3-dimensional Hopf bifurcation via averaging theory
Bibliographic record
Abstract
We consider the Lorenz system $\dot x = \s (y-x)$, $\dot y =rx -y-xz$ and $\dot z = -bz + xy$; and the Rössler system $\dot x = -(y+z)$, $\dot y = x +ay$ and $\dot z = b-cz + xz$. Here, we study the Hopf bifurcation which takes place at $q_{\pm}=(\pm\sqrt{br-b},\pm\sqrt{br-b},r-1),$ in the Lorenz case, and at $s_{\pm}=(\frac{c+\sqrt{c^2-4ab}}{2},-\frac{c+\sqrt{c^2-4ab}}{2a}, \frac{c\pm\sqrt{c^2-4ab}}{2a})$ in the Rössler case. As usual this Hopf bifurcation is in the sense that an one -parameter family in ε of limit cycles bifurcates from the singular point when ε=0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two $2$-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other $3$- or $n$-dimensional differential systems.
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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 | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 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
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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".