Dynamical mechanism behind ghosts unveiled in a map complexification

• Published in: Chaos, solitons and fractals Vol. 156; p. 111780
• Main Authors: Canela, Jordi, Alsedà, Lluís, Fagella, Núria, Sardanyés, Josep
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2022
• Subjects: Complexification; Discrete dynamics; Ghosts; Holomorphic dynamics; Saddle-node bifurcation; Scaling laws; Tansients; Complexification; Ghosts; Scaling laws; Tansients; Holomorphic dynamics; Discrete dynamics; Saddle-node bifurcation
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2021.111780

•Long transients and scaling phenomena arise close to saddle-node bifurcations.•We have applied holomorphic dynamics to study transient delays after this bifurcation.•Two repelling spirals in the complex phase space cause such delays.•The inverse square-root scaling law is derived from the multipliers of these spirals. Complex systems such as ecosystems, electronic circuits, lasers, or chemical reactions can be modelled by dynamical systems which typically experience bifurcations. It is known that transients become extremely long close to bifurcations, also following well-defined scaling laws as the bifurcation parameter gets closer the bifurcation value. For saddle-node bifurcations, the dynamical mechanism responsible for these delays, tangible at the real numbers phase space (so-called ghosts), occurs at the complex phase space. To study this phenomenon we have complexified an ecological map with a saddle-node bifurcation. We have investigated the complex (as opposed to real) dynamics after this bifurcation, identifying the fundamental mechanism causing such long delays, given by the presence of 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. We analytically provide the relation between the well-known inverse square-root scaling law of transient times and the multipliers of these repellers. We finally prove that the same phenomenon occurs for more general i.e. non-necessarily polynomial, models.