Evidence of a Critical Phase Transition in Purely Temporal Dynamics with Long-Delayed Feedback

We wish to thank S. Lepri and A. Politi for useful discussions. MF and FG acknowledge support from EU Marie Curie ITN grant n. 64256 (COSMOS).Peer reviewedPublisher PD

Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence

Dynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms

Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence

Dynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms

THE UPPER PLEISTOCENE “ISOLA DI COLTANO SANDS” 1 (ARNO COASTAL PLAIN, TUSCANY ITALY): REVIEW OF STRATIGRAPHIC DATA ANDTECTONIC IMPLICATIONS FOR THE SOUTHERN MARGIN OF THE VIAREGGIO BASIN

We present and discuss previously published stratigraphic and chronological (mainly archaeological remains) data about the “Isola di Coltano Sands”(ICS), with the support of unpublished core stratigraphies and taking into account the geological frame of the Arno coastal plain. ICS outcrops in the southern portion of the extensional Viareggio Basin, forming three isolated small-sized reliefs rising up to 15 m above the present-day Arno coastal plain on both sides of the Arno River. We document that the deposits outcropping north of the Arno River (Palazzetto site) reasonably belong to the Holocene prograding beach-ridge system to which they are physically juxtaposed. Indeed, both sedimentological and morphological characteristics indicate that the Palazzetto sands were exclusively formed by wind-related processes, likely occurred during the late Holocene according to the presence of Eneolithic artefacts. Conversely, the common presence of Mousterian artefacts at the Castagnolo and Coltano sites, located south of the Arno River, documents an age older than 40 kyr (upper Pleistocene) for these reliefs. Moreover, new stratigraphic data show that ICS are constituted by alluvial deposits with evidences of repetitive fluvial erosion episodes. All these features,, indicate that ICS can be reasonably included into the Late Pleistocene Vicarello Formation, widely outcropping along the southern margin of the Leghorn Hills. In this context, an estimated age ranging between MIS 6 and MIS 3 can be hypothesized for the ICS. However, the occurrence of Upper Pleistocene reliefs formed by alluvial deposits (Coltano and Castagnolo sites) in the southern portion of the Arno coastal plain seems to conflict with the acknowledged interpretation of the area as an extensional, subsiding setting. Moreover, the sharp morphological boundary dividing the flat Holocene coastal plain from the Quaternary Leghorn Hills is roughly coincident with the SW-NE transpressive fault (Sillaro line) that subdivides the subsiding area (Viareggio Basin, to which the Arno plain belongs) from the uplifting area (Leghorn mounts). Thus, our review of the available stratigraphic and chronological data strongly suggests the occurrence of a geological connection between the southern portion of the Arno coastal plain, specifically of the Castagnolo and Coltano reliefs, and the Leghorns Hills where the Vicarello Formation outcrops. This connection, which may have strong consequences on the geotectonic interpretation of the study area, and the formation 50 age of ICS needs to be better investigated in the future with new high-resolution tectonic and absolute chronological data