Chaos, Fractals and Bifurcation in Real Dynamics of Two-Parameter Family Associated to Logarithmic Functions
Bibliographic record
Abstract
The purpose of this article is to provide bifurcation diagrams and observe chaotic behaviour in the real dynamics of two-parameter family of function \(\Phi(x)=x+(1-\lambda x)\ln(ax): x>0, \lambda>0, a>0\). We consider here parameter a is a positive and continuous real parameter while λ is positive but a discrete real parameter. The dynamical properties of this nonlinear system family analyze numerically as well as graphically by using fixed point iterative method. Bifurcation diagrams for the real dynamics of the function are plotted by varying the values of the parameter which are fractals in nature. Also, we show that chaos exists in the dynamics of the function by looking at period-doubling in the bifurcation diagram. Further, chaotic behaviour studies by simulation of the positive Lyapunov exponents which observe by varying the parameters similar to the bifurcation diagrams.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".