Jtam-A4.dvi


JOURNAL OF THEORETICAL SHORT RESEARCH COMMUNICATION

AND APPLIED MECHANICS

52, 1, pp. 281-284, Warsaw 2014

COUPLING MULTISTABLE SYSTEMS: UNCERTAINTY DUE TO THE

INITIAL POSITIONS ON THE ATTRACTORS

Patrycja Kuzma

Technical University of Lodz, Division of Dynamics, Poland

Marcin Kapitaniak

Technical University of Lodz, Division of Dynamics, Łódź, Poland and

Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Scotland

Tomasz Kapitaniak

Technical University of Lodz, Division of Dynamics, Łódź, Poland

e-mail: tomasz.kapitaniak@p.lodz.pl

We consider the coupling of multistable nonidentical systems. For small values of the co-
upling coefficient the behavior of the coupled system stronglydepends on the actual position
of trajectories on their attractors in the moment when the coupling is introduced. After re-
aching the coupling threshold value, this dependence disappears.We give an evidence that
this behavior is robust as it exists for a wide range of parameters and different types of
coupling. We argue why this behavior cannot be considered as a dependence on the initial
conditions.

Key words: coupled systems, synchronization, attractors

Multistability, e.g. the existence of several attractors for a given set of system parameters, is
common in: weakly dissipative systems, systems involving a delay, and coupled systems (Feudel
et al., 1998). It has been observed in a large variety of systems inmany areas of science (Feudel,
2008), namely, nonlinear optics, neuroscience, climate dynamics, laser physics, and electronic
oscillators. Most studies performed so far dealt with systems which being uncoupled are mono-
stable, with some exceptions (Pisarchik et al., 2006).

We consider the dynamical system xn+1 = f(xn,p) or dx/dt = f(x,p) (x ∈ R
n, p ∈ R

describes the systemcontrol parameter).Let for p∈ D ⊂ R the considered systembemultistable.
Assume that m of such systems (characterized by different values of p) become the subsystems
of one coupled system. Let for t = 0 (i.e. at the moment when the coupling is introduced)
each subsystem operates on a different co-existing attractor. For small values of the coupling
coefficient ε, the behavior of the coupled systemdepends not only on the types of the attractors
onwhich m subsystems operate but also on the actual positions of the subsystem trajectories on
their attractors at the moment when the coupling is introduced. We call this type of behavior
the uncertainty due to the positions on the attractors. Increasing the coupling strength after
reaching the particular threshold value of the coupling coefficient ε, this dependence is no
longer observed. This behavior is not simply the dependence on the initial conditions. Imagine
that in the experimental case we have n multistable systems operating on different attractors.
At one moment (t = 0), all n systems are coupled together creating one augmented system.
As typically in the experiments, one cannot estimate the exact values of the systems state at
t=0 and the behavior for ε> 0 cannot be predicted. The behavior of coupled systems can be
predicted and controlled only for a large coupling.



282 P. Kuzma et al.

To illustrate our finding, we use examples: the coupled Hénon maps (discrete system) and
coupled excited van der Pol-Duffing oscillators (continuous system). Firstly, we focus on the
Hénon maps

x
(i)
n+1 =1−pix

(i)
n

2
+y(i)
n
+ε(y(i−1)

n
−y(i)
n
)

y
(i)
n+1 =−bx

(i)
n

(1)

where i=1,2,3, pi and b are constant and ε is the coupling coefficient. For pi ∈ [1.480,1.485]
and b = 0.138, each map exhibits multistability (Casas and Rech, 2012; Mart́ınez-Zérega and
Pisarchik, 2012; Sausedo-Solorio and Pisarchik, 2011). In our numerical simulations, we consi-
der the following values p1 = 1.4807 (uncoupled map has period-2 and period-6 attractors),
p2 = 1.4820 (uncoupled map has period-4 and period-12 attractors) and p3 = 1.4847 (unco-
upled map has period-4 and chaotic attractors). At the moment when the coupling has been
introduced, the Hénon maps evolve respectively on period-2 (x1,y1-map) and period-4 (x2,y2
and x3,y3-maps) shown in the boxes in Fig. 1. The initial position of the subsystem trajectories

Fig. 1. Bifurcation diagrams of system (1): b=0.138, p1 =1.4807, p2 =1.4820 and p3 =1.4847.
Subsystems operates on different attractors as can be seen on the Poincarémaps shown

in the small boxes

is indicated in light-grey color. In the example of Fig. 1a, the coupled system switches to the
period-4 attractors for ε 6=0 and stays on it then the whole considered interval of the coupling
coefficient. In Fig. 1b, the bifurcation scenario is different. The evolution of the coupled system
switches to the different period-4 orbit which with the increase of ε undergoes period doubling
route to chaos. After reaching the threshold value εth, the chaotic attractor disappears and the
system evolves on the same period-4 orbit as in the example of Fig. 1a. The same uncertainty
effects have been observed for a larger number of the coupled Hénon maps (we consider up to
100 maps). The threshold value of the coupling coefficient εth seems to be independent of the
number of maps.
As an example of the continuous system, we consider a ring of unidirectionally coupled

externally exited van der Pol-Duffing oscillators

ẍi−α(1−x
2
i
)ẋi+x

3
i
=F sin(pit)+ε(xi−xi−1) (2)



Short Research Communication – Coupling multistable systems: uncertainty due to... 283

where i=1,2,3, α, pi and F are constant.We assumed α=0.2,F =1, p1 =0.975, p2 =0.962,
p3 = 0.955 and consider ε as a control parameter. In Chudzik et al. (2011), it was found that
for pi ∈ (0.93,0.98) each oscillator possessed plethora co-existing attractors of different types.

At the moment when the coupling is introduced (t = 0) the subsystem trajectories have
been respectively on 7, 18 and 31 periodic attractors shown in the boxes in Fig. 2. The actual
position of the trajectories on the attractors is indicated in light-grey color. In Figs. 2a and 2b,
we present the bifurcation diagrams of system (2) along the coupling coefficient ε. The results

Fig. 2. Bifurcation diagrams of system (2): α=0.2, F =1, p1 =0.9750, p2 =0.9621, p3 =0.9552.
Subsystems operates on different attractors as indicated on the Poincarémaps shown in the small boxes

shown in Fig. 2a differ from those in Fig. 2b only by the subsystem trajectories positions on
the attractors at t = 0. Comparing both bifurcations diagrams, one notices the differences
(indicated in light-grey color). The dependence on the positions of the initial attractors is visible
in small intervals around ε1 and ε2 and in the larger interval [ε3,εth]. Notice that in the
interval [ε4,εth], we observe the coexistence of different types of attractors. After the passage
of the threshold value εth, the uncertainty disappears and the coupled systems evolve on the
same chaotic attractor. The described uncertainty has been observed for a larger number of the
coupled oscillators (we consider up to 100 oscillators) and also for different types of coupling
(mutual, global). The threshold value of the coupling coefficient εth seems to be independent of
the number of oscillators but differs for different types of coupling.

To summarize, we have shown that the coupling of multistable systems which operate on
different attractors, we cannot predict the behavior of the system. For small coupling, the co-
upled system can operate on different coexisting attractors. The uncertainty due to the initial
positions on the attractors seems to be common for the class of coupled systemswithmultistable
subsystems.We observed it in a number of systems independently of the number of subsystems
and type of coupling. This uncertainty may have practical implications. For some applications,
the coexistence of attractors can be considered as undesired behavior. For example, for small
values of the coupling coefficient (ε < εth), undesired behavior can be observed after the tem-
poral breakdown of the coupling. After reestablishing of the coupling, one has to apply a special
control procedure to reach the desired behavior again. From the point of view of the experimen-
tal systems, the described behavior cannot be simply considered as a dependence on the initial
conditions.



284 P. Kuzma et al.

Acknowledgment

This work has been supported by the Foundation for Polish Science, Team Programme – Project

No. TEAM/2010/5/5.

References

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Manuscript received July 14, 2013; accepted for print October 7, 2013