Recall: Pendulum
Unstable Pendulum Exponential growth dominates. Equilibrium is unstable.
Recall: Finding eigvals and eigvecs
Nonlinear systems: the qualitative theory Day 8: Mon Sep 20 1.How we solve it (the basic idea). 2.Why it matters. 3.How we solve it (details, examples). Systems of 1st-order, linear, homogeneous equations
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
Equilibria
Linearizing about an equilibrium 2 nd -order (quadratic) nonlinearity
Linearizing about an equilibrium 2 nd -order (quadratic) nonlinearity small really small
The linearized system cancel
The linearized system Phase plane diagram
The “other” equilibrium Section 6 Problem 4 ?
Linear, homogeneous systems
Solution
Interpreting σ
General solution
N=2 case Recall
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 Interpreting two σ’s both real
d. limit cycle e. unstable spiral f. stable spiral Interpreting two σ’s: complex conjugate pair
b. repellora. attractor c. saddle d. limit cycle e. unstable spiral f. stable spiral Interpreting two σ’s
The mathematics of love affairs R(t)= Romeo’s affection for Juliet J(t) = Juliet’s affection for Romeo Response to own feelings (><0) Response to other person (><0) Strogatz, S., 1988, Math. Magazine 61, 35.
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 R J
Decaying case: a>b R J
Saddle: a<b R J
R J
Homework Sec. 6, p. 89 #4: Sketch the full phase diagram: ? ? #6: Optional
Why a saddle is unstable R J No matter where you start, things eventually blow up.