Are multiple kinds of attractors (chaotic and otherwise) possible within a single system of differential equations?

Solution 1:

There is a straightforward (but somewhat tedious to execute) way to construct such a system from scratch. I talk in terms of flows (instead of systems of differential equations) since I find this more intuitive for this procedure.

1. Take some same-dimensional flows with whatever attractors (red) you want. I illustrate them as mirrored limit cycles because I am lazy and due to the limits of two dimensions, but the procedure is general:

   

2. Move the flows so that the attractors do not overlap. In the illustration I only move the right flow to the right:

   

3. Create a new flow that is defined piecewise such that in a neighbourhood of each attractor (blue background), the corresponding flow applies:

   

4. Extend this piecewise definition to fill the rest of the phase space and connect the neighbourhoods of the attractors in a manner that fulfils your desires of continuity:

   

Now, for each of your attractors, there is a neighbourhood for which nothing changed (except the offset), so it’s still an attractor.

Any perturbation that will bring the state over the separatrix between the two basins of attraction can bring the system from one attractor to the other. In our example, this separatrix is the straight vertical arrows in the middle in the illustration and the blue arrow illustrates one such perturbation:

If your perturbation comes from noise, this is called noise-induced attractor hopping. The general phenomenon of multiple attractors is called multistability, though this says nothing about the type of attractors differing.