*Lou Pecora, Naval Research Laboratory*

**Cluster Synchronization of Chaotic Systems in Complex Networks**

The concept of synchronized systems has been around for centuries with one of the earliest studies being on the synchronization of clocks by Christiaan Huygens in the mid 1700’s. By the mid 1900’s it was well-known how to mathematically model the synchronization of systems that oscillated periodically or regularly, i.e. in a steady, repeatable way. But as a new type of motion called chaos started to be studied in the 1970’s the notion of synchronization of such systems was difficult to grasp since chaotic systems never repeated or had regular periodic motion. I’ll show how to synchronize an entire network of chaotic oscillators. This will lead into a theory of cluster synchronization in networks of oscillators where the network settles into subsets of oscillators that are synchronized with other oscillators in their subset, but not in other subsets, even though all are interconnected. This will lead to some applications of finite symmetry groups to explain the mathematics behind the phenomena and expose some non-intuitive behavior.