see also:
- Criticality
- Curvature
- Critical Points
- Attractor Basins
- Singularities
- Information Theory
- Complexity Theory
- Chaos Theory
- Learning Dynamics
- Grokking
- Fixed Points
- Topological Dynamics
tell me about attractors, singularities (information theory), their potential connection to phase change learning dynamics as observed with “grocking” phenomena, boundary effects, and connections to chaos theory
Attractors in Dynamical Systems
An attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. In the context of neural networks and deep learning, attractors can represent stable states or patterns that the network learns to recognize or generate. For instance, in recurrent neural networks (RNNs) used for sequence processing, attractors can correspond to stable sequences or patterns that the network converges to over time.
An attractor is a set towards which a dynamical system evolves over time from a wide range of starting conditions. Attractors can be points, curves, surfaces, or more complex sets known as strange attractors, depending on the system’s complexity.
- Point Attractors: Represent stable equilibrium states where the system settles into a steady state. For example, a damped pendulum eventually comes to rest in a vertical position, regardless of its initial swing.
- Periodic Attractors: Correspond to cycles or orbits that the system repeatedly follows, such as the periodic motion of a pendulum with small oscillations or the regular cycle of predator-prey populations in a simplified ecological model.
- Strange Attractors: Are associated with chaotic dynamics, where the system’s path is highly sensitive to initial conditions, leading to complex, non-repeating patterns in phase space. The Lorenz attractor is a famous example, arising in simplified models of atmospheric convection.