Switching Ratios in Discretized Hysteresis Dynamics

Definitions and mathematical statement

The supplied record concerns the discrete evolution equation \[ \dot u_n(t)=\frac{u_{n-1}(t)-2u_n(t)+u_{n+1}(t)}{h^2}+\mathcal H(u_n)(t), \qquad u_n(0)=-c(hn)^2, \] with $c>0$ and constants $H_1>0>H_{-1}$. Here $\dot u_n$ denotes the time derivative, and $h$ denotes the spatial mesh parameter, conventionally taken positive. The notation $\mathcal H(u_n)(t)$ represents the unspecified hysteresis term appearing as $\mathcal H(u_n)$ in the supplied statement; it does not specify a memory rule. The record calls $H_1,H_{-1}$ thresholds but does not define how they enter this term.

The source question asks whether the proportion of switched points relative to nonswitched points satisfies \[ \frac{\#\{\text{switched points}\}}{\#\{\text{nonswitched points}\}} \approx \left|\frac{H_1}{H_{-1}}\right|. \] The absolute value is part of the corrected supplied formulation. Since $H_1>0>H_{-1}$, its right-hand side is positive; the signed ratio in the original extraction is negative.

To expose the missing counting conventions, suppose an observation time $t$ and a finite set $A$ of lattice indices have been specified, together with a precise switching rule. Let $S(A,t)$ be the indices in $A$ classified as switched under that rule, and let $N(A,t)=A\setminus S(A,t)$. The corresponding count ratio would be \[ R(A,t)=\frac{\#S(A,t)}{\#N(A,t)}, \qquad \#N(A,t)>0. \] This notation is conditional bookkeeping, not an assertion that this observation convention is the intended one. The record does not specify whether switching means ever having switched by time $t$, occupying a particular hysteresis state at time $t$, or some other condition. It also does not specify the lattice domain, boundary conditions, observation sets, observation times, or a limiting regime or error bound interpreting $\approx$. Thus it does not determine a precise quantified assertion relating $R(A,t)$ to $\lvert H_1/H_{-1}\rvert$.

The hysteresis law, its initial memory state, its behavior at switching thresholds, any additional parameter restrictions, and the quantifiers over parameters must be recovered before the question becomes a fully specified mathematical conjecture. The supplied background also mentions a multidimensional extension, but provides no equation or precise conjecture for it.

Short English statement

The question concerns whether a discretized hysteresis evolution with negative quadratic initial data produces switched and nonswitched points in a ratio approximately equal to $\lvert H_1/H_{-1}\rvert$. The supplied record leaves the switching law, counting region, and meaning of approximation unspecified, so a precise quantified conjecture still requires clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002253$, title "Switching Ratios in Discretized Hysteresis Dynamics," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2012 and names no proposer.
• Supplied report citation: Dynamics of Patterns (2013), Oberwolfach Reports, pp. 3549--3633, DOI 10.4171/owr/2012/60. https://doi.org/10.4171/owr/2012/60. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "rattling primary paper": https://doi.org/10.1137/16M106039X.
• Supplied literature source, labeled "spatially discrete hysteresis paper": https://www.numdam.org/articles/10.1016/j.anihpc.2017.09.006/.
• Supplied literature source, labeled "Oberwolfach report": https://ems.press/content/serial-article-files/46431.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports later results on the rattling mechanism, propagation behavior, and detailed one-dimensional microstructure, while describing the exact switched-point ratio conjecture and its multidimensional extension as not fully resolved. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and supplies the verification claim that the official report states the switching-ratio conjecture and that the original extraction omitted an essential absolute value. This attribution and verification claim are supplied metadata; the report has not been independently inspected for this draft.