Exploration of phase space

Two views for a function: at the bottom, the usual graph for a function [math]\text{y(t)}[/math] as the set of points with coordinates [math]\text{(t,y(t))}[/math]. At the top, associated with it, the so called [i] phase space[/i] where the function is represented by the set of points (y(t),y'(t)) for [math]t\in\mathbb{R}[/math]. In particular, the graphs of functions fulfilling [math]f\left(x\right)=g\left(x+h\right)[/math] are [i]translated[/i] horizontally in the bottom graph but are [i]equal[/i] in the phase space![br][br]You can choose the starting point in the bottom graph but the main drawing tool here is to pilot the small colour point in the phase space and see what the integration result gives in the bottom graph. You pilot the function through its value [math]y[/math] and its derivative [math]y'[/math]. Therefore your movement is constrained: when [math]y'>0[/math], the value [math]y[/math] can only increase while when in the lower half plane, it can only decrease.

Arrows

Along which arrows can you go when in the phase space?

[img]https://i.imgur.com/undKGSn.png[/img] [img]https://i.imgur.com/GSnU6Fb.png[/img] [img]https://i.imgur.com/VotstOn.png[/img] [img]https://i.imgur.com/VEY90qt.png[/img] [img]https://i.imgur.com/QOGdODO.png[/img]

A horizontal line in the phase space

is associated in the second graph with

the graph of a vertical line the graph of a horizontal line (the constant function) a single point the graph of an oblique line (an affine function)

A vertical line in the phase space

is associated in the second graph with

the graph of an oblique line (an affine function) the graph of a vertical line a single point the graph of a horizontal line (the constant function)

The constant function

is represented in the phase space by

a single point on the ordinate axis a single point on the abscissa axis a single point somewhere a horizontal line a vertical line a linear line an affine line

A linear function

is represented in the phase space by

a linear line a vertical line an affine line a single point on the abscissa axis a horizontal line a single point somewhere a single point on the ordinate axis

A periodic function

is represented in the phase space by

a periodic graph a closed loop an oblique line

The exponential function

is represented in the phase space by

a vertical line a horizontal line an affine half-line an affine line a single point on the abscissa axis a single point somewhere a single point on the ordinate axis a linear half-line a linear line

The quadrants

A diverging exponential function

such that [math]\lim_{t\to+\infty}f\left(t\right)=\pm\infty[/math] is represented in the phase space by

an affine half-line in the second quadrant an affine half-line in the fourth quadrant a linear half-line in the second quadrant a linear half-line in the fourth quadrant a linear half-line in the third quadrant an affine half-line in the first quadrant an affine half-line in the third quadrant a linear half-line in the first quadrant

A converging exponential function

such that [math]\lim_{t\to+\infty}f\left(t\right)=0[/math] is represented in the phase space by

an affine half-line in the fourth quadrant an affine half-line in the second quadrant an affine half-line in the third quadrant a linear half-line in the fourth quadrant a linear half-line in the third quadrant a linear half-line in the first quadrant an affine half-line in the first quadrant a linear half-line in the second quadrant

Exploration of phase space

Arrows

A horizontal line in the phase space

A vertical line in the phase space

The constant function

A linear function

A periodic function

The exponential function

The quadrants

A diverging exponential function

A converging exponential function

Information: Exploration of phase space