# Nonlinear Dynamics ## Overview Nonlinear dynamics studies systems where outputs are not proportional to inputs, giving rise to phenomena such as chaos, bifurcations, limit cycles, and strange attractors. In the context of the Free Energy Principle, nonlinear dynamics are essential for understanding how biological and cognitive systems exhibit multistability, phase transitions, and the spontaneous formation of structured behavior through self-organization. ## Key Concepts ### Attractors - Fixed points, limit cycles, and strange attractors - Attracting sets as the long-run behavior of dissipative systems - FEP interpretation: attracting sets as the states a system "expects" to occupy ### Bifurcations - Qualitative changes in system behavior as parameters vary - Saddle-node, Hopf, and pitchfork bifurcations - Connection to phase transitions in [[complex_systems]] ### Chaos - Sensitive dependence on initial conditions in deterministic systems - Lyapunov exponents and fractal dimensions - Implications for prediction and control in cognitive systems ### Multistability - Coexistence of multiple attracting states - Transitions between states as perceptual switching or decision-making - Related to [[knowledge_base/cognitive/decision_making|decision making]] under the FEP ## Applications - Neural dynamics and brain state transitions - Perceptual multistability (e.g., binocular rivalry) - Ecosystem regime shifts - Social tipping points ## Related Topics - [[dynamical_systems|Dynamical Systems]] - [[complex_systems|Complex Systems]] - [[emergence|Emergence]] - [[synergetics|Synergetics]] - [[knowledge_base/mathematics/numerical_methods|Numerical Methods]] --- > [!note] Open Source and Licensing > Repository: [ActiveInferenceInstitute/cognitive](https://github.com/ActiveInferenceInstitute/cognitive) > - Documentation and knowledge base content: [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) > - Code and examples: MIT License (see `LICENSE`)