How do I know if a system of differential equations has a global attractor?
I am an undergraduate physics student. I am trying to understand what is the domain of applicability of Convergent Cross Mapping. I tested it on nonlinear and linear systems of equations and gives counter-intuitive results. My hypothesis is that it applies only to dynamical systems of differential equations which have a global attractor. But I am new to this area. How do I know if system of diff eqs has a global attractor? Are there criteria for it?
I understood the definition of the global attractor from here and maybe here. But I am interested only to know if it has or not. Not what the global attractor is.
Thanks in advance!
PS: I know this is a soft question and I am not sure if I posted it in the right place.
Best Answer
There seem to be multiple definitions of global attractor, so I cover them separately.
Case 1: Global attractors are attractors
Here, I assume that a global attractor is an attractor whose basin of attraction is the entire phase space. In particular, a system with a global attractor is never multistable.
I am not aware of any straightforward criteria that allow you determine the existence of global attractors and are applicable to a relevant number of cases. For most complex dynamical systems the existence of attractors and their number is usually determined by simulations or thinking very hard™.
That being said, there are a few classes of systems which are known to have global attractors and, in certain cases it is easy to show that a system does not have global attractors. Some examples:
Linear differential-equation systems have a known solution, which has at most one attractor.
Hamiltonian systems do not have attractors at all (only dissipative systems have).
If your differential equation is $\dot{x} = f(x)$, a point $x$ is a stable fixed point (a certain type of attractor) if and only if $f(x)=0$ and the real parts of all eigenvalues $\nabla f(x)$ are negative. If you can find two such stable fixed points, your system does not have a global attractor. Mind that the inverse does not hold as not all attractors are stable fixed points.
Ecological or metabolic systems are known to need a positive feedback loop to exhibit multistability.
Case 2: Global attractors contain all attractors
Here, I assume that a global attractor is a set whose basin of attraction is the entire phase space. Such a global attractor would be the union of all attractors. Most confusingly, such a global attractor would not need to be an attractor itself, since it is not minimal. Finally note that the union of all attractors of a system needs not be a global attractor.
In this case, all you need to show is that your dynamics is bounded, i.e., there are no trajectories going towards infinity. Again there are some cases where it has been shown that such global attractors exist and in some cases it can be easy to show that the system has such a global attractor. For example, if you know that your system’s dynamics is usually evolving around the origin, it may be easy to show that for points that have a certain distance to the origin, this distance is decreasing over time.