# Contributions to the Study of Autonomous Chaotic Circuits and Cellular Automata

### Bharathwaj Muthuswamy

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2009-164

December 9, 2009

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2009/EECS-2009-164.pdf

This work focuses on autonomous chaotic circuits and cellular automata. In the realm of chaotic systems, it is often difficult to rigorously prove the existence of chaos. For example, the Lorenz system that was discovered in 1963 was rigorously proved to be chaotic only in 1999, after a span of 36 years. Hence, the first part of this thesis concerns rigorous proofs of chaos. The first approach uses a combination of linear dynamics, trajectory analysis in the Jordan space and describing function techniques for period-doubling bifurcations. This approach is applied to a rigorous proof of chaos in the Four-Element Chua's circuit, the simplest chaotic circuit. To our knowledge, this is the first rigorous proof of chaos in the Four-Element Chua's circuit. The second approach involves topological horseshoe theory and is applied to memristor based chaotic systems. This thesis also proposes a realization of a memristor based chaotic system on the breadboard. To our knowledge, this is the first analog realization of a memristor which is not based on designing a mutator for converting a v-i curve into a phi-q curve and also the first rigorous verification of chaos in a memristor based chaotic system. Both these approaches provide the reader with mathematical tools for investigating the behavior of continuous time chaotic systems. In the second part of this thesis, the relationship between bit length and attractor periods of totalistic one dimensional cellular automata are classified. Specifically, the relationship between integer factorization and dynamics of totalistic one dimensional cellular automata is explored for the first time.

The organization of this thesis is: in Chapter 1 we discuss background material necessary for understanding this thesis. Chapter 2 discusses the rigorous proof of chaos in the Four-Element Chua's circuit. Chapter 3 involves memristor based (higher dimensional) chaotic circuits and topological horseshoe theory. Chapter 4 explores the relationship between integer factorization and cellular automata. This is followed by conclusions with suggestions for future work, bibliography and appendices with simulation code.

**Advisor:** Leon O. Chua

BibTeX citation:

@phdthesis{Muthuswamy:EECS-2009-164, Author = {Muthuswamy, Bharathwaj}, Title = {Contributions to the Study of Autonomous Chaotic Circuits and Cellular Automata}, School = {EECS Department, University of California, Berkeley}, Year = {2009}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2009/EECS-2009-164.html}, Number = {UCB/EECS-2009-164}, Abstract = {This work focuses on autonomous chaotic circuits and cellular automata. In the realm of chaotic systems, it is often difficult to rigorously prove the existence of chaos. For example, the Lorenz system that was discovered in 1963 was rigorously proved to be chaotic only in 1999, after a span of 36 years. Hence, the first part of this thesis concerns rigorous proofs of chaos. The first approach uses a combination of linear dynamics, trajectory analysis in the Jordan space and describing function techniques for period-doubling bifurcations. This approach is applied to a rigorous proof of chaos in the Four-Element Chua's circuit, the simplest chaotic circuit. To our knowledge, this is the first rigorous proof of chaos in the Four-Element Chua's circuit. The second approach involves topological horseshoe theory and is applied to memristor based chaotic systems. This thesis also proposes a realization of a memristor based chaotic system on the breadboard. To our knowledge, this is the first analog realization of a memristor which is not based on designing a mutator for converting a v-i curve into a phi-q curve and also the first rigorous verification of chaos in a memristor based chaotic system. Both these approaches provide the reader with mathematical tools for investigating the behavior of continuous time chaotic systems. In the second part of this thesis, the relationship between bit length and attractor periods of totalistic one dimensional cellular automata are classified. Specifically, the relationship between integer factorization and dynamics of totalistic one dimensional cellular automata is explored for the first time. The organization of this thesis is: in Chapter 1 we discuss background material necessary for understanding this thesis. Chapter 2 discusses the rigorous proof of chaos in the Four-Element Chua's circuit. Chapter 3 involves memristor based (higher dimensional) chaotic circuits and topological horseshoe theory. Chapter 4 explores the relationship between integer factorization and cellular automata. This is followed by conclusions with suggestions for future work, bibliography and appendices with simulation code.} }

EndNote citation:

%0 Thesis %A Muthuswamy, Bharathwaj %T Contributions to the Study of Autonomous Chaotic Circuits and Cellular Automata %I EECS Department, University of California, Berkeley %D 2009 %8 December 9 %@ UCB/EECS-2009-164 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2009/EECS-2009-164.html %F Muthuswamy:EECS-2009-164