1.3 & 1.4

Question 1

Bernoulli Equation has the form:

Answer 1

x’ + p(t)x = q(t) * x^b

Question 2

How do you solve Bernoulli Equation (i.e. x’ + p(t)x = q(t) * x^b)?

Answer 2

It is linear in the variable y = x^ 1-b
1. Take derivative of that to get dy/dt = (1-b) * x^-b * dx/dt
2. Sub in dx/dt from the initial question
3. When possible, substitute y in for any x^1-b
4. Then you have a linear ODE, so solve using integrating factor method

Question 3

An autonomous equation is one which:

Answer 3

does not depend on t, so that x’ = f(x).

Question 4

An equilibrium point is an x* such that given x’ = f(x)

Answer 4

f(x*) = 0

Question 5

Thm: Assume x* is an equilibrium and x(t) is any solution of the equation. If x( t* ) = x* for any t*, then:

Answer 5

x(t) = x* for all t. AKA, you cannot cross an equilibrium point.

Question 6

Suppose x(t) is a solution and x* is an equilibrium point. If x(t) > x* for any t, then:

Answer 6

x(t) > x* for all t. AKA, if you’re above an equilibrium point, you cannot touch it or cross it.

Question 7

Suppose x(t) is a solution which is not a constant. Then:

Answer 7

then x(t) is either strictly increasing or strictly decreasing

Question 8

Suppose x(t) is a solution to x’ = f(x) and x(t.0) = x.0. Then one of the following is true as t gets large

Answer 8

1. x(t) is unbounded for t > t.0
2. x(t) exists for all t > t.0 and lim( x(t) ) = x* for some equilibrium point

Question 9

How to tell if x* equilibrium point is stable or unstable given x’ = f(x)

Answer 9

if f ‘ (x) < 0, stable
if f ‘ (x) > 0, unstable

Question 10

How to find the linearized version of a solution around a equilibrium point x* = x* given x’ = f(x)

Answer 10

x’ = f ‘ (x* ) * (x - x*)
such that x(0) = x.0
so also solve for x.0

Question 11

Phase diagram equivalence

Answer 11

When 2 phase diagrams have the same number and arrows all point in the same direction, doesn’t matter what the points are exactly

Question 12

A bifurcation point is when:

Answer 12

The point p in the parameters when the phase diagram changes (not equivalent). AKA, the phase diagram is unstable at point p because it looks different in the immediate area around it.

Question 13

A bifurcation diagram is a plot of

Answer 13

a plot of equilibrium points as a function of the parameters

Question 14

How to draw a bifurcation diagram.

Answer 14

1. Solve for x* in terms of p, then see how p affects the x* in each case
2. Plug in each x* into f ‘ (x) to find stability of each line

Question 15

Saddle node bifurcation

Answer 15

No x* on one side, and has one x* at bifurcation point, then splits into 2 x*, one stable and one unstable

Question 16

Pitchfork Bifurcation

Answer 16

One side has 1 x, then splits into 3 at bifurcation point. Top and bottom share stability properties with the 1 x on the other side, and the middle one flips stability properties.

Question 17

Transcritical Bifurcation

Answer 17

2 x* pass through each other at bifurcation point, and swap stability properties

Question 18

How to solve for impossible bifurcation diagrams:

Answer 18

Instead of solving for x* in terms of p, solve for p in terms of x*, and the flip graph over the y=x line