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A phase diagram or phase space is a useful construct used in mathematics and physics to demonstrate and visualise the changes in
a given system.
Every degree of freedom or parameter of the system is represented as an axis of a multidimensional space. Time is not considered to be one
of the dimensions but rather is allowed to flow letting the system (which is represented as a single point moving through the
phase space) evolve. A phase space may contain very many dimensions, For instance a gas containing many molecules may require a
separate dimension for each particle's x, y and z positions and velocities as well as any number of
other properties.
For simple systems (consisting of a single particle moving in one dimension, for example) there may be as few as two degrees
of freedom (typically, position and velocity) and a sketch of the phase portrait may give qualitative information about the
dynamics of system, such as the limit-cycle of the Van Der Pol Oscillator
shown in the diagram.
Here, the horizontal axis gives the position and vertical axis the velocity. As the system evolves, its state follows one of
the lines (trajectories) on the phase diagram.
An classic example of a phase diagram from chaos theory is the Lorenz attractor.
See also Hamiltonian mechanics, Phases of matter, vector
field, divergence.
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