Editing Autonomous system (mathematics) (section)

=== Higher orders ===

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance [[linear differential equation|linearity]] or dependence of the right side of the equation on the dependent variable only<ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0503.pdf Third order autonomous equation] at [[eqworld]].</ref><ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0506.pdf Fourth order autonomous equation] at [[eqworld]].</ref> (i.e., not its derivatives). This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly [[chaos theory|chaotic]] behavior such as the [[Lorenz attractor]] and the [[Rössler attractor]].

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum.<ref>{{cite book | author = Blanchard | author2 = Devaney | author2-link = Robert L. Devaney | author3 = Hall | title = Differential Equations | publisher = Brooks/Cole Publishing Co | year = 2005 | pages = 540–543 | isbn = 0-495-01265-3}}</ref>