An invariant-manifold-based method for chaos control

In this paper, we extend the OGY chaos-control method to be one based on the invariant manifold theory and the sliding mode control concept. This extended-control method not only can deal with higher order chaotic systems in the same spirit of the OGY method, but also can remove the reliance of the control on eigenvalues and eigenvectors of the system Jacobians, resulting in an even simpler but more effective controller. The novelty of the new design lies in the construction of suitable invariant manifolds according to the desired dynamic properties. The controller is then forcing the system state to lie on the intersection of the selected invariant manifolds, so that once the invariant manifolds are reached, the chaotic system will be guided toward a desired fixed point that corresponds to an originally targeted unstable periodic orbit of the given system. Such an idea is directly relevant to the sliding mode control approach. This new method is particularly useful for controlling higher order chaotic systems, especially in the case where some of the eigenvalues of the system Jacobian are complex conjugates. The effectiveness of the proposed method is tested by numerical examples of the third-order continuous-time Lorenz system and the fourth-order discrete-time double rotor map.