A class of initials-dependent dynamical systems

Abstract

Nonlinear term is critical for emergence of chaos in autonomous dynamical systems. The sampled time series in chaotic system are dependent on the initial selection of variables, while the attractors are invariant for fixed parameters. In this paper, the dynamical behavior of a class of dynamical system is investigated at fixed parameter region. It is found that the state selection is dependent on the initials and the potential mechanism is discussed. It is confirmed that the system can be switched between stable state, periodical state and even chaotic state by selecting appropriate initials even the parameters are fixed. We think that nonlinear cross terms with higher order could account for the emergence of this behavior. It indicates that initial selection and resetting can be also effective to control some chaotic systems, and these chaotic systems could enhance security for possible secure communication because the chaotic attractor depends on the parameter and initials selection as well. In the case of secure communication, the reconstruction of phase space becomes more difficult because the attractors are changed arbitrarily, thus the safety for secure keys is enhanced. For chaos control, when the initials are reset, the controller can be removed and the system can develop to step into the desired target by itself.