University of New Hampshire
Mentor: Dr. John Gibson, Department of Mathematics
Examining The Periodic Orbit Of The Lorenz Systems And Double Pendulum System
Periodic orbits provide an underlying structure to chaotic attractors, which provides order and low-dimensionality to complex, chaotic systems. The trajectory of chaotic systems proceeds to infinity for an arbitrary initial condition; for this reason, periodic orbits are useful as a finite set of finite mathematical objects to characterize the infinite, complex behavior of a chaotic system. In particular, the mathematical framework of chaos, chaotic attractors, and unstable periodic orbits allow for a new understanding of self-organization in complex physical systems like turbulence. In this study periodic orbits as an organizing principle will be examined in the Lorenz system and the double pendulum system. This study will find the unstable, periodic orbits of the Lorenz system and the double pendulum system using the multivariate Newton method. The expected results of the study are that periodic orbits of both systems will both be found. This study presents an opportunity for an understanding of chaotic behavior which will lead to engagement with the current research on turbulence. The goal of this study to be a stepping stone to more challenging research projects in the future with the end goal being engagement with research done by Dr. John F. Gibson on turbulence. The purpose of this study is to develop the researcher’s understanding of periodic orbits in relation to the concepts of dynamical systems and the researcher’s proficiency of the technical skills needed to compute these periodic orbits by computer algorithms.