Chaotic bursting in semiconductor lasers

Abstract

We investigate the dynamic mechanisms for low frequency fluctuations in semiconductor lasers subject to delayed optical feedback, using the Lang-Kobayashi model. This system of delay differential equations displays pronounced envelope dynamics, ranging from erratic, so called low frequency fluctuations to regular pulse packages, if the time scales of fast oscillations and envelope dynamics are well separated. We investigate the parameter regions where low frequency fluctuations occur and compute their Lyapunov spectrum. Using geometric singular perturbation theory, we study this intermittent chaotic behavior and characterize these solutions as bursting slow-fast oscillations.

Absolute stability and absolute hyperbolicity in systems with discrete time-delays

“…, This hints that the stability of the system can be obtained at a formal level by a discrete dynamical system. There are many publications devoted to relations between the DDE (2.2) and the singular map (2.3), see [37,[61][62][63][64][65][66][67][68][69][70]. In fact, in order to obtain equivalent stability conditions, one should consider an extended singular map (2.4) x(T ) = (iωI − A 0 ) −1 A 1 e iϕ x(T − 1).…”

Delay master stability of inertial oscillator networks

“…(v) The adjacency matrix A and the Jacobian D 2 i j g(x * i , x * j ), evaluated on the phase synchronization manifold Z, factorize into the direct product A ⊗ G (2) of an effective N × N adjacency matrix A and a universal 2 × 2 Jacobian G (2) , such that the local matrix G (2) is the same for all nodes and only A depends on the indices i, j of the network.…”

“…Time delays, caused by finite propagation speeds or processing times, induce retarded reactions of variables to changes in the system. For example, delays occur in lasers because of the finite speed of light [1,2]; population dynamics depends on maturation and gestation times [3], and the exchange of information between neurons requires time for both signal transmission as well as processing [4].…”

Delay master stability of inertial oscillator networks

“…Replacing assumption 4, we assume: 5. The adjacency matrix A and the Jacobian D 2 ij g(x * i , x * j ), evaluated on the phase synchronization manifold Z, factorize into the direct product A ⊗ G (2) of an effective N × N adjacency matrix A and a universal 2 × 2 Jacobian G (2) , such that the local matrix G (2) is the same for all nodes and only A depends on the indices i, j of the network. Similarly, A and D 1 ij g(x * i , x * j ) factorize into the direct product A ⊗ G (1) .…”

“…Time delays, caused by finite propagation speeds or processing times, induce retarded reactions of variables to changes in the system. For example, delays occur in lasers because of the finite speed of light [1,2]; population dynamics depend on maturation and gestation times [3], and the exchange of information between neurons requires time for both signal transmission as well as processing [4].…”