Editing Dynamical system (section)

== Construction of dynamical systems ==
The concept of ''evolution in time'' is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of [[classical mechanics|classical mechanical systems]]. But a system of [[ordinary differential equation]]s must be solved before it becomes a dynamic system. For example, consider an [[initial value problem]] such as the following:

:<math>\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})</math>
:<math>\boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0</math>

where 
*<math>\dot{\boldsymbol{x}}</math> represents the [[velocity]] of the material point '''x''' 
*''M'' is a finite dimensional manifold
*'''v''': ''T'' × ''M'' → ''TM'' is a [[vector field]] in '''R'''<sup>''n''</sup> or '''C'''<sup>''n''</sup> and represents the change of [[velocity]] induced by the known [[force]]s acting on the given material point in the phase space ''M''. The change is not a vector in the phase space&nbsp;''M'', but is instead in the [[tangent space]] ''TM''.

There is no need for higher order derivatives in the equation, nor for the parameter ''t'' in ''v''(''t'',''x''), because these can be eliminated by considering systems of higher dimensions.

Depending on the properties of this vector field, the mechanical system is called 
*'''autonomous''', when '''v'''(''t'', '''x''') = '''v'''('''x''')
*'''homogeneous''' when '''v'''(''t'', '''0''') = 0 for all ''t''

The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above

:<math>\boldsymbol{{x}}(t)=\Phi(t,\boldsymbol{{x}}_0)</math>

The dynamical system is then (''T'', ''M'', Φ).

Some formal manipulation of the system of [[differential equation]]s shown above gives a more general form of equations a dynamical system must satisfy

:<math>\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak{{G}}\left(t,\Phi(t,\boldsymbol{{x}}_0)\right)=0</math>

where <math>\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}</math> is a [[functional (mathematics)|functional]] from the set of evolution functions to the field of the complex numbers.

This equation is useful when modeling mechanical systems with complicated constraints.

Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally [[Banach space]]s—in which case the differential equations are [[partial differential equation]]s.