Demonstrations of the complex behavior of nonlinear dynamical systems appear in a variety of disciplines, including chemistry, fluid dynamics, meteorology and many others. Nonlinear dynamics tools, such as bifurcation theory, provide a common framework to characterize two- and three-dimensional patterns in these systems.

One way of applying these techniques to nonspatially extended systems, where patterns have a structure close to the system’s size, is by using time delays. Output from a system is fed back in after a time delay, creating a spatio-temporal relationship. Using a second delay creates two pseudo-space dimensions.

In their article, Brunner et al. investigate the dynamics of a dual-delay system using highly asymmetric feedback delays and reveal emergent patterns. They subject a semiconductor laser to a short, delayed feedback and then to a second delay which exceeds the first by two orders of magnitude. They transmit the signal through a Fabry-Perot resonator that introduces nonlinearity and allows for parameter adjustments.

The authors found either chimera states or dissipative solitons by varying parameters in the resonator. Chimeras are states that arise in networks of coupled oscillators and are the result of spontaneous symmetry breaking; in a chimera state, one subgroup of oscillators are synchronized and locked into a coherent mode, while another subgroup of oscillators exhibit complex, unsynchronized behavior. In this paper, a resulting 3-D diagram shows chimeras where steady behavior is surrounded by a “sea” of chaos or vice versa — chaos surrounded by steady behavior.

Dissipative solitons arise in excitable systems, such as neural or cardiac tissue. In another parameter regime for this experiment, the authors also observed solitons: steady-state periods interrupted by spikelike dynamics, behavior that is reminiscent of neuronal firing patterns. The authors showed that numerous solitons can be excited in the system, creating an optical memory and suggesting possible uses in neuromorphic computing.

Source: “Two-dimensional spatio-temporal complexity in dual-delayed nonlinear feedback systems: Chimeras and dissipative solitons,” by D. Brunner, B. Penkovsky, R. Levchenko, E. Scholl, L. Larger, and Y. Maistrenko, Chaos (2018). The article can be accessed at