Spatiotemporal Patterns in a Lengyel–Epstein Model Near a Turing–Hopf Singular Point
Bibliographic record
Abstract
Abstract. In this paper, a study is carried out on the spatiotemporal dynamics of a Lengyel–Epstein model describing the chlorite-iodine-malonic-acid (CIMA) reaction with time delay and the Neumann boundary condition in a two-dimensional region. The existences for Turing, Hopf, Turing–Turing, Turing–Hopf, and Bogdanov–Takens bifurcations are derived by analyzing the dispersion relation between eigenvalues and wave numbers. In particular, to study the dynamics around a Turing–Hopf bifurcation singularity, the amplitude equations near a codimension-two bifurcation point are derived by employing the weakly nonlinear analysis method. Different spatiotemporal patterns for the system in parameter space are classified and various patterns identified, including spatially homogeneous periodic solutions, mixed mode, coexistence mode, bistable phenomenon, square, hexagon, black eye, two-phase oscillating staggered hexagon lattice, and other complex spatiotemporal patterns. The theoretical predictions are verified by numerical simulations showing an excellent agreement with many reported experiment results not only in chemistry but also in physics and biology. Results presented in this article reveal the mechanism of generating the spatiotemporal patterns of the CIMA reaction.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".