Dynamics of Hamiltonian Systems and Memristor Circuits

Abstract

In this paper, we show that any [Formula: see text]-dimensional autonomous systems can be regarded as subsystems of [Formula: see text]-dimensional Hamiltonian systems. One of the two subsystems is identical to the [Formula: see text]-dimensional autonomous system, which is called the driving system. Another subsystem, called the response system, can exhibit interesting behaviors in the neighborhood of infinity. That is, the trajectories approach infinity with complicated nonperiodic (chaotic-like) behaviors, or periodic-like behavior. In order to show the above results, we project the trajectories of the Hamiltonian systems onto [Formula: see text]-dimensional spheres, or [Formula: see text]-dimensional balls by using the well-known central projection transformation. Another interesting behavior is that the transient regime of the subsystems can exhibit Chua corsage knots. We next show that generic memristors can be used to realize the above Hamiltonian systems. Finally, we show that the internal state of two-element memristor circuits can have the same dynamics as [Formula: see text]-dimensional autonomous systems.