dynamical-systems-paper-figures

# Dynamical Systems Paper Figures ## Aim and scope A figure should let a mathematician identify an object, see a geometric relation, or follow a difficult step in an argument. It must not suggest a conclusion stronger than the paper proves. Work only at the requested level: audit, design, production, or review. Before changing a figure, read the theorem, definitions, and surrounding prose that determine it; preserve the manuscript's claims and unrelated figures. ## Read selectively | Task | Read | |---|---| | New figure type, journal comparison, or whole-paper strategy | [published-figure-practice.md](references/published-figure-practice.md) | | Phase portrait, fast-slow diagram, canard, blow-up chart, Poincaré map, or limit cycle | [figure-semantics.md](references/figure-semantics.md) | | Production, computation, reproducibility, or final-page review | [production-and-review.md](references/production-and-review.md) | Use [figure-contract.md](templates/figure-contract.md) when several mathematical layers must be coordinated; ordinary prose is enough for a simple schematic. ## Decide whether a figure is needed Ask what the reader should see or compare and which definition, result, or proof transition that serves. If prose or an existing figure answers both questions economically, do not add another. For a paper-wide set, choose the smallest collection that shows the main object, headline phenomenon, and geometric or scale transition that cannot reasonably be held in working memory. ## Fix the mathematical semantics first For each panel or layer, record: - the system, parameters, coordinates, and time or iteration convention; - whether the content is **SCHEMATIC**, **COMPUTED**, **EXACT**, or **MIXED**; - the identity of every curve, set, marker, and direction; - which positions or scales are qualitative and which are quantified; - the misleading inference the figure must prevent. Do not choose geometry by eye when these facts are unsettled. Do not let a curve change meaning between panels. Distinguish trajectories, invariant manifolds, critical manifolds, nullclines, sections, parameter boundaries, and guides to the eye. Schematic or computed evidence illustrates the mathematics; it does not prove existence, uniqueness, stability, or a global assertion. ## Use dynamical-systems conventions Show flow or iteration direction when it matters, and do not join discrete map iterates by an unexplained smooth curve. In fast-slow systems distinguish the critical manifold from finite-parameter slow manifolds, attracting and repelling branches, singular concatenations and actual trajectories, and reduced, layer, and desingularized flows. Mark folds, contacts, fast jumps, canard segments, and entry or exit sections when they organize the argument. In planar phase portraits distinguish nullclines, vector fields, separatrices, equilibria, and periodic orbits. Treat a numerically closed curve as evidence for a periodic orbit; call it a limit cycle only when existence and any claimed isolation or stability are established analytically or by validated numerics. When a return map or scalar function carries the proof, consider displaying it alongside or instead of a phase portrait. Use a three-dimensional view only when projection loses an essential relation; introduce complicated geometry through simpler projections first. ## Visual language and captions Keep labels, line styles, markers, and colors consistent across the paper. Give color a redundant cue so distinctions survive grayscale and common color-vision deficiencies. Give each panel one focus; use arrows only when direction changes interpretation; label central objects directly and quiet secondary geometry. Preserve viewpoint, scale, and encoding in comparisons. Draft the caption before production. It states the system or parameters needed to read the figure, identifies the visual encoding, names the relation to notice, and says which elements are schematic or computed. It is not a second proof and does not repeat theorem quantifiers. ## Production and review Use the simplest reproducible route: TikZ or editable vector graphics for conceptual geometry, maps, sections, and dependency diagrams; scripts for computed curves, phase portraits, bifurcation plots, and error plots; a combined workflow when numerical geometry needs vector composition or LaTeX labels. Prefer vector PDF or SVG for line art and raster output only for genuinely dense data. Retain source code, parameters, data, solver settings, and the reproduction command. Generative image models are unsuitable for evidence-bearing mathematical geometry. Inspect the standalone figure and the rendered manuscript page. Check object identities, signs, directions, coordinates, parameters, crop, overlap, fonts, label size, grayscale meaning, caption, first citation, float placement, and consistency with nearby figures. For computed figures check closure residuals, invariant quantities, convergence, or another problem-specific diagnostic. State what was checked without treating production quality as a proof. Ask what mathematical relation the reader sees first and where ambiguity begins. Remove a layer, separate panels, change the hierarchy, or clarify the caption before adding labels. Stop when the figure answers its reader question at final size and can be reproduced from its retained source.