Well, in my understanding basin of attraction is a sub space of the state space or phase space of the system. So by definition of attractor basin, one can not just tell whether the solution has reached  a attractor basin by considering a condition like |x(T)-x*|

Just more details about correct definition given by Sabuj. For any arbitrary small eps, if time is large enough, t>T(eps), all points of dynamical system f(t) should stay at distance less than eps from the attractor.

There are 3 types of attractors: fixed point, limit cycle and strange attractor; see pictures here https://en.wikipedia.org/wiki/Attractor .

With the fixed point it will be just distance from it less than eps. With limit cycle distance is defined as the minimal distance between trajectory and this cycle.

If you have analytical calculation for those attractors, you can check the distance between reached point and attractor in your program.

While it is true, that the measure of strange attractors can be much less than the area of their minimal coverage, the last object (coverage of attractor, .i.e. all points at the distance eps from it) is typically smooth, and it plays role in the definition.

The limit of smooth sets can be not smooth. Consider a family of ellipses around two fixed focuses in the points x=-1 and x=1, for y=0. Consider a sequence of thinner ellipses with set inside that a smooth objects. The limit case is interval [-1,1], which has 2 non-smooth points in the end.

Hi Sabuj, thanks for your comments. I am sorry for my unclear expressions.

The real goal is to determine whether a given point lies in the basin of attraction of an attractor. According to the definition of the basin of attraction, if the given point leads to long-time behavior that approaches that attractor, then we say this point lies in the basin of attractor. Therefore, I consider a condition, like |x(T)-x*|

• Dear Dr. Yegorov, thanks for your comments.  I am sorry for my unclear expressions about my questions. The real goal is to determine whether a given point (in the phase space) lies in the basin of attraction of an attractor. I have corrected it and you can also see my reply to Sabuj.
• Based on your comments, as I understand it, you agree with  the idea that a condition, like |x(T)-x*|
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Dear Suokin,

I think that use can use Mathematica sotware to draw the trajectory. In most cases it will be clear what is the attractor; it is approximately what is left after you delete all evolution before some finite T1. The problem is how to set T2, since you cannot do simulation to infinite time. It is a piece of art.

Dear Dr. Yegorov,

Thanks for your suggestions. I will try it.  When I performed the numerical simulations, I set the final time enough long to ensure the system approaching the stable states.

I have only considered what might be called elementary systems and strange attractors have not arisen. I use the separatrices to determine the basin of attraction. If the system has several critical values, often there are separatrices which are determined essentially by the saddle points. These separatrices divide the phase plane into distinct regions of behavior of the system. In this sense, they become the boundary of the basin of attraction for the appropriate critical point. If this comment is to elementary or off the mark, please excuse it.

I vaguely remember a theorem or a proposition in the book of J. Smoller, Shock waves and reaction diffusion equation which explicitly gave a precise region of attraction for a hyperbolic singularity. The precise region depends on the jacobian of the vector field. As i said, it was many years ago i encountered this theorem (more than 20 years ago)so I do not remember well. but if I am not mistaken it was at chapter 12.