@phdthesis{Heiligenthal2012,
  author    = {Heiligenthal, Sven},
  title     = {Strong and Weak Chaos in Networks of Semiconductor Lasers with Time-Delayed Couplings},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-77958},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {This thesis deals with the chaotic dynamics of nonlinear networks consisting of semiconductor lasers which have time-delayed self-feedbacks or mutual couplings. These semiconductor lasers are simulated numerically by the Lang-Kobayashi equations. The central issue is how the chaoticity of the lasers, measured by the maximal Lyapunov exponent, changes when the delay time is changed. It is analysed how this change of chaoticity with increasing delay time depends on the reflectivity of the mirror for the self-feedback or the strength of the mutal coupling, respectively. The consequences of the different types of chaos for the effect of chaos synchronization of mutually coupled semiconductor lasers are deduced and discussed. At the beginning of this thesis, the master stability formalism for the stability analysis of nonlinear networks with delay is explained. After the description of the Lang-Kobayashi equations and their linearizations as a model for the numerical simulation of semiconductor lasers with time-delayed couplings, the artificial sub-Lyapunov exponent \$\lambda_{0}\$ is introduced. It is explained how the sign of the sub-Lyapunov exponent can be determined by experiments. The notions of "strong chaos" and "weak chaos" are introduced and distinguished by their different scaling properties of the maximal Lyapunov exponent with the delay time. The sign of the sub-Lyapunov exponent \$\lambda_{0}\$ is shown to determine the occurence of strong or weak chaos. The transition sequence "weak to strong chaos and back to weak chaos" upon monotonically increasing the coupling strength \$\sigma\$ of a single laser's self-feedback is shown for numerical calculations of the Lang-Kobayashi equations. At the transition between strong and weak chaos, the sub-Lyapunov exponent vanishes, \$\lambda_{0}=0\$, resulting in a special scaling behaviour of the maximal Lyapunov exponent with the delay time. Transitions between strong and weak chaos by changing \$\sigma\$ can also be found for the R{\"o}ssler and Lorenz dynamics. The connection between the sub-Lyapunov exponent and the time-dependent eigenvalues of the Jacobian for the internal laser dynamics is analysed. Counterintuitively, the difference between strong and weak chaos is not directly visible from the trajectory although the difference of the trajectories induces the transitions between the two types of chaos. In addition, it is shown that a linear measure like the auto-correlation function cannot unambiguously reveal the difference between strong and weak chaos either. Although the auto-correlations after one delay time are significantly higher for weak chaos than for strong chaos, it is not possible to detect a qualitative difference. If two time-scale separated self-feedbacks are present, the shorter feedback has to be taken into account for the definition of a new sub-Lyapunov exponent \$\lambda_{0,s}\$, which in this case determines the occurence of strong or weak chaos. If the two self-feedbacks have comparable delay times, the sub-Lyapunov exponent \$\lambda_{0}\$ remains the criterion for strong or weak chaos. It is shown that the sub-Lyapunov exponent scales with the square root of the effective pump current \$\sqrt{p-1}\$, both in its magnitude and in the position of the critical coupling strengths. For networks with several distinct sub-Lyapunov exponents, it is shown that the maximal sub-Lyapunov exponent of the network determines whether the network's maximal Lyapunov exponent scales strongly or weakly with increasing delay time. As a consequence, complete synchronization of a network is excluded for arbitrary networks which contain at least one strongly chaotic laser. Furthermore, it is demonstrated that the sub-Lyapunov exponent of a driven laser depends on the number of the incoherently superimposed inputs from unsynchronized input lasers. For networks of delay-coupled lasers operating in weak chaos, the condition \$|\gamma_{2}|<\mathrm{e}^{-\lambda_{\mathrm{m}}\,\tau}\$ for stable chaos synchronization is deduced using the master stability formalism. Hence, synchronization of any network depends only on the properties of a single laser with self-feedback and the eigenvalue gap of the coupling matrix. The characteristics of the master stability function for the Lang-Kobayashi dynamics is described, and consequently, the master stability function is refined to allow for precise practical prediction of synchronization. The prediction of synchronization with the master stability function is demonstrated for bidirectional and unidirectional networks. Furthermore, the master stability function is extended for two distinct delay times. Finally, symmetries and resonances for certain values of the ratio of the delay times are shown for the master stability function of the Lang-Kobyashi equations.},
  subject      = {Halbleiterlaser},
  language  = {en}
}
@phdthesis{Ruttor2006,
  author    = {Ruttor, Andreas},
  title     = {Neural Synchronization and Cryptography},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-23618},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2006},
  abstract  = {Neural networks can synchronize by learning from each other. For that purpose they receive common inputs and exchange their outputs. Adjusting discrete weights according to a suitable learning rule then leads to full synchronization in a finite number of steps. It is also possible to train additional neural networks by using the inputs and outputs generated during this process as examples. Several algorithms for both tasks are presented and analyzed. In the case of Tree Parity Machines the dynamics of both processes is driven by attractive and repulsive stochastic forces. Thus it can be described well by models based on random walks, which represent either the weights themselves or order parameters of their distribution. However, synchronization is much faster than learning. This effect is caused by different frequencies of attractive and repulsive steps, as only neural networks interacting with each other are able to skip unsuitable inputs. Scaling laws for the number of steps needed for full synchronization and successful learning are derived using analytical models. They indicate that the difference between both processes can be controlled by changing the synaptic depth. In the case of bidirectional interaction the synchronization time increases proportional to the square of this parameter, but it grows exponentially, if information is transmitted in one direction only. Because of this effect neural synchronization can be used to construct a cryptographic key-exchange protocol. Here the partners benefit from mutual interaction, so that a passive attacker is usually unable to learn the generated key in time. The success probabilities of different attack methods are determined by numerical simulations and scaling laws are derived from the data. If the synaptic depth is increased, the complexity of a successful attack grows exponentially, but there is only a polynomial increase of the effort needed to generate a key. Therefore the partners can reach any desired level of security by choosing suitable parameters. In addition, the entropy of the weight distribution is used to determine the effective number of keys, which are generated in different runs of the key-exchange protocol using the same sequence of input vectors. If the common random inputs are replaced with queries, synchronization is possible, too. However, the partners have more control over the difficulty of the key exchange and the attacks. Therefore they can improve the security without increasing the average synchronization time.},
  language  = {en}
}