Exploration of phase space
Two views for a function: at the bottom, the usual graph for a function as the set of points with coordinates . At the top, associated with it, the so called phase space where the function is represented by the set of points (y(t),y'(t)) for . In particular, the graphs of functions fulfilling are translated horizontally in the bottom graph but are equal in the phase space!
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 and its derivative . Therefore your movement is constrained: when , the value 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?
A horizontal line in the phase space
is associated in the second graph with
A vertical line in the phase space
is associated in the second graph with
The constant function
is represented in the phase space by
A linear function
is represented in the phase space by
A periodic function
is represented in the phase space by
The exponential function
is represented in the phase space by
The quadrants
A diverging exponential function
such that is represented in the phase space by
A converging exponential function
such that is represented in the phase space by