“bifurcation”的搜索结果

Complex systems and emergent behavior — chaos theory, fractals, neural networks, cellular automata, and computability. Analyzes dynamical systems for chaos via Lyapunov exponents, classifies bifurcations, measures fractal dimension, and models emergent phenomena. Use when analyzing chaotic systems, computing Lyapunov exponents, classifying bifurcations, measuring fractal dimension, modeling neural networks, or studying emergent behavior from simple rules.

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Hopf bifurcation detection for dynamical system state transitions with GF(3) phase portraits

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Generate bifurcation diagrams for dynamical systems. Use when visualizing parameter-dependent behavior transitions.

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Analyze nonlinear dynamical systems — phase portraits, fixed points, stability analysis, bifurcation diagrams, Poincare sections, Lyapunov exponents, and chaos detection. Use for any autonomous or non-autonomous ODE system where qualitative behavior matters.

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Think and work like an expert Dynamical Systems Theorist. Use when a task calls for Dynamical Systems Theorist judgment. Reasons from state spaces, invariant sets, bifurcations, and multiple time scales through normal-form classification, center-manifold reduction, Floquet/Poincaré maps, and continuation tools like AUTO, MatCont, and DynamicalSystems.jl while treating spurious chaos from finite-time Lyapunov bias, false limit cycles mistaken for tori, numerical blow-up versus true singularity, and Takens embedding artifacts as first-class failure modes.

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