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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. How we solve it (the basic idea). Why it matters. How we solve it (details, examples). Solution: the basic idea.
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Recall: Newton’s 2nd Law for Small Oscillations Equilibrium: F=0 ~0
Systems of 1st-order, linear, homogeneous equations How we solve it (the basic idea). Why it matters. How we solve it (details, examples).
Systems of 1st-order, linear, homogeneous equations Why important? Higher order equations can be converted to 1st order equations. A nonlinear equation can be linearized. Method extends to inhomogenous equations.
Another example Any higher order equation can be converted to a set of 1st order equations.
Nonlinear systems: qualitative solution e.g. Lorentz: 3 eqnschaos phase plane diagram • Stability of equilibria is a • linear problem • qualitative description • of solutions
2-eqns: ecosystem modeling reproduction getting eaten eating starvation
Ecosystem modeling reproduction getting eaten eating starvation Reproduction rate reduced OR: Starvation rate reduced
Linearizing about an equilibrium 2nd-order (quadratic) nonlinearity small really small small
The linearized system Phase plane diagram
N=2 case yesterday
Interpreting two σ’s a. attractor (stable) b. repellor (unstable) c. saddle (unstable) d. limit cycle (neutral) e. unstable spiral f. stable spiral
Strange Attractor Need N>3
Interpreting two σ’s a. attractor b. repellor c. saddle d. limit cycle e. unstable spiral f. stable spiral
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)
Limit cycle J R
Example: Birds of a feather both real positive if b>a negative if b<a negative b<a: both negative (romance fizzles) b>a: one positive, one negative (saddle …?) c. saddle decay eigvec growth eigvec
Example: Birds of a feather both real positive if b>a negative if b<a negative b<a: both negative (romance fizzles) b>a: one positive, one negative (saddle …?)
Example: Birds of a feather both real positive if b>a negative if b<a negative b<a: both negative (romance fizzles) b>a: one positive, one negative (saddle …?)
J R
J R
J R
Why a saddle is unstable J R No matter where you start, things eventually blow up.