Concept
Long-term behavior of time-evolving systems. Started by Poincaré (3-body problem) — fixed points, periodic orbits, chaos.
Understand it in one breath
Specify a state and a rule of change, then study trajectories, equilibria, cycles, and chaos. Pendulums, planets, neurons, and ecosystems use different equations but share geometric and topological tools. Poincaré's work on the three-body problem exposed limits to integrability and homoclinic structure, helping open modern chaos theory; it should not be reduced to a blanket claim that no closed form exists.
At a glance
System | Equation | Long-term behavior |
|---|---|---|
Simple pendulum | θ'' + ω² sin θ = 0 | Periodic |
Damped pendulum | θ'' + γθ' + ω² sin θ = 0 | Converges to a fixed point |
Lotka–Volterra (predator–prey) | x' = αx − βxy, y' = δxy − γy | Periodic orbit |
Lorenz system | x' = σ(y−x), y' = x(ρ−z) − y, z' = xy − βz | Chaotic attractor |
Neural firing (Hodgkin–Huxley) | Four state variables | Electrical pulses |
Three-body problem | Positions, velocities, and conserved quantities | Special and series solutions exist, but general behavior is complex |
Different phenomena can be compared through common questions about state spaces, fixed points, stability, and bifurcations. The range over which a model describes reality must be validated separately.
Key formula
Modern applications
Weather prediction, neuroscience, planetary orbits, population models, and reinforcement learning.
Beyond MathVoyage
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