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In this paper, we study the results of coupling multistable systems which have hidden attractors with each other. Three modified Sprott systems were coupled and their synchronization was observed. The final state of the synchronized system changes with the change in the coupling strength. This was seen for two different types of coupling, one with a single variable and the other with two system variables.

Synchronization of dynamical systems has become a field of intense interest and hence extensive study in the last decades [

The reason why synchronization of coupled systems is gaining interest is the huge real life applications that collective behavior [

Since the introduction of the concept of synchronization by Pecora and Carroll [

Now looking at behaviors of dynamical systems, a recent concept that is becoming increasingly interesting is the concept of multistability and hidden attractors [

Another interesting concept is the hidden attractor. Hidden attractors are attractors whose basins do not intersect with small neighborhoods of equilibria. This results in needing special methods to find them as standard methods become insufficient. These kinds of attractors were first observed by Yang et al. [

In our study, we have decided to take a multistable system with hidden attractors [

In this work, we study the different synchronization of one such system proposed by Wang and Chen [

This system has a hidden attractor and shows multistability and the parameter

Phase space plot for the uncoupled system (

There are a variety of ways in which the system mentioned can be coupled. We have chosen a simple unidirectional coupling but with all

It can be seen from Figure

Change of the absolute difference between the variables

Let us take the case of unidirectional

After studying the individual cases, it was seen that the final region of synchronization changes from one attractor to another at the different synchronization regions (see Figure

The final state of the variables of the first system at different coupling strength for

Now, when we try the same using

Next we tried the same unidirectional coupling, but now with two variables. For example, the equations for

Many regions of synchronization were observed separated by a desynchronized region. Like the previous case, we expected these regions to show different behavior, which is true, though the order is different, with the system going from point attractor to chaotic to period-three attractor as seen in Figure

Change of the absolute difference between the variables

The final state of the variables of the first system at different coupling strength for

The final state of the variables of the first system at different coupling strength for

In the model studied, we observed synchronization of three systems that were introduced by Sprott et al., in the case of two types of unidirectional coupling, using one variable and then two. We studied the case where the individual systems were in period-three hidden attractor. It was seen that, depending on the strength of coupling between the systems, not only was the synchronization affected, but also the final synchronized state was affected. While the one-variable coupling that gave the most interesting result was for

Our work emphasizes this observed synchronization, which, depending on the coupling strength alone, changes its final state from a stable equilibrium point to the strange and hidden attractors in that order or vice versa depending on the type of coupling. This change is interesting and we attribute this effect to the multistability of the system which made it very sensitive to perturbations.

Since the uncoupled system is very sensitive to the initial conditions, we tried to repeat the whole process with different initial condition, where the initial uncoupled system was in different attractors, including cases where each system was in a different attractor. Similar results were observed, where the system started from the point attractor and then moved to the periodic or chaotic attractor and then again moved on to a different attractor.

The authors declare that they have no conflicts of interest.

This work has been supported by the Polish National Science Centre, MAESTRO Programme, Project no. 2013/08/A/ST8/00/780.