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CHIN. PHYS. LETT. Vol.28, No.2 (2011) 020504

Spiral Wave Dynamics in a Response System Subjected to a Spiral Wave Forcing *

LI Guang-Zhao(黎广钊), CHEN Yong-Qi(陈永淇), TANG Guo-Ning(唐国宁)**, LIU Jun-Xian(刘军贤)

College o Physics and Technology, Guangxi Normal University, Guilin 541004

(Received 14 September 2010)Unidirectional linear error eedback coupling o two excitable medium systems displaying spiral waves is con-

sidered. The spiral wave in the response system is thus subjected to a spiral wave orcing. We fnd that the 

unidirectional eedback coupling can lead to richer behaviour than the mutual coupling. The spiral wave dynamics

in the response system depends on the coupling strength and requency mismatch. When the coupling strength is

small, the eedback coupling induces the drit or meander o the orced spiral wave. When the coupling strength

is large enough, the eedback coupling may lead to the transition rom spiral wave to anti-target or target-like 

wave. The generation o anti-target wave in coupled excitable media is observed or the frst time. Furthermore,

when the coupling strength is strong, the synchronization between two subsystems can be established.

PACS: 05.10.−a, 05.45.−a, 82.40.CK  DOI: 10.1088/0256-307X/28/2/020504

Spiral waves (SWs) and target waves (TWs)

are the most requently encountered patternsin two-dimensional systems driven away romequilibrium.[1−5] They have been observed in a largevariety o spatiotemporal systems, including biologi-cal systems (e.g., the cardiac muscle tissue), physicalsystems (CO oxidation on platinum), and the chemi-cal reaction difusion systems (Belousov–Zhabotinskyreaction), etc. The two types o patterns have been ex-tensively investigated in both excitable and oscillatorymedia.[6−15] The driting, meandering, multi-arm, in-wardly rotating and breathing SW patterns have beenound successively.[6−10] The breathing TW and anti-

target wave (ATW) have been observed too.[11,12]

Recently, coupled nonlinear systems have beenwidely studied theoretically and experimentally.[16−24]

In coupled excitable medium systems, or exam-ple, Hildebrand et al .[17] investigated experimentallyand theoretically the synchronization o two mutu-ally coupled Belousov–Zhabotinsky systems, and re-vealed the generalized synchronization o SWs. Yangand Yang[21] investigated the SW dynamics in linearlycoupled 2D reaction-difusion systems, and ound thatwhen synchronization o SWs in the two subsystemsis established, the two subsystems play diferent roles

in collective dynamics: one subsystem is always dom-inant and enslaves the other. However, the dynamicso SWs in an excitable medium, which is coupled uni-directionally with other excitable medium exhibitingSW behaviors, has never been studied in the litera-ture. Such a conguration is signicant in the cardiactissue. Thereore, it is quite interesting to understandwhat would happen or SW dynamics in response sys-tems i the drive and response systems have diferentparameters.

In this Letter, we investigate the SW dynamics in

the response system subjected to an SW orcing. We

consider coupled reaction-difusion systems with no-ux boundary where the dynamical equations o thesubsystem are proposed by Bär and Eiswirth.[6] Themodel in the dimensionless orm is expressed as

1

=  1(1, 1) + ∇21, (1a)

1

= (1, 1), (1b)

2

=  2(2, 2) + ∇22 + (1 − 2), (1c)

2

= (2, 2), (1d)

 1,2(, ) =1

1,2(1− )

(−

+

),

(, ) =

⎧⎪⎨⎪⎩−, < 1

3 ,

1− 6.75(− 1)2 − , 13 ≤ ≤ 1,

1− , > 1,

where ∇2 =  2/2 +  2/2. The subsystem is ex-citable or small positive and < 1+. For any given, the isolated two-dimensional system may undergotransition rom a rigidly rotating SW to SW breakupwhen 1,2 is increased.[6] The rotating requency o 

the SW always decreases as increases.[21] To numer-ically simulate the coupled systems, we use the or-ward Euler method together with a second-order ac-curate nite-diference method, with xed time step∆ = 0.02 and spatial step ∆ = ∆ = 0.4. Thesystem size is set to be  ∆× ∆, where  = 300.Throughout this study, the positive parameters and are xed at = 0.84 and = 0.07. The value o  varies within the range [0.025,0.06] so that there canexist rigidly rotating SW in the subsystem. All thenumerical results in this study are checked with ner

*Supported by the National Natural Science Foundation o China under Grant Nos 10765002 and 10947011.**Email: tangguoning@sohu.comc○ 2011 Chinese Physical Society and IOP Publishing Ltd

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CHIN. PHYS. LETT. Vol.28, No.2 (2011) 020504

space grids and smaller time step.We suppose that the subsystem exists an SW and

its tip is located at the center o the medium. Thus,the SW in the response system will be subjected toan SW orcing. We rst consider the case o  1 < 2.

Generally, the SW dynamics in the response systemdepends on the coupling strength or given 1,2.Without loss o generality, we let 1 = 0.025 and2 = 0.05. For a broad range o  ( ≤ 0.6), the ol-lowing scenario is ound. When < 0.05, the orcedSW keeps its dynamics almost unchanged in respecto its period and wavelength. When 0.05 ≤ ≤ 0.1,the tip o the orced SW drits slowly along the spiralcurve while it moves along a simple cycle (see Fig. 1).Usually, the drit velocity o SW increases as in-creases while meander o SW becomes weaker. Whenthe drit velocity is large enough, the tip o the orced

SW can drit to the boundary o the system, as shownin Fig. 2. From Fig. 2, it is observed that the SW orc-ing will produce a point vibration source (i.e., TWsource) at the location where the tip dissipates. Thepoint vibration source evolves gradually and becomesa line vibration source. The wave produced by theline vibration source is nally transormed into ATWby the SW orcing. When 0.1 < ≤ 0.25, the re-sults obtained are similar to those shown Fig. 2. Whenthe coupling strength is large enough, the SW orcing

can directly annihilate the tip o the orced SW andwill generate an irregularly circular vibration sourcearound the location where the tip dissipates. The vi-bration source can generate a TW or 0.25 < ≤ 0.35.When 0.35 < ≤ 0.55, the circular vibration source

generates an inwardly propagating wave while the cir-cular vibration source expands outward until the cir-cular vibration source reaches the boundary, leadingto the generation o ATW (see Fig. 3). When > 0.55,the global synchronization between the two SWs canoccur. It is obvious that the physical mechanismo generating ATW is diferent or diferent couplingstrength intervals. That is why the ATW region isseparated by the TW region.

1 4 0 1 6 0 1 8 0            8 0 1 2 0 1 6 0 2 0 0            

    

    

Fig. 1. Tip trajectory o the SW in the response systemwith parameters 1 = 0.025 and 2 = 0.05 or (a) = 0.05,(b) = 0.1.

(a) (b) (c) (d) (e) (f) (g) (h)

Fig. 2. The spatial contour patterns o variable o the response system at diferent time moments. The parameters1 = 0.025, 2 = 0.05 and = 0.20 are applied: (a) = 200, (b) = 400, (c) = 480, (d) = 600, (e) = 800, () = 1600, (g) = 1602, (h) = 1604. The propagation direction o the ATW is indicated by the arrow, which locatesat the same wave train.

(a) (b) (c) (d) (e) (f) (g)

Fig. 3. The spatial contour patterns o variable in the response system at diferent time moments. The parameters1 = 0.025, 2 = 0.05 and = 0.55 are taken: (a) = 100, (b) = 200, (c) = 360, (d) = 362, (e) = 500, () = 502,(g) = 504. The propagation direction o the ATW is indicated by the arrow, which locates at the same wave train.

In order to investigate the inuence o the difer-ent values o  2 on SW dynamics in the response sys-tem, we change 2 within [0.025,0.06] while keeping1 = 0.025. The phase diagram regarding diferentwave patterns is shown in Fig. 4. It is observed that

the SW orcing can lead to chaotic meander and lin-ear drit o SWs. Typical tip trajectories o meander-

ing and driting SWs are shown in Fig. 5. The drito SW may be explained as ollows: The orcing im-posed on the tip o the orced SW can be decomposedinto normal and tangential components. The normalorce component causes the spiral tip to move toward

the propagation direction o the drive wave. The tan-gential orce component results in the movement o 

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CHIN. PHYS. LETT. Vol.28, No.2 (2011) 020504

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