Editing Dynamical system (section)

===Flows===
For a [[flow (mathematics)|flow]], the vector field v(''x'') is an [[affine transformation|affine]] function of the position in the phase space, that is,
:<math> \dot{x} = v(x) = A x + b,</math>
with ''A'' a matrix, ''b'' a vector of numbers and ''x'' the position vector.  The solution to this system can be found by using the superposition principle (linearity).
The case ''b''&nbsp;≠&nbsp;0 with ''A''&nbsp;=&nbsp;0 is just a straight line in the direction of&nbsp;''b'':

: <math>\Phi^t(x_1) = x_1 + b t. </math>

When ''b'' is zero and ''A''&nbsp;≠&nbsp;0 the origin is an equilibrium (or singular) point of the flow, that is, if ''x''<sub>0</sub>&nbsp;=&nbsp;0, then the orbit remains there.
For other initial conditions, the equation of motion is given by the [[matrix exponential|exponential of a matrix]]: for an initial point ''x''<sub>0</sub>,

:  <math>\Phi^t(x_0) = e^{t A} x_0. </math>

When ''b'' = 0, the [[eigenvalue]]s of ''A'' determine the structure of the phase space.  From the eigenvalues and the [[eigenvector]]s of ''A'' it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.

The distance between two different initial conditions in the case ''A''&nbsp;≠&nbsp;0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast.  Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for [[chaos theory|chaotic behavior]].

[[File:LinearFields.png|thumb|500px|center|Linear vector fields and a few trajectories.]]
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