Combinatorial Control of Global Dynamics in a Chaotic differential equation

Controlling chaos has been an extremely active area of research in applied dynamical systems, following the introduction of the Ott, Grebogi, Yorke (OGY) technique in 1990 [Ott et al., 1990], but most of this research based on parametric feedback control uses local techniques. Associated with a dynamical system which pushes forward initial conditions in time, transfer operators, including the Frobenius–Perron operator, are associated dynamical systems which push forward ensemble distributions of initial conditions. In [Bollt, 2000a, 2000b; Bollt & Kostelich, 1998], we have shown that such global representations of a discrete dynamical system are useful in controlling certain aspects of a chaotic dynamical system which could only be accessible through such a global representation. Such aspects include invariant measure targeting, as well as orbit targeting. In this paper, we develop techniques to show that our previously discrete time techniques are accessible also to a differential equation. We focus on the Duffing oscillator as an example. We also show that a recent extension of our techniques by Góra and Boyarsky [1999] can be further simplified and represented in a convenient and compact way by using a tensor product.