Neimark–Sacker bifurcations in a non-standard numerical scheme for a class of positivity-preserving ODEs
Bibliographic record
Abstract
We discuss the nature of Neimark–Sacker bifurcations occurring in a non-standard numerical scheme, for a class of positivity-preserving systems of ordinary differential equations (ODEs) which undergoes a corresponding Hopf bifurcation. Extending previous work (Alexander & Moghadas 2005 a Electron. J. Diff. Eqn. Conf . 12 , 9–19), it is shown that the type of Neimark–Sacker bifurcation (supercritical or subcritical) may be affected by the integration time-step . The general form of the scheme in the vicinity of a fixed point is given, from which the expression for the first Lyapunov coefficient for two-dimensional systems, valid for arbitrary time-step, is explicitly derived. The analysis shows that this coefficient undergoes an shift with respect to the corresponding coefficient of the original ODE. This could lead to a type of bifurcation which differs from the corresponding Hopf bifurcation in the ODE, due to changes in the sign of the first Lyapunov coefficient as varies. This is especially problematic in the vicinity of certain types of degenerate Hopf bifurcation, at which this coefficient may vanish. We also present a general method to eliminate the possible shift in the bifurcation parameter of the scheme; however, the first Lyapunov coefficient may still be subjected to an shift, leading to a possibly erroneous type of bifurcation. Examples are given to illustrate the theoretical results of the paper with applications to mathematical biology.
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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 | 0.001 | 0.003 |
| 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 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".