For the problem u'=f(t,u), one can numerically solve the zero-finding problem F(u) = u(t=T)-u=0 to obtain the periodic solution u(t+T)=u(t).
Now, how can one obtain quasi-periodic solutions?
If there are periods,T_1 and T_2, i.e. solutions to the (infinite) system of equations u(t+T)=u(t), such that T_1/T_2 isn’t a rational number, then the linear combination of such solutions will be quasiperiodic.
However finding the period(s) isn’t, simply, a numerical problem-it depends on the properties of f(t,u). Not all systems of diferential equations have periodic solutions or quasiperiodic solutions.
Yes, there are methods to numerically calculate quasi-periodic solutions to a system of ordinary differential equations of the form u'=f(t,u).
One common method is the Harmonic Balance Method (HBM). The idea behind HBM is to assume that the solution can be written as a sum of a finite number of harmonic functions with unknown amplitudes and phases. The number of harmonics required depends on the complexity of the solution.
Using this assumption, one can then formulate a system of algebraic equations, called the nonlinear algebraic equations, by substituting the assumed solution into the original differential equations. The unknown amplitudes and phases can then be found by solving this system of equations.
Another method is the Averaging Method. This method involves averaging the differential equations over a period of time, which reduces the original differential equations to a set of algebraic equations. The resulting equations can then be solved numerically to obtain the quasi-periodic solutions.
Finally, the Method of Multiple Scales can also be used to find quasi-periodic solutions. This method involves assuming that the solution can be expressed as a power series in a small parameter and using this assumption to derive a set of equations that describe the dynamics of the system.
All of these methods require some level of analytical manipulation and are often used in conjunction with numerical methods such as Runge-Kutta or the shooting method.
Here is a reference that might be helpful:
Article Invariant tori in dissipative hyperchaos
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