Dynamical mechanism behind ghosts unveiled in a map complexification

Abstract

Complex systems such as ecosystems, electronic circuits, lasers or chemical reactions can be modelled by dynamical systems which typically experience bifurcations. Transients typically suffer extremely long delays at the vicinity of bifurcations and it is also known that these transients follow scaling laws as the bifurcation parameter gets closer the bifurcation value in deterministic systems. The mechanisms involved in local bifurcations are well-known. However, for saddle-node bifurcations, the relevant dynamics after the bifurcation occur in the complex phase space. Hence, the mechanism responsible for the delays and the associated inverse-square root scaling law for this bifurcation can be better understood by looking at the dynamics in the complex space. We follow this approach and complexify a simple ecological system undergoing a saddle-node bifurcation. The discrete model describes a biological system with facilitation (cooperation) under habitat destruction for species with non-overlapping generations. We study the complex (as opposed to real) dynamics once the bifurcation has occurred. We identify the fundamental mechanism causing these long delays (called ghosts), given by two repellers in the complex space. Such repellers appear to be extremely close to the real line, thus forming a narrow channel close to the two new fixed points and responsible for the slow passage of the orbits, which remains tangible in the real numbers phase space. We analytically provide the relation between the inverse square-root scaling law and the multipliers of these repellers. We finally prove that the same phenomenon occurs for more general i.e., non-necessarily polynomial, models.