[bifurcation-basic] Bifurcation Diagram for Dynamical Systems

[MarkusNeusinger]

Description

A bifurcation diagram shows how the steady-state behavior of a dynamical system changes as a control parameter varies. It reveals transitions from stable fixed points through period-doubling cascades to chaos. The classic example is the logistic map.

Applications

• Studying onset of chaos in nonlinear dynamical systems
• Analyzing population dynamics in ecology
• Understanding period-doubling routes in laser physics
• Teaching nonlinear dynamics and chaos theory

Data

• parameter (numeric) — bifurcation parameter (e.g., r in logistic map, range 2.5 to 4.0)
• state (numeric) — steady-state or periodic orbit values
• Size: 100,000+ points (many states per parameter value)

Notes

• Use very small point size (1px) with transparency for density visualization
• Show clear period-doubling cascade leading to chaos
• Use logistic map x(n+1) = r·x(n)·(1-x(n)) as default example
• Label key bifurcation points

[claude]

✅ Spec ID: bifurcation-basic

Rationale: A bifurcation diagram is a distinct, well-established visualization in nonlinear dynamics — no existing spec covers this (the related phase-diagram shows state-space trajectories, not parameter-dependent steady-state behavior).

This is a valid and useful plot type, commonly used in chaos theory and dynamical systems education. The logistic map is an excellent default example.

[github-actions]

✅ Specification Ready: bifurcation-basic

PR: #4441

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specification.md

bifurcation-basic: Bifurcation Diagram for Dynamical Systems

Description

A bifurcation diagram shows how the steady-state behavior of a dynamical system changes as a control parameter varies. By plotting the long-term values of a state variable against a continuously varied parameter, it reveals transitions from stable fixed points through period-doubling cascades to chaotic regimes. The classic example is the logistic map, where the route to chaos is clearly visible as the growth rate parameter increases.

Applications

• Studying the onset of chaos through period-doubling cascades in nonlinear dynamical systems
• Analyzing population dynamics and carrying capacity transitions in ecology
• Understanding period-doubling routes and mode-locking in laser physics
• Teaching nonlinear dynamics and chaos theory with the logistic map as a canonical example

Data

• parameter (numeric) — Bifurcation parameter (e.g., r in the logistic map), typically ranging from 2.5 to 4.0
• state (numeric) — Steady-state or periodic orbit values of the iterated variable after transients are discarded
• Size: 100,000+ points (many state values per parameter value, sampled after discarding transient iterations)
• Example: Logistic map x(n+1) = r * x(n) * (1 - x(n)), iterating for each r value and recording the last N states

Notes

• Use very small point size (1px or smaller) with low alpha transparency to create density-based visualization
• Show the full period-doubling cascade from period-1 through period-2, period-4, etc., leading into chaos
• Use the logistic map x(n+1) = r * x(n) * (1 - x(n)) as the default example system
• Label key bifurcation points where period-doubling occurs (e.g., r ~ 3.0, 3.449, 3.544)
• For each parameter value, discard initial transient iterations (e.g., first 200) and plot the subsequent values (e.g., next 100)
• Parameter axis should span at least 2.5 to 4.0 to capture the full route from stability to chaos

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[github-actions]

🚀 Specification Merged!

Specification: bifurcation-basic
PR: #4441 (merged)

The specification is now in main and synced to PostgreSQL.

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