Abstract
Small log-periodic oscillations have been observed in many systems and have previously been studied via renormalization-group approaches in the context of critical phenomena [Gluzman and Sornette, Phys. Rev. E 65, 036142 (2002); Derrida and Giacomin, J. Stat. Phys. 154, 286 (2014)]. Here we report their appearance in a random walk model with damaged memory, and we develop an exact discrete-time solution, free from adjustable parameters. Our results shed light on log-periodicity and how it arises. We also discuss continuous-time approaches to the solution along with their limitations and advantages. We show that, as a direct consequence of memory damage, the first moment for the model acquires piecewise ballistic behavior. The piecewise segments are separated by regularly placed singular points. Log-periodicity in this model is seen to be due to memory damage. Remarkably, piecewise ballistic behavior is only observed if one uses the discrete-time solution, because the continuous-time solution does not correctly account for the model's discrete-time dynamics.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
