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In this thesis we study various aspects of chaos synchronization of time-delayed coupled chaotic maps. A network of identical nonlinear units interacting by time-delayed couplings can synchronize to a common chaotic trajectory. Even for large delay times the system can completely synchronize without any time shift. In the first part we study chaotic systems with multiple time delays that range over several orders of magnitude. We show that these time scales emerge in the Lyapunov spectrum: Different parts of the spectrum scale with the different delays. We define various types of chaos depending on the scaling of the maximum exponent. The type of chaos determines the synchronization ability of coupled networks. This is, in particular, relevant for the synchronization properties of networks of networks where time delays within a subnetwork are shorter than the corresponding time delays between the different subnetworks. If the maximum Lyapunov exponent scales with the short intra-network delay, only the elements within a subnetwork can synchronize. If, however, the maximum Lyapunov exponent scales with the long inter-network connection, complete synchronization of all elements is possible. The results are illustrated analytically for Bernoulli maps and numerically for tent maps. In the second part the attractor dimension at the transition to complete chaos synchronization is investigated. In particular, we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents for iterated maps. We argue that the Kaplan-Yorke dimension must be discontinuous at the transition and compare it to the correlation dimension. For a system of Bernoulli maps we indeed find a jump in the correlation dimension. The magnitude of the discontinuity in the Kaplan-Yorke dimension is calculated for networks of Bernoulli units as a function of the network size. Furthermore the scaling of the Kaplan-Yorke dimension as well as of the Kolmogorov entropy with system size and time delay is investigated. Finally, we study the change in the attractor dimension for systems with parameter mismatch. In the third and last part the linear response of synchronized chaotic systems to small external perturbations is studied. The distribution of the distances from the synchronization manifold, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transmitted signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments. The linear response is also quantified by means of the bit error rate of a transmitted binary message which perturbs the synchronized system. The bit error rate is given by an integral over the distribution of distances and is studied numerically for Bernoulli, tent and logistic maps. It displays a complex nonmonotonic behavior in the region of synchronization. For special cases the distribution of distances has a fractal structure leading to a devil's staircase for the bit error rate as a function of coupling strength. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. A bi-directionally coupled chain of three units can completely filter out the perturbation. Thus the second moment and the bit error rate become zero.
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 Rössler 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.
Die vorliegende Arbeit beschäftigt sich mit der Chaossynchronisation in Netzwerken mit zeitverzögerten Kopplungen. Ein Netzwerk chaotischer Einheiten kann isochron und vollständig synchronisieren, auch wenn der Austausch der Signale einer oder mehreren Verzögerungszeiten unterliegt. In einem Netzwerk identischer Einheiten hat sich als Stabilitätsanalyse die Methode der Master Stability Funktion von Pecora und Carroll etabliert. Diese entspricht für ein Netzwerk gekoppelter iterativer Bernoulli-Abbildungen Polynomen vom Grade der größten Verzögerungszeit. Das Stabilitätsproblem reduziert sich somit auf die Untersuchung der Nullstellen dieser Polynome hinsichtlich ihrer Lage bezüglich des Einheitskreises. Eine solche Untersuchung kann beispielsweise numerisch mit dem Schur-Cohn-Theorem erfolgen, doch auch analytische Ergebnisse lassen sich erzielen. In der vorliegenden Arbeit werden Bernoulli-Netzwerke mit einer oder mehreren zeitverzögerten Kopplungen und/oder Rückkopplungen untersucht. Hierbei werden Aussagen über Teile des Stabilitätsgebietes getroffen, welche unabhängig von den Verzögerungszeiten sind. Des Weiteren werden Aussagen zu Systemen gemacht, welche sehr große Verzögerungszeiten aufweisen. Insbesondere wird gezeigt, dass in einem Bernoulli-Netzwerk keine stabile Chaossynchronisation möglich ist, wenn die vorhandene Verzögerungszeit sehr viel größer ist als die Zeitskala der lokalen Dynamik, bzw. der Lyapunovzeit. Außerdem wird in bestimmten Systemen mit mehreren Verzögerungszeiten anhand von Symmetriebetrachtungen stabile Chaossynchronisation ausgeschlossen, wenn die Verzögerungszeiten in bestimmten Verhältnissen zueinander stehen. So ist in einem doppelt bidirektional gekoppeltem Paar ohne Rückkopplung und mit zwei verschiedenen Verzögerungszeiten stabile Chaossynchronisation nicht möglich, wenn die Verzögerungszeiten in einem Verhältnis von teilerfremden ungeraden ganzen Zahlen zueinander stehen. Es kann zudem Chaossynchronisation ausgeschlossen werden, wenn in einem bipartiten Netzwerk mit zwei großen Verzögerungszeiten zwischen diesen eine kleine Differenz herrscht. Schließlich wird ein selbstkonsistentes Argument vorgestellt, das das Auftreten von Chaossynchronisation durch die Mischung der Signale der einzelnen Einheiten interpretiert und sich unter anderem auf die Teilerfremdheit der Zyklen eines Netzes stützt. Abschließend wird untersucht, ob einige der durch die Bernoulli-Netzwerke gefundenen Ergebnisse sich auf andere chaotische Netzwerke übertragen lassen. Hervorzuheben ist die sehr gute Übereinstimmung der Ergebnisse eines Bernoulli-Netzwerkes mit den Ergebnissen eines gleichartigen Netzwerkes gekoppelter Halbleiterlasergleichungen, sowie die Übereinstimmungen mit experimentellen Ergebnissen eines Systems von Halbleiterlasern.