A numerical framework for designing periodic orbits embedded in chaotic attractors

Abstract

This paper proposes a framework for the numerical design of continuous-time dynamical systems that bind desirably configured unstable periodic orbits (UPOs) into a chaotic attractor. The proposed numerical framework is comprised of the following four steps:(a) construction of a chaos-generating template structure consisting of a set of trajectories including explicitly embedded UPOs,(b) topology-preserving deformation of the template structure according to the desired configuration of UPOs,(c) assignment of attracting properties to the deformed template structure, and(d) function approximation that yields a vector-field function of a dynamical system that generates a chaotic attractor with desired UPOs. This paper elaborates on each step of the framework and presents a reference implementation, supported with numerical examples. From the viewpoint of numerical vector-field design, our proposal intends to extend the functionality of (stable) limit-cycle generators by introducing transitivity among configured UPOs that is intrinsic to chaotic systems.