Researchers from the International Laboratory of Dynamical Systems and Applications at the HSE Campus in Nizhny Novgorod have discovered that the use of systems with hyperbolic solenoid attractors and repellers can lead to issues in the transmission of communication signals. The findings of this study have been published in Results in Mathematics. The research was financed by a substantial grant from the Russian Government as part of the 'Science and Universities' National Project and a further grant from the Russian Science Foundation.
Attractors and repellers are mathematical concepts that describe and visualise complex dynamical processes occurring both in nature and technology. Attractors draw the trajectories of the system towards them, while repellers push them away, much like magnets with opposing charges. To effectively control and predict various processes, it is crucial for the system's attractor and repeller to be structurally stable. This ensures the preservation of dynamics even with minor changes in the system’s parameters.
One can envision a structurally stable attractor as a point of attraction or a state to which the system converges and remains there even when subjected to minor perturbations. It is like a ball placed into a funnel: regardless of the force or angle at which you throw a ball into a funnel, it will move around initially but ultimately come to rest at the bottom of the funnel.
Hyperbolic attractors can exhibit a chaotic structure, meaning that the trajectories inside the attractor scatter widely over a short period, covering a considerable distance. This behaviour opens up new possibilities for predicting and controlling complex systems, including communication systems. However, despite being locally structurally stable, systems with coexisting attractors and repellers may exhibit global instability as a whole. In such cases, even the slightest interference or distortion can alter the trajectory of the signal or result in information loss during transmission.
One example of a structurally stable attractor or repeller is the Smale-Williams solenoid: it consists of an (countless) set of 'threads' wound around a torus, a shape resembling a bagel. The interest in Smale-Williams solenoids stems from the fact that their structural stability was mathematically proven back in the 1960s. Recently, it has been established that the Smale-Williams attractor and repeller cannot coexist in any structurally stable system. The discovered effect is mainly explained by the fact that the solenoid sets are not even locally flat.
Mathematicians from the HSE in Nizhny Novgorod International Laboratory of Dynamical Systems and Applications have proven that systems with a globally flat solenoid attractor and repeller are also not structurally stable.
Our discovery reveals that in the case of chaos generation by a system with a hyperbolic solenoid attractor and repeller for signal transmission, such as in encryption systems or secure communications, this system will be susceptible to interference, losses, and distortions. This implies that such systems may encounter issues with the reliability of information transmission.
According to the authors, the findings from their study could assist in minimising the impact of interference and ensuring a more stable signal transmission in communication systems.