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DYNAMICAL STUDIES OF A HYBRID OPTICAL BISTABLE SYSTEM
B. Hawdon, J. O’Gorman, D. Heffernan
To cite this version:
B. Hawdon, J. O’Gorman, D. Heffernan. DYNAMICAL STUDIES OF A HYBRID OPTI- CAL BISTABLE SYSTEM. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-419-C2-422.
�10.1051/jphyscol:19882100�. �jpa-00227610�
DYNAMICAL STUDIES OF A HYBRID OPTICAL BISTABLE SYSTEM
B.J. HAWDON, J. O'GORMAN and D.M. HEFFERNAN*
Department of Pure and Applied Physics, Trinity College, IR-Dublin 2, Ireland
* ~ c h o 6 l of Physical Sciences, NIHE, IR-Dublin 9, Ireland and School of Theoretical Physics, DIAS, IR-Dublin 4, Ireland
A b s t r a c t
We have numerically studied an equation which models the hybrid optical bistable device. We have found, in agreement with previous studies, regions of linearly increasing dimension and periodic windows as the system's delay parameter was increased. We discover both frequency-locked and periodic waveforms in these windows with which is associated metastable chaotic and quasiperiodic transients. This type of behaviour can be interpreted as arising from attractor crisis.
Introduction
The hybrid optical bistable device has attracted considerable attention recently because it is a nonlinear system showing extremely rich dynamical behaviour including transitions via bifurcations from periodicity to chaos.
This system, which incorporates a delay has shown good agreement between theory and experiment. Ikeda et a1 [I] were the first to predict bifurcations and chaos in this system while Gibbs et a1 [2] were the first to verify these predictions experimentally. The period doubling sequence to chaos observed was well described by the discrete map model. This difference equation, in fact, predicts periodicity windows in the chaotic domain and these windows were subsequently observed by VallBe et a1 [3] along with frequency-locked (FL) waveforms which had previously been reported [4]. Derstine et a1 [4]
have described the different transitions that can occur between the stable and unstable FL waveforms pointing out that the stability properties of the FL waveforms are complicated although they were not sure whether this was due to system drifts or an intrinsic property of the chaos. VallCe et a1 [3] also state that the experimental transition to a periodic orbit is characterized by a hysteresis such that by comparison to a usual bistable system, the chaotic orbit corresponds to the lower branch. This phenomenon is not observed theoretically.
In this paper we study crisis phenomena which has been studied by Grebogi et a1 [6,7]. The increase in the (average) transients necessary as one goes from one type of waveform to another, in the observed periodic windows, seems to indicate that a boundary crisis is being observed.
Discussion
The equation which we consider is given by
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882100
C2-420 JOURNAL DE PHYSIQUE
H e r e p,
5
and XB me constants and are assumed to take on values 1.01, 0.96 and -z/2 respectively. zmed and tR are the risetime of the photodetector and the rountrip time in the fibre (in the experimental setup of Derstine et a1 [4]).We let y = tR/zmed. Equation (1) is solved numerically for the range 6 c y c 22 where we are not oinly concerned with the final waveform predicted by this equation after transients have decayed but also in the metastable states through which this final waveform is attained. Over this interval, the dimension of the attractors shows regions of roughly linear increase (in agreement with the work of Le Berre et a1 [5]) punctuated by dips of low dimension. Examination of the time series associated with these dips, shows that they are in fact well defined periodic windows. With each of these regions there corres;ponds a mode of oscillation which can be determined from the fast oscillation frequency component of these waveforms [3].
Figure1 summarizes the behaviour across the first dip. The (average) number of transients necessary across this periodic window (i.e. the number of
Figure 1. Diagram of the final waveforms found across the range 10 < y < 13.
computed roundtrips) before getting a stable waveform varies, being greatest at the points nearest the transitions between the different waveforms. This phenomenon can be observed in Figure 2 where the convergence of the largest Lyapunov exponent is plotted versus the number of computed rountrips for y =10.2. This graph would be different for different initial
Chaos
Metastable Transients
. .-.
Frequency
Periodic Frequency
Locked
"A
.
\.---
. . Locked OrbitI
Chaos
0.00
0 2000 4000 6 0 0 0 8 0 0 0 1 0 0 0 0 Rountrips
Figure 2. Convergence of the largest Lyapunov exponent for y = 10.2.
quantity. This type of behaviour is a signature of boundary crisis. A boundary crisis manifests itself through the increase in the length of chaotic transients in the transition zone. Typical trajectories initialised in the region occupied by the chaotic attractor appear to move about in this region in a chaotic fashion, as it does before the crisis, but only for a finite time T after which the orbit leaves rapidly. This time T is shown by Grebogi et a1 [6,7] to follow the relation:
where yc is the crisis delay value and a is the critical exponent although this relationship has not been confirmed by us yet. The dimension of the metastable chaotic transients is high for this transition (as would be expected from y < yc dimension measurements) confirming to a certain extent our observation. In the transition from the frequency-locked to periodic waveform regions the dimension of the metastable transient state is low-dimensional. It therefore seems that the dimension of the metastable transients can indicate the type of boundary crisis being observed. Figure 3 shows the sequence of transitions for y =11.20 for one particular set of initial conditions.
Quasifrequency
Locked
+
Chaotic--+
PeriodicFigure 3. The waveforms predicted by equation (1) for y =11.20 after lo3, l o 4 and lo5 roundtrips respectively. The dimension of the aperiodic waveforms are low-dimensional.
Conclusion
We have numerically investigated a differential difference equation which models a hybrid optical device. Our results are, for the most part, in agreement with those of other works. We find an overall increase in dimension of the chaotic attractor as the delay is increased, accompanied by regions of low integer dimension where the motions of the system are complicated but not chaotic. We have found that these stable motions are accompanied by long-lived chaotic transients and have tentatively suggested that these may be the signatures of crises driven dynamics in this system.
Acknowledgements
We wish to thank P Jenkins for valuable assistance and for bringing the work of Vallee and Delisle to our attention.
References
[I] K Ikeda, H Daido and 0 Akimoto, Phys Rev Lett 45(1980)709
[2] H Gibbs, F Hopf, Kaplan and Shoemaker, Phys Rev Lea 46(1981)474
C2-422 JOURNAL DE PHYSIQUE
[31 R Vall6e and C Delisle, Phys Rev 34A(1986)309
[4] M Derstine, H Gibbs, F Hopf and D Kaplan, Phys Rev 27A(1983)3200 [5] M Le Berre, E Ressayre, A Tallet, H Gibbs. D Kaplan and M Rose,
Phys Rev 35A(1987)4020
[6] C Grobogi, E Ott and J Yorke, Phys Rev Lett 57(1986)1284
[7] C Grebogi, E Ott and F Romerias and J Yorke, Phys Rev 36A(1987)5365
