Georgia Institute of Technology College of Sciences

Over recent years, a great deal of analytical studies and modeling simulations have been brought together to identify the key signatures that allow dynamically similar nonlinear systems from diverse origins to be united into a single class. Among these key structures are bifurcations of homoclinic and heteroclinic connections of saddle equilibria and periodic orbits. Such homoclinic structures are the primary cause for high sensitivity and instability of deterministic chaos in various systems. Development of effective, intelligent and yet simple algorithms and tools is an imperative task for studies of complex dynamics in generic nonlinear systems. The core of our approach is the reduction of the time evolution of a characteristic observable in a system to its symbolic representation to conjugate or differentiate between similar behaviors. Of our particular consideration are the Lorenz-like systems and systems with spiral chaos due to the Shilnikov saddle-focus. The proposed approach and tools will let one detect homoclinic and heteroclinic orbits, and carry out state of the art studies homoclinic bifurcations in parameterized systems of diverse origins.