Over the last two decades, chaos in engineering systems such as nonlinear circuits has gradually moved from being simply a curious phenomenon to one with practical significance and applications. Chaos has been found to be useful or have great potential in many disciplines such as in thorough liquid mixing with low power consumption, high-performance circuit design for telecommunication, collapse prevention of power systems, biomedical engineering applications to the human brain and heart, to name just a few [Chen 1999; Chen & Dong 1998].
Creating chaos, therefore, becomes a key issue in such applications where chaos is important and useful. Given a system or process, which may be linear or nonlinear but is originally non-chaotic or even stable, the question of whether or not one can generate chaos (and, if so, how) by means of designing a simple and implementable controller (e.g., a parameter tuner or a state feedback controller) is known as anti-control of chaos or chaotification. This problem is theoretically attractive and yet technically very challenging, as can be imagined from the complicated behavior of chaos and its association with various bifurcations.
Nevertheless, tremendous efforts have been devoted to trying to achieve this goal not only via computer simulations for the task but also by developing complete and rigorous mathematical theories to support the task. In this endeavor, anti-control of discrete chaos has seen success whereby chaos is generated in an ``arbitrarily given'' system via a simple nonlinear state feedback controller - more precisely, for any given finite-dimensional discrete-time system with a bounded Jacobian, chaotification can be achieved by a linear state feedback plus a modulo operation [Chen & Lai, 1998] or a piecewise-linear sawtooth function [Wang & Chen, 1999]. Moreover, it was rigorously justified in these papers that the generated chaos satisfied the mathematical definition of chaos.
Moving forward from the discreate-time systems to the continuous-time setting, a simple linear partial state-feedback controller was found to be able to derive the Lorenz system, currently not in the chaotic region, to be chaotic. In fact, it led to the discovery of a new chaotic system, which is competitive with the Lorenz system in the structure (it is a three-dimensional autonomous equation with only two quadratic terms), topologically not equivalent (there does not exist a homeormophism that can take one system to the other), and yet has even more complex dynamical behavior than the Lorenz system [Chen & Ueta, 1999]. This paper is devoted to a more detailed analysis of this new chaotic system -- the chaotic Chen's equation (see also [Yu & Xia, 2000].