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Exploration of phase space

Author:
Christian Mercat
Topic:
Derivative, Differential Calculus, Differential Equation, Exponential Functions, Functions, Function Graph, Integral Calculus
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?

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A horizontal line in the phase space

is associated in the second graph with

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A vertical line in the phase space

is associated in the second graph with

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The constant function

is represented in the phase space by

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A linear function

is represented in the phase space by

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A periodic function

is represented in the phase space by

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The exponential function

is represented in the phase space by

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The quadrants

The quadrants

A diverging exponential function

such that is represented in the phase space by

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A converging exponential function

such that is represented in the phase space by

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