Control of chaos by oscillating feedback

Parametric feedback control of chaos relies on detailed knowledge of the locations of unstable periodic orbits. We show that unstable periodic orbits of dynamical systems with unknown locations but known periodicity $\tau$ can be stabilized by an oscillating feedback term proportional to ${\varepsilon }^t$ $\left(x{\to }^t-x{\to }^{t-\tau }\right)$, where $x{\to }^t$ is the location of the trajectory at time $t$ and ${\varepsilon }^t$ is periodic in time. Periodic feedback overcomes the limitations of Giona’s theorem [Nonlinearity 4, 911 (1991)], which states that constant feedback (i.e., a time-independent $\varepsilon )$ can stabilize an unstable periodic orbit only if the stability matrix has no positive eigenvalues greater than unity. As an application of oscillating feedback, we use it to stabilize the memory patterns in an associative memory $($Hopfield [Proc. Natl. Acad. Sci. USA 79, 2554 (1982); 81, 3088 (1984)]$)$ network, thereby enhancing the total capacity of the memory device. We extend our method to high-dimensional systems described by differential equations; in this framework, it is possible to stabilize the spatiotemporal chaos generated by the Kuramoto-Sivashinsky equation [G. J. Sivashinsky and D. M. Michelson, Prog. Theor. Phys. 63, 2122 (1980)].