Dynamical system
Flows

For a flow, the vector field v(x) is an affine function of the position in the phase space, that is,

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 ≠ 0 with A = 0 is just a straight line in the direction of b:

When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x = 0, then the orbit remains there.For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x,

When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors 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 ≠ 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 chaotic behavior.
oulipo:avoid_letter[t], language:response_language[Spanish]