TY - JOUR

T1 - Dynamical mechanism behind ghosts unveiled in a map complexification

AU - Canela Sánchez, Jordi

AU - Alsedà i Soler, Lluís

AU - Fagella Rabionet, Núria

AU - Sardanyés, Josep

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Complexification

KW - Discrete dynamics

KW - Ghosts

KW - Holomorphic dynamics

KW - Saddle-node bifurcation

KW - Scaling laws

KW - Tansients

U2 - 10.1016/j.chaos.2021.111780

DO - 10.1016/j.chaos.2021.111780

M3 - Article

SN - 0960-0779

VL - 156

JO - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

ER -