High-dimensional chaotic behavior in systems with time-delayed feedback

The nature of high-dimensional chaos exhibited by a class of delay-differential equation is investigated by various methods. This delay-differential equation models systems with time delayed feedback such as nonlinear optical resonators. We first describe briefly the bifurcation phenomena exhibited by the system: With an increase in a control parameter representing the energy flow rate, periodic states bi(multi)furcate themselves successively, forming a hierarchy of multistable periodic states with increasing complexities. Finally each branch of multistable periodic states makes transition to chaos. The possibility of applying such a multistability as a memory device for complicated information is discussed. In the chaotic regime each of the bifurcated states, in turn, merge successively into fewer sets of states with larger attractor dimensions, and finally a single developed chaos with a very large attractor dimension is formed. Lyapunov analysis is introduced to study the high-dimensional chaotic states. The Lyapunov vectors as well as Lyapunov spectrum are shown to be very useful to understand the underlying mechanism of the successive merging process mentioned above. Characteristics of developed chaos are investigated by high-pass filtered time series (HFTS). The intermittency characteristics of the HFTS changes markedly at a certain frequency. The Lyapunov analysis reveals that this frequency corresponds to a characteristic Lyapunov mode number. This characteristic number can be looked upon as the dimension of subspace in which active chaotic information is generated and is different from the attractor dimension in the customary sense.