Finite Dimensional Integrable Nonlinear Dynamical Systems

We have seen in the earlier chapters that dynamical equations of typical nonlinear systems are not amenable to exact solutions in terms of known functions, except for isolated cases like the equations of motion of the undamped and unforced Duffing oscillator, the pendulum, the Kepler particle and so on. Generally speaking, finite dimensional nonlinear systems of the type (3.1), for n > 2, are not explicitly solvable, and they are often chaotic depending upon the values of the control parameters. Integrating the equations of motion completely, obtaining analytic solutions and finding acceptable constants of motion/integrals of motion/invariants for such nonlinear systems seem to be rare. Yet they do exist in isolated, but a large number of cases; one says that they are measure zero systems among the totality of nonlinear systems.

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