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“Variational Principles for Periodic Orbits of Dynamical Systems”

Bruce Boghosian
Mathematics Department
Tufts University
April 7, 2006
 
Periodic orbits are interesting features of a wide variety of dynamical systems. Driven dissipative dynamical systems, such as the Lorenz equations, are well known to exhibit attractors that are dense with unstable periodic orbits (UPO's); that is, these attractors may be thought of as the closure of the set of all their UPO's. This means that UPO's provide a countable sequence of closed orbits whose limiting behavior may be used to understand chaotic dynamics on an attractor.
 
This approach is valid even for infinite-dimensional dynamical systems, such as the driven Navier-Stokes equations of viscous fluid dynamics, for which it has been proven that the attracting set is finite-dimensional with a dimension that scales as a power law in Reynolds number. An improved understanding of UPO's may therefore lead to improved short-term prediction of turbulent flow.
 
In this talk, we describe a new variational principle for locating periodic orbits of all kinds, including UPO's. Since we have no a priori way to know the orbit period, we leave it as another quantity that must be varied; that is, we define a functional of both the orbit path and its period, and we demand that the variation of this functional with respect to both of these dependencies vanish. The Euler-Lagrange equations corresponding to this variational principle lead to a second-order integrodifferential equation that must be satisfied along any periodic orbit.
 
Finally, we demonstrate that, in addition to its theoretical interest, this variational principle may be used as the basis of a numerical algorithm for finding periodic orbits. Numerical simulations to find periodic orbits for the Lorenz attractor are demonstrated, and the computational resources necessary to carry out a similar program for the Navier-Stokes equations are estimated.
 

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