Cubic model
Linear growth, external forcing, and nonlinear terms compete.
du/dt=f+au−4u²−4u³
Simple Nonlinear ODEs
Multiple equilibria, thresholds, and multistability
Nonlinearity allows several equilibria to coexist, so long-term behavior can depend on the initial condition.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: stable states and unstable thresholds are local features of the nonlinear vector field.
One Concept · One Example
Simple Nonlinear ODEs
Presented by Charlotte Moser · Coming soon
What to notice
- A cubic can have one or three real equilibria.
- Stable roots attract nearby trajectories.
- An unstable root separates basins of attraction.
The mathematical structure
Stability test
The slope of the vector field determines local attraction or repulsion.
F′(u*)<0 stable; F′(u*)>0 unstable
Create or remove multiple stable states
Match the slides by comparing a=4 with f=10 and f=2, then choose an initial condition and follow its trajectory.
Nonlinearity can create multiple possible futures separated by unstable thresholds.