International Journal of Bifurcation and Chaos, Vol. 13, No. 9 (2003) 2733–2737 c World Scientific Publishing Company
NUCLEATION FROM THE BOUNDARIES IN EXCITABLE MEDIA ´ M. DECASTRO and M. GOMEZ-GESTEIRA Department of Applied Physics, Faculty of Sciences, University of Vigo, 32004 Ourense, Spain ∗ ´ ˜ M. N. LORENZO and V. PEREZ-MU NUZURI Group of Nonlinear Physics, Faculty of Physics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain ∗
[email protected]
Received September 27, 2002; Revised October 2, 2002 Nucleation from a boundary is experimentally and numerically studied in a one-dimensional array in two excitable media consisting of Chua’s circuits and the Oregonator model, respectively. Forcing from a boundary with a pulse of constant amplitude and infinite duration gives rise to a periodic wave train propagating through the array. As the amplitude of the pulse increases, wave period evolves to chaos through a period doubling cascade. Keywords: Mechanism of nucleation; excitable media; periodic emission of wave trains; chaos.
1. Introduction Nucleation plays a key role in the abrupt change among different states. In a lot of systems nucleation is related to the formation of bubbles of the most stable state that grow inside the less stable state. In general, these bubbles tend to develop near the boundaries, where inhomogeneities favor the appearance of the most stable state. Once this new state has been formed near the boundaries, bubbles can detach and be transported away. This gives rise to the appearance of a new bubble, which leads to a periodic emission from the boundary. This emission mechanism has been observed to depend strongly on the system. Well-known examples are the fluid heated from below [Drazin & Reid, 1981] vortex emission behind an obstacle in a classical fluid [Tritton, 1988] or in a superfluid [Hakim, 1997; Frisch et al., 1992; Eggers, 1997], the dripping faucet ∗
[Austin, 1991; Martien et al., 1985; Y´epez et al., 1989], or the undesirable effect of wave emission from the boundaries in the Belousov–Zhabotinsky reaction [Zaikin & Zhabotinskii, 1970]. Moreover, the irregular emission of drops in the leaky faucet has been well investigated [Austin, 1991; Martien et al., 1985; Y´epez et al., 1989], which appears to be a prototypical example of low dimensional chaos. In most cases cited above, nucleation takes place after some threshold is surpassed giving rise to the periodic emission of bubbles. For example, vortex detachment occurs above some critical Reynolds number, and the periodic emission of waves in the BZ reaction takes place for certain conditions of excitability, or in the presence of large inhomogeneities in the medium. In excitable media, periodic emission of wave trains in a homogeneous one-dimensional media gives rise to complex patterns if propagation occurs
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through channeled domains. Thus, for example, propagation through rectangular narrow channels gives rise to resonance patterns characterized by firing numbers corresponding to a Farey tree [Toth et al., 1994], and if propagation occurs through a channel with sinusoidal boundaries, then wave trains adopt a quasiperiodic spatial configuration that repeats periodically in time [Sendi˜ na-Nadal et al., 2001]. The aim of this Letter is to study the mechanism of nucleation and then periodic emission of wave trains from a boundary in a free, homogeneous, one-dimensional excitable medium. Two systems have been studied, namely; a one-dimensional array of cells described by the Oregonator model, and a chain of electronic circuits. The arrays were forced at the boundary with a pulse of constant amplitude resulting in a periodic emission of waves. Then, as in the dripping faucet experiment, irregular, even chaotic emission of waves has been observed as the forcing amplitude increases.
2. Model Numerically, the two-variable Oregonator model has traditionally been used as a paradigm for a excitable medium like the Belousov–Zhabotinsky reaction [Krug et al., 1990], u−q 1 ∂u u − u2 − (f v + φ) = ∂t ε u+q + Du
∂2u ∂x2
(1)
∂v = (u − v) ∂t where u (resp. v) describe HBrO2 (resp. catalyst) concentrations. Du is a scaled diffusion coefficient, and q and ε are parameters related to the BZ kinetics, f is an adjustable stoichiometric factor, and φ is a parameter proportional to the light-induced
Table 1. Experimental Chua’s circuit parameters. Parameter C1 C2 L R r0
Value 1 12 10 270 10
nF nF mH Ω Ω
Tolerance 5% 5% 10% 1% 1%
Rc
Rc
R L r0
C2
+
+
v2
v1 -
NR
+ vR
C1
-
iL Fig. 1. Schematic diagram of an experimental Chua’s circuit coupled bidirectionally with the nearest neighbors through resistor Rc . Component values are in Table 1.
flow of Br− . Equations (1) were numerically inteFIGURE (demethod Castrowith et al., 2001) grated using an 1 Euler a time step of −3 10 dimensionless time units (t.u.) per iteration. Initially all cells were set to the stationary value u0 = v0 = 0.004, a value close to the fixed point of the local dynamics. Free ends were considered at the right boundary of the array, while a pulse of constant amplitude was considered at the left boundary. The pulse was generated by setting u = u th and v = v0 during all numerical simulation. With the aim of showing the nucleation effect from a perturbed boundary in another system, a one-dimensional array of Chua’s circuits was built. Chua’s circuit is a nonlinear circuit [Madan, 1993; Chua, 1992] which works in an excitable state with the set of parameters shown in Table 1 [G´omezGesteira et al., 1999; deCastro et al., 1998; deCastro et al., 1999]. Initially, all these circuits were adjusted to have the same initial stable state within the tolerances commercially allowed. Figure 1 shows one of the cells in the 1D array. Every circuit in the array was connected by a resistor, Rc , to its nearest neighbors. Two different coupling resistances were used in the experiment. Note that the effective diffusion between adjacent circuits depends on the resistance between circuits D ∼ 1/Rc . The first circuit in the array was connected to a wave form generator (Hewlett-Packard 33120A). A negative pulse of infinite duration was delivered at node V1 of this circuit (see Fig. 1). The amplitude of this pulse V1th was the control parameter in our experiments. Some of the circuits in the array (circuits 2 and 12) were sampled with
Nucleation From the Boundaries in Excitable Media
a digital oscilloscope (Hewlett-Packard 54601) with a sampling rate of 4 MHz and 8 bits A/D resolution, which allows the automatic measurement of the wave periods corresponding to the waves emitted by the boundary. Period data were acquired by this oscilloscope and stored in a personal computer. Each period was calculated as the average of at least 1000 samples. The shape and amplitude of the waves, which depend on medium properties, hardly vary with the forcing amplitude.
3. Results The period between consecutive peaks in terms of the experimental forcing amplitude is represented in Fig. 2 for the two systems used here. In both cases, there is some critical forcing value, below which no wave train is obtained. This threshold increases slightly with the coupling resistance for the one-dimensional array of Chua’s circuits [Fig. 2(b)] and, for any forcing amplitude, the period between consecutive waves increases with R c . Both with the Oregonator and Chua models as the forcing amplitude increases, the wave train period tends to an asymptotic value which corresponds to the refractory period of the medium, time elapsed after an excitation below which an exter-
nal perturbation cannot generate a new propagating wave in a excitable system [Keener & Phelps, 1989]. After this critical value, in the Oregonator model a period-2 bifurcation arises as indicated by the two branches. As uth increases, a cascade of further period doublings occurs, yielding period-4, period-8, and so on, until at uth = u∞ th ≈ 2.4, the period wave train emission from the boundary becomes chaotic. By carefully measuring the values of u th at the period doubling bifurcations, the Feigenbaum’s constant was measured, δ = 4.8 ± 0.1, in reasonable agreement with the theoretical result δ ≈ 4.669. In the Chua’s 1D array, we were unable to obtain the equivalent period doubling cascade, probably due to experimental uncertainties in the determination of multiple periods. Far from the perturbed boundary, the period of the system accommodates to a value average of those emitted by the source. The cases where a single period was obtained were observed to depend on the forcing amplitude as, A1 T = A0 + √ A − Ac
(2)
where A = |V1th | (resp. A = uth ) and Ac = |V1c | (resp. Ac = ucth ) being |V1c | (resp. ucth ) the critical amplitude, above which wave trains are generated
4.9
125
Wave period (µs)
4.8 4.7 4.6 4.5 4.4
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0
0.5
1
1.5
2
2.5
120
115
110
105
2
4
6
8
10
|Forcing amplitude| (V)
12
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FIGURE 3 (de Castro et al., 2001)
4.8
4.8
FIGURE 3 (de Castro et al., 2001)
4.5 4.6
u =2.0
Ti+1
4.6 4.7
4.6 4.7
i+1
4.7 4.8
4.5 4.6
T
T
i+1
T
i+1
4.7 4.8
th
4.4 4.5 4.4
th
4.5
4.6
u =2.0 th
4.7
4.4 4.5 4.4
4.8
T 4.5
(a)
4.5
u =2.2
4.6
th
i
4.4 4.4 4.8
u =2.2
4.6
4.7
4.4 4.4 4.8
4.8
4.5
(b)
4.8
4.7
4.8
4.7
4.8
i
Ti+1
Ti+1
4.6 4.7
i+1
4.5 4.6
Ti+1
4.7 4.8
4.6 4.7
4.5 4.6
T
4.6
T
4.7 4.8
u =2.4
uth=2.5
th
4.5
4.6
uth=2.4
4.7
4.8
4.4 4.5 4.4
4.5
4.5
4.6
4.6
uTth=2.5 i
T
i
4.7
4.8
4.4 4.4
4.5
4.6
T
T
i
(c) Fig. 3. figure.
4.7
i
i
4.4 4.4
4.8
T
T
4.4 4.5 4.4
4.7
i
(d)
Return map mosaic for the period doubling cascade obtained with the Oregonator model shown in the preceding
from the boundary. A0 , A1 and Ac are free fitting parameters, and the constraint Ac > 0 has been imposed. Equation (2) was analytically predicted by Argentina [1999] for a generic reaction–diffusion system. Fitted curves are represented in Figs. 2(a) and 2(b). Note that T goes to infinity more rapidly than the numerical data for values of u th ≈ ucth , while the experimental data fit quite well to the theoretical expression. Equation (2) suggests a very sharp behavior of the wave period as A → A c which makes difficult to attain numerical values of T near
the discontinuity. Data do not follow exactly the theoretical equation when A → Ac due to the discrete nature of the experimental medium and to discretization effects when solving the Oregonator model. For the period doubling cascade obtained for the Oregonator model, Figs. 3(a) and 3(d) show a return map mosaic for the behavior of the system. In this, return map points corresponding to two consecutive periods for 2T, 4T and chaos regions are plotted.
Nucleation From the Boundaries in Excitable Media
4. Conclusions In summary, we have observed, both numerically and experimentally, that nucleation can give rise to a periodic wave train starting from a boundary. The period of that wave train was observed to decrease with the forcing amplitude following Eq. (2) which was theoretically predicted for a general reaction– diffusion model. This process was experimentally observed in complex systems as an array of excitable electronic circuits, or a lattice of cells described in terms of the Oregonator model. In spite of the different nature of both media, the validity of the theoretical prediction has proved to be independent of the particular features of the medium. Theoretical predictions [Argentina, 1999] also consider the existence of a multiperiod region when approaching the refractory period of the medium but such prediction was only obtained numerically.
Acknowledgments We want to thank M. Argentina for fruitful discussions and a careful reading of the first version of the paper. The support by MCyT under Research Grant BFM2000–0348 is gratefully acknowledged.
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