# Global Unfolding of Chua's Circuits

### Leon O. Chua

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M93/7

1993

By adding a small linear resistor in series with the inductor in Chua's circuit, we obtain a circuit whose state equation is topologically conjugate (i.e., equivalent) to a 21-parameter family C of continuous odd-symmetric piecewise- linear equations in R 3. In particular, every system or vector field belonging to the family C, can be mapped via an explicit non-singular linear transformation into this circuit, which is uniquely determined by 7 parameters. Since no circuit with less than 7 parameters has this property, this augmented circuit is called an unfolding of Chua's circuit--it is analogous to that of "unfolding a vector field" in a small neighborhood of a singular point. Our unfolding, however, is global since it applies to the entire state space R 3. The significance of the unfolded Chua's Circuit is that the qualitative dynamics of every autonomous 3rd-order chaotic circuit, system, and differential equation, containing one odd-symmetric 3-segment piecewise-linear function can be mapped into this circuit, thereby making their separate analysis unnecessary. This immense power of unification reduces the investigation of the many heretofore unrelated publications on chaotic circuits and systems to the analysis of only one canonical circuit. This unified approach is illustrated by many examples selected from a zoo of more than 30 strange attractors extracted from the literature. In addition, a gallery of 18 strange attractors in full color is included to demonstrate the immensely rich and complex dynamics of this simplest among all chaotic circuits.

BibTeX citation:

@techreport{Chua:M93/7, Author = {Chua, Leon O.}, Title = {Global Unfolding of Chua's Circuits}, Institution = {EECS Department, University of California, Berkeley}, Year = {1993}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2272.html}, Number = {UCB/ERL M93/7}, Abstract = {By adding a small linear resistor in series with the inductor in Chua's circuit, we obtain a circuit whose state equation is topologically conjugate (i.e., equivalent) to a 21-parameter family C of continuous odd-symmetric piecewise- linear equations in R 3. In particular, every system or vector field belonging to the family C, can be mapped via an explicit non-singular linear transformation into this circuit, which is uniquely determined by 7 parameters. Since no circuit with less than 7 parameters has this property, this augmented circuit is called an unfolding of Chua's circuit--it is analogous to that of "unfolding a vector field" in a small neighborhood of a singular point. Our unfolding, however, is global since it applies to the entire state space R 3. The significance of the unfolded Chua's Circuit is that the qualitative dynamics of every autonomous 3rd-order chaotic circuit, system, and differential equation, containing one odd-symmetric 3-segment piecewise-linear function can be mapped into this circuit, thereby making their separate analysis unnecessary. This immense power of unification reduces the investigation of the many heretofore unrelated publications on chaotic circuits and systems to the analysis of only one canonical circuit. This unified approach is illustrated by many examples selected from a zoo of more than 30 strange attractors extracted from the literature. In addition, a gallery of 18 strange attractors in full color is included to demonstrate the immensely rich and complex dynamics of this simplest among all chaotic circuits.} }

EndNote citation:

%0 Report %A Chua, Leon O. %T Global Unfolding of Chua's Circuits %I EECS Department, University of California, Berkeley %D 1993 %@ UCB/ERL M93/7 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2272.html %F Chua:M93/7