Comment on “How to obtain extreme multistability in coupled dynamical systems”

Abstract

We note that extreme multistability of the type described in the referenced paper can be achieved in virtually any dynamical system by adding extraneous variables and using their initial conditions in place of the existing parameters or as additional parameters. We show several simple examples of this and show how the referenced examples are similar.

Reply to “Comment on ‘How to obtain extreme multistability in coupled dynamical systems’ ”

“…(8)-(12) in Wang et al [3] are very similar to Eq. (5) in the Comment [1]. The same mathematical form can also occur in coupled systems, as discussed in the first coupled autocatalator system analyzed in [4].…”

“…In turn, this corresponds to a foliation of the state space into infinitely many such manifolds each of them containing at least one attractor. Though this has been only demonstrated with coupled systems in [4], the consequence of this statement is much more general and includes all the simpler examples discussed in the Comment [1]. As a consequence, the Comment does not add any new insights into the emergence of extreme multistability.…”

“…Second, we show that the design of controllers to achieve extreme multistability in coupled systems is as general as described in Phys. Rev In this Reply we address the two issues raised in the Comment [1]. (1) The authors of the Comment illustrate that extreme multistability can occur in simple, noncoupled nonlinear dynamical systems by adding an additional variable corresponding to an extraneous driving.…”

“…Rev In this Reply we address the two issues raised in the Comment [1]. (1) The authors of the Comment illustrate that extreme multistability can occur in simple, noncoupled nonlinear dynamical systems by adding an additional variable corresponding to an extraneous driving. (2) The authors state that the method outlined in [2] would lack generality since it would lead to a master-slave dynamics.…”

“…Therefore it will, of course, also work for more complicated Lyapunov functions and controllers, which do not yield a master-slave dynamics. To demonstrate this we take the autocatalator example used by Sprott and Li in their Comment [1] and design the controllers in a different way. The original governing equations with controllers (u ij ,i = 1,2 and, j = 1,2,3) are,…”