Stable and Unstable EPs in 1 and 2D

Summary --- Stable and Unstable EPs in 1 and 2D

These scientific concepts of equilibrium are modeled by the mathematical notion of an equilibrium point of a vector field.

• review of stable and unstable equilibrium point
• all EPs are linear EPS (HGT)
• definition of a linear function in 1D
• equilibrium points of linear Eqs in 1D: \( X' = kX \) and \( X'=-kX \)

One Dimension

• Go back to 1D: EPs in 1D come in 2 flavors: stable and unstable
• These are solutions to diff eqs \( X' = -kX \) and \( X'=kX \)
• All EPs are linear EPs (Hartman-Grobman Theorem)
• What do nonlinear models do? They can have multiple EPs
• example of logistic differential equations: 2 EPs: an unstable and a stable

2 Dimensions

• already talked about the 6 types of linear EPs in 2D. Review them and their differential equations.
• Pure stable, pure unstable, saddle point: what are their corresponding differential equations?
• Pure center (neutral), unstable spiral and stable spiral and their differential equations.
• And by HGT, since all EPs are linear EPs, this is also all EPs in nonlinear systems