Impact of step size on convergence in swarmalator systems

Group robotics is one of the key areas of the development of robotic systems. This is due to the fact that for a wide class of practical tasks, the use of a group of relatively simple robots is much more efficient than using a single large multi-purpose device. The modern development of computer technology and communication systems opens up wide opportunities for the construction of such systems. The most progressive and effective approach is the implementation of the collective behavior of robots according to the swarm principle, when each of them interacts only with neighboring individuals, synchronously exchanging the collected information about the environment and their condition. Such a group compensates for the weakness of its detection and communication devices by joining a team. The problems of introducing group robotics into the modern world are studied in this thesis. If they combine two concepts, synchronization and swarming, they are called a swarmalator. In swarmalator systems, the movement of the robots is governed by differential equations. These equations are solved with the Euler method, where the location and phase are determined. The Euler method is time-discrete and allows the integration of first-order differential equations. Therefore, there is a step size to be chosen. The main task is to study group movement, which is based on transmitting information with a definite step size. The step value affects how often the swarmalators share their location and phase. Three main conclusions are made. The first research is what happens when varying the step size - is it most optimal to use with small step sizes? The second conclusion is that when increasing the step size with a small increment or using randomization of the step size. Such methods are typically, more optimal to use with a gradual increase in the step size because the convergence time is lower. The third is when decreasing the step size using a small increment. The results showed that this method is optimal to use when the step size exceeds 1. The states converge at a rather large interval, compared with previous results, but at the same time with a large value of the convergence time. The values of optimal step sizes are presented and analyzed. As performance criteria, we consider the computational power that is required, the average convergence time, the coupling probability and the step size. The behavior of all parameters is graphically represented in plots. The conclusions are based on the simulations done for the results.