Calculation of homoclinic and heteroclinic orbits in 1D maps

Abstract Homoclinic orbits and heteroclinic connections are important in several contexts, in particular for a proof of the existence of chaos and for the description of bifurcations of chaotic attractors. In this work we discuss an algorithm for their numerical detection in smooth or piecewise smooth, continuous or discontinuous maps. The algorithm is based on the convergence of orbits in backward time and is therefore applicable to expanding fixed points and cycles. For simplicity, we present the algorithm using 1D maps

Multi-parametric bifurcations in a piecewise-linear discontinuous map

In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained

Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios

We investigate the structure of the chaotic domain of a specific one-dimensional piecewise linear map with one discontinuity. In this system, the region of ``robust" chaos is embedded between two periodic domains. One of them is organized by the period-adding scenario whereas the other one by the period-increment scenario with coexisting attractors. In the chaotic domain, the influence of both adjacent periodic domains leads to the coexistence of the recently discovered bandcount adding and bandcount-increment scenarios. In this work, we focus on the explanation of the overall structure of the chaotic domain and a description of the bandcount adding and bandcount increment scenarios

Do We Run Large-scale Multi-Robot Systems on the Edge? More Evidence for Two-Phase Performance in System Size Scaling

With increasing numbers of mobile robots arriving in real-world applications, more robots coexist in the same space, interact, and possibly collaborate. Methods to provide such systems with system size scalability are known, for example, from swarm robotics. Example strategies are self-organizing behavior, a strict decentralized approach, and limiting the robot-robot communication. Despite applying such strategies, any multi-robot system breaks above a certain critical system size (i.e., number of robots) as too many robots share a resource (e.g., space, communication channel). We provide additional evidence based on simulations, that at these critical system sizes, the system performance separates into two phases: nearly optimal and minimal performance. We speculate that in real-world applications that are configured for optimal system size, the supposedly high-performing system may actually live on borrowed time as it is on a transient to breakdown. We provide two modeling options (based on queueing theory and a population model) that may help to support this reasoning.Comment: Submitted to the 2024 IEEE International Conference on Robotics and Automation (ICRA 2024