12 terms

mechanic21PLUS

Notes: (please read, press (i) on set to read the entire thing)
This set is revised for clarity and making sure all the terms flow in a straightforward way rather than making them more complex to make it easier to understand.
Note that a value is PURELY ___ means the value must meet the standard requirements of being a ____ type of value (nothing more or less), for example, PURELY imaginary value means it MUST have an imaginary component, no more or no less.
Phase portrait solution classificati…

Stable Node

2 **distinct** (nonequal) real negative eigenvalues.

This phase portrait has 1 or more nullclines.

This phase portrait has 1 or more nullclines.

Unstable Node

2 **distinct** (nonequal) real positive eigenvalues.

This phase portrait has 1 or more nullclines.

This phase portrait has 1 or more nullclines.

Unstable Saddle

2 **distinct** (nonequal) real eigenvalues **with opposite signs**

It is always unstable.

The phase portrait contains nullclines.

It is always unstable.

The phase portrait contains nullclines.

Stable Star

PURELY Real eigenvalues that are REPEATED

Both eigenvalues are negative.

(repeated real e.vals. means/implies you have 2 or more LINEARLY DEPENDENT eigenvectors (only one of them is linearly independent).)

Both eigenvalues are negative.

(repeated real e.vals. means/implies you have 2 or more LINEARLY DEPENDENT eigenvectors (only one of them is linearly independent).)

Unstable Star

PURELY Real eigenvalues that are REPEATED

Both eigenvalues are positive.

(repeated real e.vals. means/implies you have 2 or more LINEARLY DEPENDENT eigenvectors (only one of them is linearly independent).)

Both eigenvalues are positive.

(repeated real e.vals. means/implies you have 2 or more LINEARLY DEPENDENT eigenvectors (only one of them is linearly independent).)

Stable Center

1 or more PURELY imaginary eigenvalues.

A center is ALWAYS stable.

A center is ALWAYS stable.

Stable spiral

Eigenvalues are PURELY complex and distinct (nonequal).

The real part of all eigenvalues are negative.

The real part of all eigenvalues are negative.

Unstable spiral

Eigenvalues are PURELY complex and distinct (nonequal).

The real part of all eigenvalues are positive.

The real part of all eigenvalues are positive.

Generalization: Usually if eigenvalues are negative, or have negative real components, then the solutions are called ____ (stable or unstable) solutions and direction of travel is ____

stable, counterclockwise

(info: see the stable/unstable spiral cases)

(info: see the stable/unstable spiral cases)

Generalization: Usually if eigenvalues are negative, or have positive real components, then the solutions are called ____ (stable or unstable) solutions and direction of travel is ____

unstable, clockwise

(info: see the stable/unstable spiral cases)

(info: see the stable/unstable spiral cases)

Another name for all solutions to the DE going towards the origin

Stable solutions (or a sink (a type of phase portrait))

Another name for all solutions to the DE going away from the origin

Unstable solutions (or a source (a type of phase portrait))