arXiv subjects

Saksham Malik

Publications and source records attributed to Saksham Malik.

3 recordsLinked to original sources

arXiv2026

Proximal Relations and Maximal Equicontinuous Factors for Non-autonomous Dynamical Systems

This paper studies proximal-type relations and maximal equicontinuous factors for non-autonomous dynamical systems on compact metric spaces. Two classes of systems are considered: systems generated by uniformly convergent sequences of maps and periodic systems. For uniformly convergent generators, collective convergence and uniform rigidity imply that the regional proximal relation for the non-autonomous system is equal to the regional proximal relation for the limit autonomous system and if the limit system is equicontinuous, then the proximality relation for the non-autonomous system is equal to the proximal relation for the autonomous limit system. In the same setting, the kernel of the maximal equicontinuous factor is identified as the smallest closed equivalence relation containing the autonomous equicontinuous structure relation of the corresponding autonomous system and preserved by all generators. For periodic systems, regional proximality and Banach proximality are determined by the period map, yielding a criterion in terms of mean equicontinuity and the relation of sensitivity in the mean. Examples show that the hypotheses used in these results are necessary.

math.DS↗

arXiv2025

Auslander-Yorke Dichotomy and Its Generalizations for Non-Autonomous Dynamical Systems

We investigate the dynamics of periodic non-autonomous discrete dynamical systems on uniform spaces and topological spaces, focusing on the extension of the classical Auslander-Yorke dichotomy to these settings. We prove various dichotomy theorems in the uniform space framework, showing that a minimal periodic non-autonomous system is either sensitive or equicontinuous, and prove some more refined versions involving syndetic equicontinuity and thick sensitivity and eventual sensitivity versus equicontinuity on compact uniform spaces. We further introduce topological analogues like topological equicontinuity, Hausdorff sensitivity, and their syndetic and multi-sensitive variants and prove corresponding Auslander-Yorke-type dichotomies on T3 spaces.

math.DS↗

arXiv2025

Colloidal hydrodynamic interactions in viscoelastic fluids

The motion of suspended colloidal particles generates fluid disturbances in the surrounding medium that create interparticle interactions. While such colloidal hydrodynamic interactions (HIs) have been extensively studied in viscous Newtonian media, comprehensive understanding of HIs in viscoelastic fluids is lacking. We develop a framework to quantify HIs in viscoelastic fluids with high spatiotemporal precision by trapping colloids and inducing translation-rotation hydrodynamic coupling. Using solutions of wormlike micelles (WLMs) as a case study, we discover that HIs are strongly time-dependent and depend on the structural memory generated in the viscoelastic fluid, in contrast to "instantaneous" HIs in viscous Newtonian fluids. We directly measure time-dependent HIs between a stationary probe and a driven particle during transient start-up, developing on the WLM relaxation timescale. Following the sudden cessation of the driven particle, we observe an intriguing flow reversal in the opposing direction, lasting for a time about ten times larger than the WLM relaxation time. We corroborate our observations with analytical microhydrodynamic theory, direct numerical solutions of a continuum model, and particle-based Stokesian dynamics simulations. We find that the structural recovery of the WLMs from a nonlinear strain can generate anisotropic and heterogeneous stresses that produce flow reversals and hydrodynamic attraction among colloids. Measured heterogeneities indicate a breakdown of standard continuum models for constitutive relations when the size of colloids is comparable to the length scales of the polymeric constituents and their entanglement lengths.

cond-mat.soft↗