### About

# Phase space diagram

Phase space characterizes the interdependence of the variables and their
derivatives in a given function *y = y(x*), for example *y = y
(x, y´,y´´). *2 dimensional phase space diagrams are
projections of phase space onto a plane, as *y = y (y*´) or *y´=
y´(y´´)*. In *3D* visualizations the projection
can be spatial as in* y = y( y´,y´´). *

Phase space projections often characterize a function very evidently. In this simple simulation this is demonstrated for the periodicity of the sine function.

The left window shows the function *y *= *sin (n x) *itself in
blue, its first derivative *y´= ncos (nx) *in green*. *The
zero line is in magenta*. *With the** **slider** n** you can
change the parameter *n*, which defines the number of periods in an *x*
interval of 2π. **Reset** defines *n = 1. *

A second slider **x range** defines the range of *x* in the left
window between *0.1* and* 5 *times *2π*. Default
value is *0.5*, limiting the range of the function to one half
period.

The right window shows* *the phase projection *y´(y) *in
red. A blue point characterizes the relation *y´(y)* at the
end of the *x *range. At default (*n = 1 *and* xrange/2π=0.5*)
the phase space diagram is a half circle, corresponding to a half period
of the function within the *xrange*. When *xrange/2π* is
decreased, the phase space curve will contract.* xrange/2π=1.0*
leads to a full circle.

With *n ≠ 1* the derivative has an amplitude *n* times
that of the function. Correspondingly the phase diagram becomes an
ellipse with an elongation of *n. *It has as many revolutions as
the function has periods in the interval.

Periodicity is characterized by a closed space phase diagram with as many cycles as the function has periods.

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH- Foundation; Fremont Teng; Loo Kang Wee

### end faq

### Sample Learning Goals

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### For Teachers

## Phase Space Diagram JavaScript Simulation Applet HTML5

### Instructions

#### x-range and n Sliders

#### Toggling Full Screen

#### Reset Button

Research

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### Video

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### Other Resources

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