We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant.

We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant.

Systeme mit Zeitverzögerung sind dadurch charakterisiert, dass deren zukünftige Entwicklung durch den Zustand zum aktuellen Zeitpunkt nicht eindeutig festgelegt ist. Die Historie des Zustands muss in einem Zeitraum bekannt sein, dessen Länge Totzeit genannt wird und die Gedächtnislänge festlegt. In dieser Arbeit werden fundamentale Effekte untersucht, die sich ergeben, wenn die Totzeit zeitlich variiert wird. Im ersten Teil werden zwei Klassen periodischer Totzeitvariationen eingeführt. Da diese von den dynamischen Eigenschaften einer eindimensionalen iterierten Abbildung abgeleitet werden, die über die Totzeit definiert wird, werden die Klassen entsprechend der zugehörigen Dynamik konservativ oder dissipativ genannt. Systeme mit konservativer Totzeit können in Systeme mit konstanter Totzeit transformiert werden und besitzen gleiche charakteristische Eigenschaften. Dagegen weisen Systeme mit dissipativer Totzeit fundamentale Unterschiede z.B. in der Tangentialraumdynamik auf. Im zweiten Teil werden diese Ergebnisse auf Systeme angewendet, deren Totzeit im Vergleich zur internen Relaxationszeit des Systems groß ist. Es zeigt sich, dass ein durch dissipative Totzeitvariationen induzierter Mechanismus, genannt resonanter Dopplereffekt, unter anderem zu neuen Arten chaotischer Dynamik führt. Diese sind im Vergleich zur bekannten chaotischen Dynamik in Systemen mit konstanter Totzeit sehr niedrig-dimensional. Als Spezialfall wird das so genannte laminare Chaos betrachtet, dessen Zeitreihen durch nahezu konstante Phasen periodischer Dauer gekennzeichnet sind, deren Amplitude chaotisch variiert. Im dritten Teil dieser Arbeit wird auf der Basis experimenteller Daten und durch die Analyse einer nichtlinearen retardierten Langevin-Gleichung gezeigt, dass laminares Chaos robust gegenüber Störungen wie zum Beispiel Rauschen ist und experimentell realisiert werden kann. Es werden Methoden zur Zeitreihenanalyse entwickelt, um laminares Chaos in experimentellen Daten ohne Kenntnis des erzeugenden Systems zu detektieren. Mit diesen Methoden ist selbst dann eine Detektion möglich, wenn das Rauschen so stark ist, dass laminares Chaos mit bloßem Auge nur schwer erkennbar ist.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix / In systems with time-delay, the evolution of a system is not uniquely determined by the state at the current time. The history of the state must be known for a time period of finite duration, where the duration is called delay and determines the memory length of the system. In this work, fundamental effects arising from a temporal variation of the time-delay are investigated. In the first part, two classes of periodically time-varying delays are introduced. They are related to a specific dynamics of a one-dimensional iterated map that is defined by the time-varying delay. Referring to the related map dynamics the classes are called conservative or dissipative. Systems with conservative delay can be transformed into systems with constant delay, and thus have the same characteristic properties as constant delay systems. In contrast, there are fundamental differences, for instance, in the tangent space dynamics, between systems with dissipative delay and systems with constant delay. In the second part, these results are applied to systems with a delay that is considered large compared to the internal relaxation time of the system. It is shown that a mechanism induced by dissipative delays leads to new kinds of regular and chaotic dynamics. The dynamics caused by the so-called resonant Doppler effect is fundamentally different from the behavior known from systems with constant delay. For instance, the chaotic attractors in systems with dissipative delay are very low-dimensional compared to typical ones arising in systems with constant delay. An example of this new kind of low-dimensional dynamics is given by the so-called Laminar Chaos. It is characterized by nearly constant laminar phases of periodic duration, where the amplitude varies chaotically. In the third part of this work, it is shown that Laminar Chaos is a robust phenomenon, which survives perturbations such as noise and can be observed experimentally. Therefore experimental data is provided and a nonlinear delayed Langevin equation is analyzed. Using the robust features that characterize Laminar Chaos, methods for time series analysis are developed, which enable us to detect Laminar Chaos without the knowledge of the specific system that has generated the time series. By these methods Laminar Chaos can be detected even for comparably large noise strengths, where the characteristic properties are nearly invisible to the eye.:1. Introduction 2. Dissipative and conservative delays in systems with time-varying delay 3. Laminar Chaos and the resonant Doppler effect 4. Laminar Chaos: a robust phenomenon 5. Summary and concluding remarks A. Appendix