Recall: Pendulum

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# Recall: Pendulum - PowerPoint PPT Presentation

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. Systems of 1st-order, linear, homogeneous equations. How we solve it (the basic idea).

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Presentation Transcript
Unstable Pendulum

Exponential growth dominates.

Equilibrium is unstable.

Nonlinear systems: the qualitative theoryDay 8: Mon Sep 20

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

1.

3.

2.

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 eqnschaos

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

small

really

small

small

The linearized system

Phase plane

diagram

The “other” equilibrium

Section 6

Problem 4

?

N=2 case

Recall

Interpreting two σ’s

a. attractor (stable)

b. repellor (unstable)

d. limit cycle (neutral)

e. unstable spiral

f. stable spiral

Interpreting two σ’sboth real

a. attractor

b. repellor

Interpreting two σ’s:complex conjugate pair

d. limit cycle

e. unstable spiral

f. stable spiral

Interpreting two σ’s

a. attractor

b. repellor

d. limit cycle

e. unstable spiral

f. stable spiral

The mathematics of love affairs

Strogatz, S., 1988, Math. Magazine61, 35.

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: 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 …?)

decay eigvec

growth eigvec

J

R

Homework

Sec. 6, p. 89

#4: Sketch the full phase diagram:

?

?

#6: Optional