Attractors in Duffing equation The Duffing equation in its full form is
$$\ddot{x} + \delta \dot{x} -ax + \beta x^3 = \gamma \cos(\omega t)$$
Now for specific values of the parameters several attractors exist (or not). Let's assume that $\alpha = \beta = \omega = 1$, $\delta = 0.15$, while $\gamma = 0.2445$. For these values of the parameters the system has two fixed-point attractors and a period-3 attractor. The period-1 attractors are located at about (0.815, 0.242) and (−0.933, 0.299). The period-3 attractor is located at about (−1.412,−0.137), (−0.354,−0.614), and (0.645,−0.464).
My question is the following: How do we obtain the values of the positions of the attractors? Is there a theoretical way or is it done purely numerically? And if so, how?
Many thanks in advance!
 A: For dynamical systems in general:


*

*Numerically, you simply start with a point that eventually evolves to the attractor (i.e., that's inside its basin of attraction) and record its coordinates. That means running the simulation for a time arguably "long enough" until the phase space trajectory settles, within a desired precision, into the periodic state.

*Theoretically, in order to find a period-N solution of a map $x_{n+1}=f(x_n)$, you must solve the equation $x=f^N(x)$, where $f^N$ denotes the composition of $F(x)$ with itself $N$ times. In the case of differential equations, in general on has to rely on approximate methods, though the non-forced ($\gamma=0$) Duffing is solvable by quadratures.
Note that the periodic orbit is being given in the Original Post not as a continuous closed orbit, but as a pair of coordinates. That means a Poincaré map is being considered, instead of the original differential equations. A Poincaré map $P(x)$ describes the crossing points $x$ of the EDO trajectories with a given surface of section of dimensionality lower than that of the original system. An illustration of a 3-D curve intersecting a 2-D surface of section at two points is given below.

Image source.
