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

Question

I’m looking for any $n$-dimensional system of first order differential equations where depending on the initial conditions you can end up in a number of attractors, for example multiple chaotic orbits, fixed points, periodic etc. More specifically, I’m looking for a system that has two or more different chaotic attractors with optionally period orbits or fixed points.

I have only ever seen one system that happened to be three-dimensional which had two chaotic attractors and they had the property of being identical – reflected across the origin. I can't remember what it was.

One thing I’m currently exploring is random matrices, as they appear to have some peculiar properties (complex uniformly distributed eigenvectors). If you use them as a linkage/weighting matrix in a dynamical system, chaotic behavior is trivial to achieve, but I’m not certain that they have multiple distinct orbits.

Answer 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.

Answer 2

This answer provided to an earlier edition of the question, which asked if "you can end up in a number of attractors - for example multiple chaotic orbits, fixed points, . . . "

Here we provide a class of first order ordinary differential equations on $\Bbb R^n$ the members of which may have any positive integral number of attracting point equilibria:

Consider the one-dimensional system

$\dot x_1 = x_1 - x_1^3 = x_1(1 - x_1^2); \tag 1$

it is easy to see that this system has attractors at $x_1 = \pm 1$ and a repelling point at $x_1 = 0$. Now consider the $n$-dimensional system

$\dot x_1 = x_1 - x_1^3, \tag 2$

$\dot x_i = k_i x_i, \; k_i < 0, \; 2 \le i \le n; \tag 3$

this system also has two attractors, at

$x_1 = \pm 1, \; x_i = 0, \; 2 \le i \le n; \tag 4$

now replace $x_1 - x_1^3$ by a polynomial of the form

$p(x_1) = \displaystyle \prod_{i = 1}^m (\alpha_i - x_1), \tag 5$

where $m$ is a positive odd integer and

$\alpha_1 < \alpha_2 < \ldots < \alpha_{m - 1} < \alpha_m, \tag 6$

that is, $\alpha_i < \alpha_j$ for $i < j$. The system

$\dot x_1 = p(x_1) \tag 7$

has $(m - 1) / 2 + 1$ attractors and $(m - 1) / 2$ repellors. If we now replace (2) by (7) in the $n$-dimensional case, we obtain a system on $\Bbb R^n$ with $(m - 1) / 2 + 1$ attracting points. The non-attracting equilibria are "saddles" or hyperbolic equilibria with an $n - 1$ dimensional space of attracting directions and one repelling direction at each one, these directions corresponding to the eigenvalues of the Jacobean of the vector field given by (7), (3).