller than internal time scales; • the coupling time constants are prime numbers; • The same mapping is used; • Works for small perturbations in the Bernouilli mapping. Don’t panic