Numerical Modeling of Coexistence, Competition and Collapse
of Rotating Spiral Waves in ThreeLevel Excitable Media
with
Discrete Active Centers and Absorbing Boundaries
Abstract
Spatiotemporal dynamics of excitable media with discrete threelevel active centers (ACs) and absorbing boundaries is studied numerically by means of a deterministic threelevel model (see S. D. Makovetskiy and D. N. Makovetskii, http://arxiv.org/abs/condmat/0410460 ), which is a generalization of the ZykovMikhailov model (see Sov. Phys. – Doklady, 1986, Vol.31, No.1, P.51) for the case of twochannel diffusion of excitations. In particular, we revealed some qualitatively new features of coexistence, competition and collapse of rotating spiral waves (RSWs) in threelevel excitable media under conditions of strong influence of the second channel of diffusion. Part of these features are caused by unusual mechanism of RSWs evolution when RSW’s cores get into the surface layer of an active medium (i. e. the layer of ACs resided at the absorbing boundary). Instead of well known scenario of RSW collapse, which takes place after collision of RSW’s core with absorbing boundary, we observed complicated transformations of the core leading to nonlinear “reflection” of the RSW from the boundary or even to birth of several new RSWs in the surface layer. To our knowledge, such nonlinear “reflections” of RSWs and resulting die hard vorticity in excitable media with absorbing boundaries were unknown earlier.
ACM classes: F.1.1 Models of Computation (Automata), I.6 Simulation and Modeling (I.6.3 Applications), J.2 Physical Sciences and Engineering (Chemistry, Physics)
pacs:
05.65.+b, 07.05.Tp, 82.20.WtI Introduction
Rotating spiral waves (RSWs) are typical robust selforganized structures (autowaves) in various dissipative systems, ranging from chemistry, biology and ecology Field1985 ; Kapral1995 ; Medvinskii2002 to nonlinear optics and lasers Weiss1999 .
Evolution of these spatiotemporal structures is now a subject of extensive theoretical studies, experimental investigations and computer modeling (see, e.g., Lo2003 ; Oya2005 ; Provata2003 ; Szabo2002 ; Vanag2004 and references therein).
Many mathematical models in this field are based on concept of excitability Murray2002 . An important approach in modern computer investigations of excitable systems is based on using of discrete parallel models with local interactions, mostly by cellular automata (CA).
In the framework of the CA approach Bandman2006 ; Wolfram2002 , the threelevel representation of the excitable media states is an effective tool for carrying computer experiments with multiparticle systems possessing longtime nonstationary dynamics (see SDMDNMcondmat0410460 and references therein). Parallelization of updating of the active medium states and locality of interactions between elementary parts of the medium are the fundamental principles of the CA approach. Using of this approach admits to fulfill modeling of large systems by emulation of parallel matrix transformations.
The essence of threelevel representation of excitable medium Wiener1946 ; Zykov1986 ; SDMDNMcondmat0410460 is as follows. Each elementary part or active center (AC) of an excitable system has the single stable ground state, say , and at least two metastable (upper) states, say and .
The higher metastable state is the excited one, it may be reached only after certain hard perturbation (from neighboring ACs or from external source) of the ground state of the considered AC. A group of excited ACs may excite another ACs etc. Due to metastabilty, the lifetime of the excited state is a finite value, after which every excited AC spontaneously reaches the intermediate refractory state .
Refractority means “sleeping” state at which no excitation of the considered AC is possible both from neighboring ACs or from an external source. Moreover, the AC in refractory state can’t participate in excitation of another ACs (lying at ). So, after a refractority lifetime , the considered AC reaches the ground state . Usually only the cyclic transitions () are permitted.
The concept of excitability was developed in biology, but it is widely used now in chemistry and physics. In particular, this concept have been applied to solve many problems arising in modern nonlinear optics, laser physics etc. Weiss1993 ; Staliunas1995 ; Weiss2004 ; HagbergDiss1994 ; GiudiciTredicce1997 ; Piwonskietc ; Gomila2004
In SDMDNMcondmat0410460 we have considered even more close laserlike analog of excitable system, namely the threelevel active medium of the microwave phonon laser (phaser) with dipoledipole interactions between AC. The only significant difference of phaser medium from the usual excitable one is the second channel of diffusion of excitations. This system was experimentally studied in Institute of RadioPhysics and Electronics (Kharkov, Ukraine) DNMDiss1983 ; TPL2001 ; arxselforg2003 ; TP2004 ; arx04adestab2004 ; UFZh2006 , demonstrating selforganization, bottlenecked cooperative transient processes and other nonlinear phenomena under extremely low level of intrinsic quantum noise (phaser has 15 orders lower intensity of spontaneous emission in comparison to usual opticalrange lasers).
In preceding publications SDMDNMcondmat0410460 ; SDMForum2005 ; SDMBScThesis2005 ; SDMForum2006 we carried out a series of computer experiments on RSWs dynamics in autonomous excitable systems with various control parameters and random initial excitations. The most interesting phenomena of selforganized vorticity observed in computer experiments SDMDNMcondmat0410460 ; SDMForum2005 ; SDMBScThesis2005 ; SDMForum2006 were as follows: (a) spatiotemporal transient chaos in form of highly bottlenecked collective evolution of excitations by RSWs with variable topological charges; (b) competition of lefthanded and righthanded RSWs with unexpected features, including selfinduced alteration of integral effective topological charge; (c) transient chimera states, i. e. coexistence of regular and chaotic domains in excitable media; (d) branching of an excitable medium states with different symmetry which may lead to full restoring of symmetry of imperfect starting pattern. Phenomena (a) and (c) are directly related to microwave phonon laser dynamics features observed earlier in real experiments at liquid helium temperatures on corundum crystals doped by irongroup ions DNMDiss1983 ; TPL2001 ; arxselforg2003 ; TP2004 ; arx04adestab2004 ; UFZh2006 .
In the present paper, we report some qualitatively new features of coexistence, competition and collapse of RSWs in threelevel model SDMDNMcondmat0410460 ; SDMForum2005 ; SDMBScThesis2005 ; SDMForum2006 of excitable media under conditions of strong influence of the second channel of diffusion of excitations. Part of these features are caused by unusual mechanism of RSWs evolution when RSW’s cores get into the surface layer of an active medium (i. e. the layer of ACs resided at the absorbing boundary). Instead of well known scenario of RSW collapse, which takes place after collision of RSW’s core with absorbing boundary, we observed complicated transformations of the core leading to “reflection” of the RSW from the boundary or even to birth of several new RSWs in the surface layer. To our knowledge, such nonlinear “reflections” of RSWs and resulting die hard vorticity in excitable media with absorbing boundaries were unknown earlier.
Ii Some Remarks on ZykovMikhailov Model and Its TwoChannel Modification
The ZykovMikhailov (ZM) model Zykov1986 is a discrete parallel mapping with local interactions, which may be defined as threelevel CA Loskutov1990 . Let an active discretized medium has the form of rectangular lattice. Each cell of the lattice contains the single threelevel AC with coordinates , where , . All the ACs in the are identical, and they interact by the same set of rules (ZM model, as well as most of the CA models, is homogeneous and isotropic in the von Neumann sense Neumann1966 ). Each AC has the single stable ground state , and two metastable states and .
The upgrade of states of all ACs (i. e. their intralevel or interlevel transitions) are carrying out synchronously at each step during the system evolution. All intralevel (, , ) and some of interlevel (namely , , ) transitions are permitted. The final step may be either predefined or it will be searched during CA evolution as time (i.e. quantity of discrete steps) for reaching an attractor of this CA. The conditions of upgrade Zykov1986 ; Loskutov1990 depend both on and on , where belongs to certain active neigborhood of AC at site . In the framework of the ZM model (and in most of other CA models of excitable chemical systems), diffusion of excitations is possible only by transition. This is a singlechannel (1C) mechanism of diffusion.
A modification of ZM model was proposed by us SDMDNMcondmat0410460 ; SDMForum2005 ; SDMBScThesis2005 ; SDMForum2006 to adapt it for emulation of some aspects of dynamics of classB optical lasers Weiss1993 ; Staliunas1995 ; Weiss2004 and microwave phonon lasers DNMDiss1983 ; TPL2001 ; arxselforg2003 ; TP2004 ; arx04adestab2004 ; UFZh2006 . The relaxational properties of ACs in microwave phonon lasers are of the same type, as in classB optical lasers Tredicce1985 , and there are experimental confirmations TPL2001 ; arxselforg2003 ; TP2004 ; arx04adestab2004 of common properties of microwave phonon lasers and classB optical lasers.
In order to formulate boundary conditions for a of finite size (i. e. when ), a set of virtual cells with coordinates may be introduced SDMDNMcondmat0410460 ; SDMForum2005 ; SDMBScThesis2005 ; SDMForum2006 . Each virtual cell contains the unexcitable center having the single level . The surface layer of a bounded active system is represented by the set of ACs with coordinates . Note that our surface layer is of 1D type because active medium itself is of 2D type (we stress this circumstance to avoid misunderstandings).
The original ZM model has, as it was pointed out earlier, the single channel of diffusion Zykov1986 ; Loskutov1990 . Such 1Cdiffusion models are adequate for chemical reactiondiffusion systems Tyson1985 . In microwave phonon laser active system, the multichannel diffusion of spin excitations is the typical case, because it proceeds via (near)resonant dipoledipole () magnetic interactions between paramagnetic ions DNMDiss1983 . For a threelevel system, which is the simplest microwave phonon laser system, there are 3 possible channels of resonant diffusion. In the case when interactions are forbidden at one of three resonant frequencies of a threelevel system, the asymmetric twochannel (2C) diffusion is realized under conditions of perfect refractority of the intermediate level — see SDMDNMcondmat0410460 ; SDMForum2005 ; SDMBScThesis2005 ; SDMForum2006 . Note that asymmetric diffusion is well known in biology (see, e.g., the book of D. A. FrankKamenetzky Frank1984 ). Recently a kind of asymmetric diffusion was proposed by N. Packard and R. Shaw Packard2004 for a mechanical system.
Iii ThreeLevel Model of Excitable System With TwoChannel Diffusion (A Modified ZykovMikhailov Model)
iii.1 States of active centers and branches of evolution operator
Let is a rectangular 2D lattice containing threelevel ACs. The upgrade of state of each AC is carried out synchronously during the system evolution. The excited (upper) level has the time of relaxation , and the refractory (intermediate) level has the time of relaxation . Both and are integer numbers. This model of excitable system is a kind of threelevel CA. The original ZM model Zykov1986 ; Loskutov1990 was formulated as threelevel CA with 1C diffusion.
In our threelevel model of excitable system (TLM), the second channel of diffusion of excitations is included. A detailed argumentation for using of the 2C diffusion mechanism one can find in SDMDNMcondmat0410460 .
In the TLM, the first channel of diffusion accelerates the transitions for a given AC, and the second channel of diffusion (which is absent in the ZM model) accelerates the transitions . The complete description of AC’s states in such the TLM model includes one type of global attributes (the phase counters ) and two types of partial attributes and for each individual AC in . Full description of all AC’s possible states is as follows:
(1) 
(2) 
(3) 
In the framework of the TLM model, the phase counters lie in the interval . The following correspondences between and take place (by definition) for all the ACs in at all steps of evolution:
(4)  
(5)  
(6) 
These correspondences are of key significance for all multisteprelaxation models (Reshodko model Bogach1973 ; Bogach1979 ; Reshodkoetc , ZM model Zykov1986 ; Loskutov1990 etc.): there are only three discrete levels, and relaxation of each AC is considered as intralevel transitions (or transitions between virtual sublevels).
The evolution of each individual AC proceeds by sequential cyclic transitions (where and ), induced by the Kolmogorov evolution operator Kolmogorovetc . In the TLM model, the evolution operator has three orthogonal branches , and , which we call ground, excited and refractory branches respectively. The choice of the branch at iteration is dictated only by the global attribute of the AC at step , namely:
(7) 
iii.2 Ground branch of the evolution operator
At step , the branch by definition fulfills operations only over those ACs, which have at step . These operations are precisely the same, as in ZM model Zykov1986 ; Loskutov1990 , namely:
(8) 
(9) 
where is the accumulating term for the agent, which is an analog of chemical activator Zykov1986 ; Loskutov1990 ; is the firstchannel diffusion term; is the threshold for the agent (); is the accumulation factor for the agent (; is the active neighborhood of the AC at site ; and the agent arrives to ACs with only from ACs with in :
(10) 
The definition of the diffusion term in Eqn. (9) is very flexible. Apart of wellknown neighborhoods of the Moore () and the von Neumann () types:
(11) 
(12) 
one can easy define another neighborhoods, e. g. of the box type or of the diamond type Fisch1991 (which are the straightforward generalization of the Moore and the von Neumann neighborhoods), etc.
On the other hand, the ZM definition Zykov1986 ; Loskutov1990 of the weight factors (10) may be extended out of the binary set to model various distancedependent phenomena within, e. g., .
In this work, however, we restricted ourselves by the Moore neighborhood and by weight factors of the form (10). Another types of neighborhoods, weight factors and some other modifications of the model will be studied in subsequent papers.
iii.3 Excited branch of the evolution operator
At the same step , the branch fulfills operations only over those ACs, which have at step :
(13) 
(14) 
where is the secondchannel diffusion term; is the threshold for the agent (), and we assume that agent arrives to excited AC () only from those ACs, which have in :
(15) 
One can see from (14) that the agent does not accumulate during successive iterations. In other words, the branch at step produces ”memoryless” values of partial attributes for ACs having at step (in contrary to the agent for ACs having at step ). The agent may accelerate transitions from excited ACs to refractory ones. This is the most important difference between our 2C model of threelevel excitable medium and the original 1C model of ZM Zykov1986 ; Loskutov1990 .
iii.4 Refractory branch of the evolution operator
The branch does not produce/change any partial attributes at all (because the intermediate level is in the state of refractority). It fulfils the operations only over those ACs, which have at step :
(16) 
Generally speaking, there are many examples of active media with weak refractority (when the unit at the intermediate level is not absolutely isolated from its neighborhooding units). But in this work we restrict ourselves by the case of perfect refractority (16), which is valid, e. g., for the microwave phonon laser systems of the type SDMDNMcondmat0410460 .
iii.5 Reduction of the TLM model to the ZM automaton
The secondchannel diffusion gives the contribution to the TLM dynamics if , i.e. if for , for and so on.
At our TLM model is of ZMlike (i.e. 1C) type, and at it becomes equivalent to the original ZM model Zykov1986 ; Loskutov1990 , which (with slight rearrangement of cases) is as follows:
(17) 
iii.6 Geometry of active media, boundary conditions and transients in TLM
Pattern evolution is very sensitive to geometry of active medium and boundary conditions even in the framework of operation of the same CA. There are three types of geometry and boundary condition (GBC) combinations most commonly used in computer experiments with CA (see Table I). Let us consider them in details.
The GBC1 type is defined as follows:
(18) 
where
(19) 
(20) 
Here , and the coordinates belong to the border set of excitable area, i. e. ) if .
The GBC2 type can be easily defined by using an extended interval for the phase counter . Let every cell in contains one unexcitable unit. All these unexcitable units have frozen at all without reference to states of , and
(21) 
where, or . These unexcitable units in “absorb” excitations at the border of in contrary to the case of GBC1, where excitations at the border are reinjected in active medium. In many cases this difference leads to qualitatively different behaviour of the whole TLM. Note that by this way one may also define TLM with inhomogeneous active medium, where some of ACs are changed to unexcitable units, i. e. unexcitable units are placed not only in , but in too. This may be easily done by introducing an additional orthogonal branch into the evolution operator . The pointed additional branch is activated at only and is simply the identity operator , i.e.: . And even more complex behaviour of such “impurity” units may be defined using analogous approach (i. e. by introducing additional orthogonal branches in an evolution operator): there may be pacemakers Loskutov1990 or another special units embedded in active medium and described by phase counter in extended areas or .
The GBC3 type is, strictly speaking, a case of borderless system. But the starting pattern has, of course, bounded quantity of ACs with , located in bounded part of active medium. Hence, during the whole evolution () the front of growing excited area will meet the unexcited (but excitable!) ACs only. For the case GBC3, there is neither absorption of excitations at boundaries (as for GBC2), nor feedback by reinjection of excitations in the active system (as for GBC1). From this point of view CA of GBC3 type are “simpler”, than CA of GBC1 and GBC2 types. On the other hand, CA of GBC3 type are potentially infinite discrete systems where true aperiodic (irregular, chaotic) motions are possible, in contrast to finite discrete systems, possessing, of course, only periodic trajectories in certain phase space after ending of transient stage Lorenzetc .
On the other hand, there is also very important difference between GBC2 and other two types of GBC. The system of GBC2 type is the single from the three ones under consideration which interacts with the external world dynamically (besides of relaxation). Sure enough, GBC1 and GBC3 are connected to this outer world only by relaxation channels (the and are the measures of this connection). In contrary, a system of the GBC2 type interacts with surroundings through the real boundary, which is in fact absent in toroidal finitesize active medium of GBC1 type and it is absent by definition for a system of GBC3 type. Despite of elementary mode of such interaction (boundary simply ”absorbs” the outsidedirected flow of excitations from , a system of the GBC2 type may demonstrate very special behaviour. The main attention in this work is devoted just to TLM of GBC2 type as the most realistic model of microwave phonon laser active system, where both the mechanisms of interaction (dynamical and relaxational) of an active medium with the outer world are essential. Dissipation in a microwave phonon laser active medium (highly perfect single crystal at liquid helium temperatures) is caused by two main mechanisms: (a) dynamical, by coherent microwave phonon and photon emission directly through crystal boundaries and (b) relaxational, by thermal phonon emission. There are, of course, many more or less important differences between the TLM model and the real microwave phonon laser systems TPL2001 ; arxselforg2003 ; TP2004 ; arx04adestab2004 , but in any case autonomous microwave phonon laser has only these two mechanisms of interaction with the outer world.
Type  Geometry  Boundary conditions 

GBC1  Bounded, Toroidal  Cyclic 
GBC2  Bounded, Flat  Zeroflow 
GBC3  Unbounded, Flat  (free space) 
iii.7 Initial conditions
Patterns in ZM are described in terms of levels (or “colors”) of ACs, and initial conditions are formulated simply as matrix of ACs levels . But states of ACs in such the automata as ZM or TLM are fully defined not only by levels itself. There are global attributes which must be predefined before TLM evolution is started. Some of partial attributes must be predefined too. These points are essential for reproducing of the results of computer experiments with ZM, TLM, etc.
In our work such the initial conditions for the global attributes are used:
(22) 
Initial conditions for must be defined for groundstate AC’s () only. In this work, we suppose
(23) 
where means “if and only if”. Initial conditions for are undefined for all because agent is not defined at , see Eqn. (2). As the result, initial conditions will be defined in our TLM model as the matrix (where ) with additional condition given by Eq. (23).
Iv Results and Discussion
A bounded solitary domain of excited ACs having appropriate relaxation times and placed far from grid boundaries (or at unbounded grid) may evolve to RSWs if and only if there is an adjoined (but not a surrounding) domain of refractory ACs Balakhovskii1965 ; Loskutov1990 ; Fisch1991 ; Selfridge1948 . In this case RSWs apear by pair with opposite and integral topological charge is obviously conserved (by infinity in time if grid is unbounded). If such solitary excitedrefractory area is placed in a starting pattern near the grid boundary and the GBC2 conditions take place, the single RSW appears and evolves and is, of course, not conserved in this case. These simplest and well known examples illustrate possibilities of coexistence and competition of one or two RSWs in excitable media, and possible scenarios of the RSW(s) evolution may be easily forecasted.
An evolution of complex patterns with multiple, irregularly appearing, chaoticallylike drifting and colliding RSWs (or, possibly, another spatiotemporal structures) is in essence unpredictable without direct computing of the whole transient stage. So, the best way to investigate cellular automaton (TLM in particular) is to run it, because, as S. Wolfram pointed out, ”their own evolution is effectively the most efficient procedure for determining their future” (see Wolfram1985 , page 737).
Here we present results of our computer experiments with 2C model of excitable media described in Section III. The main tool used in these experiments was the crossplatform software package “ThreeLevel Model of Excitable System” (TLM) c⃝ 2006 S. D. Makovetskiy SDMForum2006 . It is based on extended and improved algotithms of discrete modeling of threelevel manyparticle excitable systems, proposed by us in SDMForum2005 ; SDMBScThesis2005 ; SDMForum2006 . The TLM package is written in the Java 2 language TheJavaOfficialSite using the Java 2 SDK (Standard Edition, version 1.4.2) and the Swing Library.
A discrete system with finite set of levels may have only two types of dynamically stable states at bounded lattice. The first of them is stationary state, and the second is periodic one. They may be called attractors by analogy with lumped dynamical systems (see e. g. Liu2002 ; Wuensche1998 ; Wuensche2003 ). For our cellular automaton, the first of such the attractors is spatiallyuniform and timeindependent state with , where , . In other words, the only stationary state of TLM is the state of full collapse of excitations at some step (by definition of excitable system). The second type of TLM attractors includes many various periodically repeated states (RSW is a typical but not the single case). In this case where , ; is the first step of motion at a periodic attractor; is the integernumber period of this motion, .
If starting patterns are generated with random spatial distribution by levels, then the time intervals or may be considered as times of full ordering in the system. Such irregular transients are of special interest from the point of view of nonlinear dynamics of distributed systems Awazu2003 ; Crutchfield1988 ; Hramov2004 ; Morita2003 ; Morita2004 because they may be bottlenecked by very slow, intermittent morphogenesis of spatialtemporal structures. In the next Subsections we study this collective relaxation for cases of collapse and periodic final states of TLM evolution, including an important case of lethargic transients.
iv.1 Collapse of RSWs
In this Subsection, we describe a typical scenario of the twodimensional, threelevel and twochanneldiffusion (2D3L2C) excitable system evolution leading to the full collapse of excitations (when the system reaches the single pointlike spatiotemporal attractor — stable steady state with for all ACs). Some typical stages of such the scenario of evolution are shown at Figure 1.
The system evolves at first stages () to a labyrinthic structure which contains several RSWnucleating domains (Figure 1, ).
During the next stage of the evolution, the largescale vortex is formed (Figure 1, ). This vortex pushes the labirinthic structure to the boundaries of the active medium. As the result, the vortex occupies the whole active medium (Figure 1, ).
But this giant rotating structure is unstable. Its core splits into RSWs with , and at this stage the active medium is filling by such the RSWs (Figure 1, ).
The next, relatively long stage of evolution is characterized by restless moving and strong competition of RSWs. As the matter of fact, this stage is of the “winner takes all” (WTA) type. So, at the end of this stage, the single, fully regular RSW with occupies the whole active medium (Figure 1, ).
But the core of the winner is not far from the active medium boundary. Drift of this RSW leads to collision of its core with boundary, and the core is absorbed by the latter (Figure 1, ). The residual nonspiral autowaves gradually run out the active medium (Figure 1, ) and excitation collapses fully at (not shown at Figure 1).
The described scenario of excitations collapse is very typical but not the single one. Some other ways leading to collapse will be described in a forthcoming publications. Now we will consider cases of noncollapsing systems.
iv.2 Dynamic Stabilization, Synchronization and Coexistence of RSWs
In this Subsection, we describe a typical scenario of the 2D3L2C excitable system evolution leading to dynamic stabilization and coexistence of RSWs. In this case the system reaches one of its periodic spatiotemporal attractors — fully synchronized cyclic transitions () for all ACs. Some typical stages of such the scenario of evolution are shown at Figure 2.
The system evolves at first stages () to a mixed RSWlabyrinthic structure (Figure 2, ), where nucleation of RSWs has more pronounced form, than at Figure 1 (for the same .
All the subsequent stages of evolution at Figure 2 differ more and more comparatively to Figure 1. Instead of formation of a giant vortex, which pushes the labyrinthic structure to the boundaries of the active medium (Figure 1), at Figure 2 we see multiple emergent RSWs (). These RSWs have different , but RSWs with are prevailed here (Figure 2, ).
As the result of such direction of evolution, at the next stage only negativecharged, wellformed RSWs with occupy the whole active medium (Figure 2, ) — instead of the single giant multicharged vortex with (Figure 1, ).
Due to complex interactions between these RSWs as well as between RSWs and boundaries, the processes of revival of positivecharged RSWs take place (Figure 2, ). In contrast to Figure 1, where strong competition between RSWs leads to WTA dynamics, competition is weakened in the case under consideration (Figure 2, ).
Weakening of competition and low mobility of RSWs determine qualitatively another final (Figure 2, and of evolution scenario (comparatively to Figure 1). Namely, the dynamic stabilization of RSWs over the active medium is observed at . Configuration of the system is precisely repeated (compare and at Figure 2). In other words, this is spatiotemporal limit cycle.
Fully synchronized cyclic transitions for all ACs () for this system (Figure 2) is in contrast to collapse of excitations for the previous system (Figure 1) despite of obvious transient ordering of the spatiotemporal dynamics in the course of events for both the cases (Figure 1 and Figure 2). Note that in both these cases we deal with spontaneous decreasing of effective freedom degrees quantity during the evolution of the system Haken1983 .
Transient ordering of this kind, leading to collapse of excitations (Figure 1), may be interpreted as selforganized degradation (selfdestruction) of the system. Final state of such the system, of course, is not selforganized, but it is still highly ordered. This kind of static order is known as freezing, because all ACs of the system occupies their ground levels (like usual particles of a physical system at absolute zero temperature).
In contrast, transient ordering, leading to dynamical stabilization of more or less complex spatiotemporal structures (Figure 2), may be interpreted as “normal” selforganization in autonomous dissipative system Haken1983 when most degrees of freedom are “slaved” by the small rest of ones. At this selforganized state, the system reaches the end of its evolution. But the system remains unfrozen, it is still far from equilibrium and possesses obvious order in the form of spatiotemporal limit cycle (Figure 2). All parts of this system (namely RSWs) are well defined and robust (as individual dynamical spatiotemporal autostructures). At the same time, all these parts and entirely synchronized at global level (as dynamical components of the whole system demonstrating collective rhythmic motions).
Nevertheless, both these scenarios of dissipative system evolution (selfdestruction at Figure 1 and normal selforganization at Figure 2) exhibit more or less fluent transition from initial spatial chaos to static (Figure 1) or dynamically stable (Figure 2) order.
A different scenario is exemplified in the next Subsection.
iv.3 Transient Chaos Caused by Competition of RSWs and Labyrinths
In this Subsection, we will describe another way to reach periodic state in a bounded discrete 2D3L2C excitable system. Formally, the final state for this new case is spatiotemporal limit cycle too (as in the case of Figure 2). But the time of transient process is much longer here, and the longest part of the transient proceeds in the form of lethargic competition between various spatiotemporal structures.
As it was already shown in the preceding Subsection (Figure 2), spatiotemporal structures may actively interact not only between themselves, but they interact in a nontrivial manner with boundaries of the excitable medium too. The simplest case when drift of RSWs leads to collisions of their cores with boundary, and the cores are absorbed by the latter (Figure 1, ), is not typical case for the system studied in this Subsection. On the contrary, RSWs in such the system under appropriate conditions may be ”reflected” from boundaries in a strongly nonlinear manner^{1}^{1}1Reflection of RSW from boundary, as well as reflection of any other moving dissipative structure (autostructure, autowave) is usually considered as forbidden. This is correct, commonly speaking, only in lowest (weakly nonlinear) approximations, when the properties of surface layer are assumed as close to properties of bulk active medium. But real processes in surface layer may lead to strong nonlinear phenomena of, e. g., revival (regeneration) of RSWs or even to their duplication, triplication etc. Complex phenomena of such the kind may be considered as higherorder nonlinear interactions of RSWs with boundary, even if boundary conditions itself are “simple” (e. g. zeroflow conditions, as in the present work). . Moreover, due to nonlinear processes in the surface layer, the quantity of ”reflected” RSWs may exceed the quantity of primary RSWs (phenomenon of multiplication of RSWs). A detailed description of nonlinear ”reflections” of RSWs and some accompanied phenomena will be published in separate paper.
Here we describe a particular, but important case of highly bottlenecked transient process, caused mainly by almost everlasting competition of RSWs and labyrinths. At Figure 3, a fragment of evolution is shown at for the 2D3L2C system, which differs from the previous system (Figure 2) only by two parameters: ; for Figure 3 (instead of ; for Figure 2). Starting pattern for the system at Figure 3 is precisely the same as at Figure 2 (so it is not shown at Figure 3).
Previous two systems (Figures 1 and 2) reach their attractors already at . The system under consideration (Figure 3) has very long transient time, because of lethargic competition between RSWs and labyrinthic structures under conditions of regeneration (nonlinear “reflections”) and multiplication of RSWs at boundaries. There are many fine phenomena accompanied these competition. E. g., the largest RSW with resided in bottom left corner of the active medium (Figure 3, ) moves in the North direction to the big labyrinth. Finally, the latter absorbs the core of the pointed RSW (Figure 3, ). At the same time new small RSWs are generated in the bulk active medium and regenerated (by nonlinear “reflections”) in the surface layer.
The dominant drift directions of RSWs in the system under consideration lie at NorthSouth and EastWest lines (this circumstance is evident when one see a movie of the system evolution). Drift of labyrinths is more complicated. In the fragment shown at Figure 3, the big labyrinth is slowly moving approximately in NorthEast direction. A new labyrinth is formed simultaneously. This new labyrinth pushes RSWs (Figure 3, ) etc.
The fragment of evolution shown at Figure 3 demonstrate only small part of transient phenomena observed during this system evolution. We will mention now such the typical stages of the system evolution (not shown^{2}^{2}2Additional figures to this Preprint may be requested from the author by email (see the first page of the Preprint) at Figure 3):
Nonlinear “reflection” of large RSW with from boundary accompanied by changing of and simultaneuos birth of several new small RSWs (satellites) nearby the core of large RSW ().
Rotation of giant labyrinth around the domain occupied by RSWs (); erosion of this labyrinth due to RSWs activity up to almost full dominance of RSWs over active medium (); revival of labyrinth structures and resumption of competition between RSWs and labyrinths ().
Nonlinear “reflection” of large RSW with from boundary with saving of (, upper part of active medium). Shining examples of multiplication of RSWs at boundaries (the same interval , left, right and bottom boundaries).
Birth of selforganized pacemaker (a source of concentric autowaves) due to nonlinear processes of RSW’s core transformations in the surface layer; evolution and decay of the pacemaker (, left boundary of the active medium).
Pushing of labyrinth by a group of RSWs () etc.
From the formal point of view, all these stages of the system evolution are only partial transient episodes on the long way to the final selforganized state. We know that such the state is a regular attractor (spatiotemporal limit cycle or at least pointlike attractor) because by definition our system is autonomous, it has bounded quantity of discrete elements, each element has bounded quantity of discrete states, and time is discrete too. But we do not get concrete view of the attractor until we reach it. Shortly, “transient is nothing, attractor is all”.
From an alternative point of view, each stage of the system evolution is a day in the life of this system, and the final of evolution is of minor interest, because no new events will come to pass if an attractor is already reached. Shortly, from this point of view, “transient is all, attractor is nothing”.
But this alternative is not the single one. Another alternative is as follows: “transient is all that we can see, because attractor is unachievable”.
This last alternative (for a nonCA system with continuous spectrum of states of its elementary components) was discussed in a seminal work of J. P. Crutchfield and K. Kaneko Crutchfield1988 , which was entitled: “Are Attractors Relevant to Turbulence?”. In SDMDNMcondmat0410460 , this question was slightly reformulated in the context of our studies of CA systems: “Are Attractors Relevant to Transient SpatioTemporal Chaos?” (turbulence in bounded, fully discrete system is no more than metaphor). The answer is “Yes” if an attractor may be reached for a reasonable time (in fully discrete and bounded system is always limited by quantity of all possible states of the system). But the answer is “No” if exceeds any possible duration of an experiment. In this case a system with transient spatiotemporal chaos cannot be distinguished from true chaotic system without additional testing.
As a matter of fact, there are some intermediate classes of phenomena “at the edge between order and chaos” which may appear in bounded discrete deterministic system with large phase space. Selforganization scenario which includes superslow, bottlenecked, chaoticlike stages is a signature of dominance of such an intermediate class of system dynamics in numerical experiments. Really, having limited time and computer capacity, one cannot reach final selforganized state for a system with huge dimension of phase space. Such CA (or another discrete mapping with lethargic evolution) does not permit direct forecasting of the system future without direct computation. So the computed part of transient process is the single source of available information of our fully deterministic but partially determined system.
Generally, the ideas of interconnection between chaotization, turbulence, unpredictability (at one side) and ordering, selforganization, longtime forecasting (at the opposite side) may be very fruitful at least for investigation of complex deterministic systems.
V Conclusions
In this work, we fulfill computer modeling of spatiotemporal dynamics in large dicrete systems of threelevel excitable ACs interacting by shortrange 2C diffusion. Computer experiments with this 2D3L2C system were carried out using the crossplatform software package “ThreeLevel Model of Excitable System” (TLM) c⃝ 2006 S. D. Makovetskiy SDMForum2006 .
The most typical scenario of evolution is as follows. A robust RSW or more complicated but strictly periodic in time structure is formed being a cyclic attractor at several initial conditions — see Figure 2. This is an analog of limit cycle, i. e. regular attractor known in lumped dynamical systems. Alternatively, collapse of excitations, i. e. full freezing of the excitable system takes place — see Figure 1. This is an analog of pointlike attractor in lumped dynamical systems.
Longtime evolution ( iterations) of a 2D3L2C system with slightly changed parameters demonstrates some unusual phenomena including highly bottlenecked collective relaxation of excitations. Part of these phenomena are caused by mechanism of nonlinear regeneration of RSWs in the surface layer of an active medium. Instead of well known scenario of RSW collapse, which takes place after collision of RSW’s core with absorbing boundary, we observed complicated transformations of the RSWs cores leading to nonlinear “reflections” of the RSWs from the normally absorbing boundaries or even to birth of new RSWs in the surface layer. Lethargic transient processes observed in our computer experiments are partially caused by competitions between permanently reviving RSWs and labyrinthic spatiotemporal structures. To our knowledge, phenomena of such the nonlinear “reflections” of RSWs and resulting die hard vorticity in excitable media with absorbing boundaries were unknown earlier.
The author is grateful to D. N. Makovetskii (Institute of RadioPhysics and Electronics, National Academy of Sciences of Ukraine) for stimulating discussions on various aspects of this work.
Appendix A List of Abbreviations
1C, 2C, … — OneChannel, TwoChannel, …
1D, 2D, … — OneDimensional, TwoDimensional, …
2L, 3L, … — TwoLevel, ThreeLevel, …
AC — Active Center
CA — Cellular Automaton
CP — Control Parameters
GBC — Geometry and Boundary Conditions
RSW — Rotating Spiral Wave
TLM — ThreeLevel Model (of excitable system)
ZM — ZykovMikhailov
References
 (1) A. Awazu and K. Kaneko, Phys. Rev. Lett. 92, 258302 (2004); Preprint eprint nlin/0310018 (2003).
 (2) I. S. Balakhovskii, Biofizika (USSR) 10(6), 1063 (1965), in Russian.
 (3) O. L. Bandman, CellularAutomata Models of Spatial Dynamics, in: System Informatics (Siberian Branch of RAS, Novosibirsk, 2006), Issue 10, p.57, in Russian; eprint http://ssdonline.sscc.ru/ol/canew.pdf
 (4) P. G. Bogach and L. V. Reshodko, Dopovidi Akad. Nauk Ukr. RSR (Reports of the Ukrainian Academy of Sciences), Series B, No.5, 442 (1973), in Ukrainian.
 (5) P. G. Bogach and L. V. Reshodko, Algorithmic and Automaton Models of Smooth Muscle Operation (Naukova Dumka, Kiev, 1979), in Russian.
 (6) J. P. Crutchfield and K. Kaneko, Phys. Rev. Lett. 60, 2715 (1988).
 (7) R. J. Field and M. Burger (Eds.), Oscillations and Travelling Waves in Chemical Systems (Wiley, New York etc., 1985).

(8)
R. Fisch, J. Gravner, and D. Griffeath,
Statistics and Computing 1, 23 (1991);
eprint http://psoup.math.wisc.edu/papers/tr.zip  (9) D. A. FrankKamenetzky, Diffusion in Chemical Kinetics, 3rd Edition (Nauka, Moscow, 1987), in Russian (English translation of previous edition is available).
 (10) M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, Phys. Rev. E 55, 6414 (1997).
 (11) D. Gomila, M. A. Matías, and P. Colet, Excitability Mediated by Localized Structures, Preprint eprint nlin/0411047 (2004).

(12)
A. Hagberg,
Fronts and Patterns in ReactionDiffusion Equations,
Ph. D. Thesis (Univ. of Arizona, USA, 1994);
eprint http://math.lanl.gov/hagberg/Papers
/dissertation/dissertation.pdf  (13) H. Haken, Advanced Synergetics. Instability hierarchies of selforganizing systems and devices (SpringerVerlag, Berlin and Heidelberg, 1983).
 (14) A. E. Hramov, A. E. Khramova, I. A. Khromova, and A. A. Koronovskii, Nonlin. Phenom. Complex Syst. 7(1), 1 (2004).
 (15) The Java Official Site, eprint http://java.sun.com/
 (16) R. Kapral, R. Showalter (Eds.), Chemical Waves and Patterns, (Kluwer Academic Press, Dortrecht, 1995).
 (17) A. N. Kolmogorov, Uspekhi Mathem. Nauk (USSR) 8(4), 175 (1953), in Russian; A. N. Kolmogorov and V. A. Uspenskii, Uspekhi Mathem. Nauk (USSR) 13(4), 3 (1958), in Russian; V. A. Uspenskii and A. L. Semenov, Theory of Algorithms (Nauka, Moscow, 1987), in Russian.
 (18) R. F. Liu and C. C. Chen, Preprint eprint nlin/0209005 (2002).
 (19) C.P. Lo, N. S. Nedialkov, and J.M. Yuan, Preprint eprint math/0307394 (2003).
 (20) E. N. Lorenz, Physica D 35, 299 (1989); J. L. McCauley, Phys. Scripta 20, 1 (1990); J. Palmore and C. Herring, Physica D 42, 99 (1990); P. M. B. Vitanyi, Preprint eprint condmat/0303016 (2003); Masato Ida and Nobuyuki Taniguchi, Phys. Rev. E 68, 036705 (2003); Masato Ida and Nobuyuki Taniguchi, Phys. Rev. E 69, 046701 (2004).
 (21) A. Yu. Loskutov and A. S. Mikhailov, Introduction to Synergetics (Nauka, Moscow, 1990), in Russian.
 (22) D. N. Makovetskii, Dissertation (Inst. of RadioPhys. and Electron., Ukrainian Acad. Sci., Kharkov, 1983); Diss. Summary (Inst. of Low Temper. Phys. and Engin., Ukrainian Acad. Sci., Kharkov, 1984), in Russian.

(23)
D. N. Makovetskii,
Tech. Phys. Lett. 27(6), 511 (2001)
[translated from: Pis’ma Zh. Tekh. Fiz. 27(12), 57
(2001)];
eprint http://www.ioffe.rssi.ru/journals/pjtf/2001/12
/p5764.pdf  (24) D. N. Makovetskii, SelfOrganization in Multimode Microwave Phonon Laser (Phaser): Experimental Observation of SpinPhonon Cooperative Motions, Preprint eprint condmat/0303188 (2003).

(25)
D. N. Makovetskii,
Tech. Phys. 49(2), 224 (2004);
see also eprint http://dx.doi.org/10.1134/1.1648960 [translated from: Zh. Tekh. Fiz. 74(2), 83 (2004);
eprint http://www.ioffe.rssi.ru/journals/jtf/2004/02
/p8391.pdf ]  (26) D. N. Makovetskii, Superslow SelfOrganized Motions in a Multimode Microwave Phonon Laser (Phaser) under Resonant Destabilization of Stationary Acoustic Stimulated Emission, Preprint eprint condmat/0402640 (2004).
 (27) D. N. Makovetskii, SlowingDown of Transient Processes of Fine Structure Formation in Power Spectra of Microwave Phonon Laser (Phaser), Ukrainian Journ. Phys. (2006), to be published.
 (28) S. D. Makovetskiy and D. N. Makovetskii, A Computational Study of Rotating Spiral Waves and SpatioTemporal Transient Chaos in a Deterministic ThreeLevel Active System, Preprint eprint condmat/0410460, version 2 (2005).
 (29) S. D. Makovetskiy, Program for Modeling of SpatioTemporal Structures in ThreeLevel Lasers, in: Proc. of the 9th Intl. Forum “Radioelectronics and Youth in the XXI Century” (KNURE, Kharkiv, 2005), p.348, in Russian.
 (30) S. D. Makovetskiy, Software for Modeling of SpatioTemporal Structures in ThreeLevel Lasers, B. Sc. Thesis (Kharkiv National University of Radio Electronics, Ukraine, 2005), in Russian.
 (31) S. D. Makovetskiy, A Method of Numerical Modeling of NonStationary Processes in ThreeLevel Excitable Media and its Software Implementation by the Java Language, in: Proc. of the 10th Intl. Forum “Radioelectronics and Youth in the XXI Century” (KNURE, Kharkiv, 2006), in Russian.
 (32) A. B. Medvinskii, I. A. Tikhonova, D. A. Tikhonov, G. R. Ivanitskii, S. V. Petrovskii, B.L. Li, E. Venturino, and H. Malshow, Physics  Uspekhi 45(1), 27 (2004) [translated from: Uspekhi Fiz. Nauk 172(1), 31 (2002)].
 (33) H. Morita and K. Kaneko, Europhys. Lett. 66, 198 (2004); Preprint eprint condmat/0304649 (2003).

(34)
H. Morita and K. Kaneko,
Preprint
eprint nlin/0407460
(2004).  (35) J. D. Murray, Mathematical Biology, Volumes 1–2, 3rd Edition (Springer, 2002).
 (36) J. von Neumann, Theory of SelfReproducing Automata, edited and completed by A. W. Burks (Illinois Univ. Press, Urbana & London, 1966).
 (37) T. Oya, T. Asai, T. Fukui, and Y, Amemiya, Int. J. Unconvential Computing 1, 177 (2005).
 (38) N. Packard and R. Shaw, Preprint eprint condmat/0412626 (2004).
 (39) T. Piwonski, J. Houlihan, Th. Busch, and G. Huyet, Delay Induced Excitability, Preprint eprint condmat/0411385 (2004); J. Houlihan, D. Goulding, Th. Busch, C. Masoller, and G. Huyet, Phys. Rev. Lett. 92, 050601 (2004); D. Curtin, S. P. Hegarty, D. Goulding, J. Houlihan, Th. Busch, C. Masoller, and G. Huyet, Phys. Rev. E 70, 031103 (2004).
 (40) A. Provata and G. A. Tsekouras, Phys. Rev. E 67, 056602 (2003).
 (41) L. V. Reshodko, Kybernetika (Prague) 10(5), 409 (1974); L. V. Reshodko and S. Bures, Biol. Cybern. 18(3/4), 181 (1975); L. V. Reshodko and Z. Drska, J. Theor. Biol. 69(4), 568 (1977).
 (42) O. Selfridge, Arch. Inst. Cardiologia de Mexico 53, 113 (1948).
 (43) K. Staliunas and C. O. Weiss, J. Opt. Soc. Amer. B 12, 1142 (1995).
 (44) G. Szabó and A. Szolnoki, Phys. Rev. E 65, 036115 (2002).
 (45) J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, J. Opt. Soc. Amer. B 2, 173 (1985).
 (46) J. J. Tyson, in Oscillations and Travelling Waves in Chemical Systems, ed. by R. J. Field and M. Burger (Wiley, New York etc., 1985).
 (47) V. K. Vanag, Physics  Uspekhi 47(12), 1177 (2004) [translated from: Uspekhi Fiz. Nauk (Russia) 174(9), 991 (2004)].
 (48) C. O. Weiss, H. R. Telle, K. Staliunas, and M. Brambilla, Phys. Rev. A. 47, R1616 (1993).
 (49) C. O. Weiss, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151 (1999).
 (50) C. O. Weiss, in: Proc. 6th Intl. Conf. LFNM’2004 (V. N. Karazin National University & National University of Radio Electronics, Kharkov, 2004), pp. 207208.
 (51) N. Wiener and A. Rosenblueth, Arch. Inst. Cardiologia de Mexico 16(34), 205 (1946).
 (52) S. Wolfram, Phys. Rev. Lett. 54, 735 (1985).
 (53) S. Wolfram, A New Kind of Science (Wolfram Media, Champaign, 2002); eprint http://www.wolframscience.com/

(54)
A. Wuensche,
in Complex Systems ’98
(Univ. of New South Wales, Sydney, 1998);
eprint ftp://ftp.cogs.susx.ac.uk/pub/users/andywu
/papers/complex98.ps.gz  (55) A. Wuensche, Discrete Dynamics Lab (DDLab): Software Package, ver. m04 (Discrete Dynamics, Santa Fe, 2003); eprint http://www.ddlab.com
 (56) V. S. Zykov and A. S. Mikhailov, Sov. Phys. – Doklady 31(1), 51 (1986) [translated from: Dokl. Acad. Nauk SSSR 286(2), 341 (1986)].
FIGURE CAPTIONS
to the paper of S. D. Makovetskiy
“Numerical Modeling of Coexistence, Competition and Collapse
of Rotating Spiral Waves in ThreeLevel Excitable Media
with Discrete Active Centers and Absorbing Boundaries”
(figures see as separate PNGfiles)
Figure 1: Birth, evolution and collapse of RSWs with the same (by magnitude and sign) effective topological charges. Dimensions of the active medium are . Black, gray and white pixels denote ACs in excited (), refractory (), and ground () states respectively. Starting pattern is shown at . The relaxation times of ACs are . The set of CPs is as follows: ; ; .
Figure 2: Birth, evolution, dynamic stabilization and coexistence of RSWs. Dimensions of the active medium and colors of pixels are the same as at Figure 1. Starting pattern () is not the same as at Figure 1, but statistical properties of both patterns are almost identical. The relaxation times of ACs and the set of control parameters are as follows: ; ; ; ; .
Figure 3: Typical stage of transient spatiotemporal chaos (competition of RSWs and labyrinthic structures) at Dimensions of the active medium and colors of pixels are the same as at Figures 1 and 2. Starting pattern is precisely the same as at Figure 2 (and not shown here). The relaxation times of ACs and the set of CPs are as follows: ; ; ; ; .