Recall: Finding eigvals and eigvecs

Recall: Newton’s 2 nd Law for Small Oscillations Equilibrium: F=0 ~0

Systems of 1st-order, linear, homogeneous equations 1.How we solve it (the basic idea). 2.Why it matters. 3.How we solve it (details, examples).

Solution: the basic idea

General solution

Systems of 1st-order, linear, homogeneous equations 1.Higher order equations can be converted to 1 st order equations. 2.A nonlinear equation can be linearized. 3.Method extends to inhomogenous equations. Why important?

Conversion to 1 st order

Another example Any higher order equation can be converted to a set of 1 st order equations.

Nonlinear systems: qualitative solution e.g. Lorentz: 3 eqns  chaos Stability of equilibria is a linear problem °qualitative description of solutions phase plane diagram

2-eqns: ecosystem modeling reproduction starvation eating getting eaten

Ecosystem modeling reproduction starvation eating getting eaten OR: Reproduction rate reduced Starvation rate reduced

Ecosystem modeling

Linearizing about an equilibrium 2 nd -order (quadratic) nonlinearity small really small

The linearized system Phase plane diagram

Linear, homogeneous systems

Solution

Interpreting σ

General solution

N=2 case yesterday

b. repellor (unstable)a. attractor (stable) c. saddle (unstable) d. limit cycle (neutral) e. unstable spiral f. stable spiral Interpreting two σ’s

Need N>3

b. repellora. attractor c. saddle d. limit cycle e. unstable spiral f. stable spiral Interpreting two σ’s

The mathematics of love affairs (S. Strogatz) R(t)= Romeo’s affection for Juliet J(t) = Juliet’s affection for Romeo Response to own feelings (><0) Response to other person (><0)

The mathematics of love affairs (S. Strogatz) R(t)= Romeo’s affection for Juliet J(t) = Juliet’s affection for Romeo Response to own feelings (><0) Response to other person (><0)

Example: Out of touch with feelings

Limit cycle R J

Example: Birds of a feather

negative positive if b>a negative if b<a b<a: both negative (romance fizzles) b>a: one positive, one negative (saddle …?) both real c. saddle growth eigvec decay eigvec

Example: Birds of a feather negative positive if b>a negative if b<a b<a: both negative (romance fizzles) b>a: one positive, one negative (saddle …?) both real

Example: Birds of a feather negative positive if b>a negative if b<a b<a: both negative (romance fizzles) b>a: one positive, one negative (saddle …?) both real

Example: Birds of a feather

R J

R J

R J

Why a saddle is unstable R J No matter where you start, things eventually blow up.