Controllability of spatiotemporal systems using constant pinnings

Available online at www.sciencedirect.com

Physica A 318 (2003) 200 – 212

www.elsevier.com/locate/physa

Controllability of spatiotemporal systems using constant pinnings Nita Parekha , Somdatta Sinhab;∗ a Ingenovis,

A Division of iLabs Ltd, 97, Road No: 3, Banjara Hills, Hyderabad 500 034, India for Cellular and Molecular Biology, Uppal Road, Hyderabad 500 007, India

b Centre

Abstract Most natural spatiotemporal systems are an organized ensemble of dynamical subsystems whose behaviour is regulated by nonlinearly coupled multi-variable processes. The response of each of these variables and/or combinations of them to any single or composite external perturbation can in,uence its spatiotemporal behaviour in a complex and non-intuitive manner. This paper attempts to study the dynamical response of two coupled map lattice systems having local dynamics described by (a) coupled discrete maps, and (b) coupled di-erential equations, to external perturbation (or “pinning”) applied to each variable individually or simultaneously. We show that the response of di-erent variables to external pinning is quite di-erent. Our results indicate that, though this pinning approach is useful in controlling complex dynamics both globally and locally, enhancing complexity in dynamics (“anti-control”) is variable dependent. Thus complete controllability (i.e., control and anti-control) of dynamics in these spatially extended systems having coupled multi-variable processes does not only depend on the sign and strength of the perturbation, but also on the speci3c variable being pinned. This also implies that external manipulation of state variables in natural systems can lead to quite di-erent responses depending at the level of action of the perturbation. c 2002 Elsevier Science B.V. All rights reserved.  PACS: 05.45.Gg; 05.45.Ra; 05.45.Pg Keywords: Coupled map lattice; Control; Host–parasitoid system; Lorenz system

∗

Corresponding author. E-mail address: [email protected] (S. Sinha).

c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/03/$ - see front matter
PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 4 2 8 - 0

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

201

1. Introduction Many spatially extended systems are composed of discrete entities governed by their local dynamics and are coupled to each other through di-usion, convection, conduction, etc. [1,2]. Examples are arrays of Josephson junctions [3], coupled chemical reactors [4,5], metapopulation in ecology [6,7], or tissues consisting of discrete network of individual cells with local electrical and chemical processes coupled through gap junctions in physiology [8]. On a 3ner scale, function of each subunit is the result of the interaction or coupling of several variables. In biological systems, each function is controlled by many variables. For example, electrical activity in neural or cardiac cell is the result of a large number of ionic currents and signaling events [9]. Even the simplest model of the cardiac/neural cell is a two variable Fitzhugh–Nagumo model [10]. Similarly a reasonably realistic model for the synthesis of the amino acid tryptophan in bacteria is a four variable model where several molecular reactions have been summarized to few variables [11]. The emergent dynamics or the “global” behaviour of spatially extended systems are, therefore, decided by both the nonlinearly coupled variables that describes the local dynamical system, and the interaction among these systems. The response of these higher order spatiotemporal systems, to systemic or environmental perturbations that change their normal dynamics to abnormalities or disease, can be both complex and non-intuitive. This would depend on how each of the variables of the underlying local dynamical system respond to such perturbation, and also on the nature of the global dynamics exhibited by the entire system at di-erent spatial and temporal scales. For example, elimination of spiral waves in cardiac tissue using electrical shocks can both act at the level of trans-membrane currents in the cells of the tissue, and also at a larger spatial scale (spanning several cells) of the spiral activity [12]. Possessing the ability to modify or have control over the dynamics of spatiotemporal systems has important applications. Here we use the term “control” to indicate both suppression of complex dynamics, and also to induce or maintain complexity (i.e., anti-control) in the dynamics. Several theoretical approaches have been proposed for control [13–15], and few have been applied to experimental systems [4,8,16–19]. A simple and general method developed recently [20–22] uses an externally applied perturbation or pinning to control the spatiotemporal dynamics, both globally and locally, in either direction—i.e., towards the stable or unstable manifold—by simply changing the strength and the sign of pinning. On the other hand, it has also been shown, using one-dimensional maps, that controllability of the dynamics using this approach depends on the local functional form f [23–25], and opposite signs of the pinning do not always lead to opposing types of control. This implies that local dynamical systems that are regulated by coupled processes of multiple variables may respond quite di-erently to perturbations and, hence, the eKciency of the pinning approach for controlling the system dynamics will depend crucially on the variable perturbed. In this paper we investigate the role of pinning the individual variables in two model spatially extended systems: (a) a one-dimensional lattice of the discrete host-parasitoid subpopulations interacting through migration, and (b) a di-usively coupled lattice of

202

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

Lorenz systems exhibiting chaotic dynamics—a prototype of continuous multi-variable coupled dynamical systems. In both the cases we study the global controllability by applying pinning in spatially uniform and non-uniform manner, and also show local control by pinning only a group of lattice sites. To aid our understanding of the coupled discrete processes in (a), we 3rst consider the coupled map lattice (CML) with two di-erent one-dimensional maps belonging to the same universality class—logistic and exponential [26]. We show that the response of these two maps to changing sign and strength of pinning is quite di-erent. Next, we consider the discrete host-parasitoid system, which has a logistic and an exponential function coupled to give rise to the local population dynamics. Here pinning the two variables, host and parasitoid population densities, individually a-ect the spatiotemporal dynamics quite di-erently. Similar behaviour of di-erential response to pinning is observed when the three-coupled variables (x; y; z) in the Lorenz system are perturbed independently. A chaotic Lorenz-CML can be both controlled and anti-controlled by changing the sign and strength of pinning to z, but the response of x and y to external pinning is di-erent from that of z. What is clear is that suppressing complex dynamics with pinning is achievable for all variables in the systems studied, though they may di-er in the strength required for control; but enhancement of complexity in the dynamics is not achievable with all variables. These results imply that, when external perturbations are applied to a system that is governed by nonlinear multi-variable processes, the response can di-er based on how the perturbation interacts with each variable. We also show that if perturbation is applied to all the variables simultaneously, the e-ective response of the system can be quite di-erent than perturbing the individual ones. This is important in the context of several real physical and biological systems where many di-erent factors in a complex environment can perturb each of the underlying variables di-erentially. The net observable response of the system towards a controlled dynamical state will thus not be easily predictable.

2. The method The di-usive CML model in one-dimension, representing a spatiotemporal system, is given by the following general form [27]: xk (i; n + 1) ≡ Fk [xk (i; n)] =(1 − )fk (xk (i; n)) + =2[fk (xk (i − 1; n)) + fk (xk (i + 1; n))] ;

(1)

where f de3nes the local nonlinear dynamics (described by k coupled equations) on the discrete lattice sites i = 1; 2; : : : ; L with periodic boundary conditions. The k continuous state variables x(i; n) are evaluated in discrete time steps n = 1; 2; : : : ; N , and, di-usion to the nearest neighbours is represented by the coupling parameter . Our method [20] of controlling the spatiotemporal dynamics involves applying a constant perturbation or pinning to the state variable on the lattice sites in (1) in the following manner: x(i; n + 1) = F[x(i; n)] + p(i; n) ;

(2)

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

203

where p(i; n) represents the strength of pinning on the ith site at nth time step and can assume both negative and positive values. The pinning can be applied in three di-erent manners for achieving control. For global control of the system one can use—uniform pinning—perturbation of the same strength applied to all the sites at all time steps, i.e., p(i; n) = p. Here one requires that all the sites are available for the application of the pinning signal. Non-uniform pinning—when pinning of the same strength is applied at all time steps but in a spatially inhomogeneous manner, i.e., p(i; n) = (i − mip )p, for m = 1; 2; : : : ; L=ip , such that, if (i − mip ) = 1 then those sites that are m multiples of ip are said to be “pinned” and take a 3nite value p, else p(i; n) = 0. For example, when ip = 2, every alternate site of the lattice is pinned. This approach is useful when it is not possible to have control probes over the entire spatial domain, e.g., in biological tissues where it is diKcult to probe individual cells. But global control may not always be the aim of many applications. For example, a diseased state can induce spatially localized changes in biological tissues (e.g., ectopic node in heart or epileptic focus), which in turn a-ects the normal functional dynamics [28]. The therapeutic measures involve suppressing such ectopic activities locally. This is achieved by local pinning— where pinning is applied to a small localized region only, leaving the rest of the sites unperturbed, i.e., for a lattice of size L, pinning is applied only to a group of k sites (k ⊂ L). 3. Results and discussion Here we demonstrate the role of pinning the variables in CML systems with (A) one-dimensional maps with two di-erent forms of the local map function, (B) coupled discrete maps, and (C) coupled di-erential equations. We show the results only for uniform pinning for (A), but for (B) and (C), we describe the results with non-uniform and local pinning also. 3.1. (A) Controlling CML with one-dimensional maps—the logistic and exponential maps The logistic map given by f(x) = rx(1 − x), for 1 6 r 6 4, 0 6 x 6 1, and the exponential map given by f(x) = x exp [r(1 − x)], for x ¿ 0; r ¿ 0, both belong to the same class of one-dimensional single hump maps, and show the universal bifurcation structure of period-doubling route to chaos with increasing nonlinear parameter r [26]. The corresponding di-usively coupled map lattices, logistic-CML (LCML) and exponential-CML (ECML), exhibit a wide variety of novel and complex spatiotemporal behaviours including spatiotemporal chaos for di-erent values of r and coupling strength [27]. Here we compare the response of these two CMLs to uniform pinning only. For details see [20–22]. 3.1.1. Uniform pinning Fig. 1 summarizes the basic features of the controllability of LCML and ECML using the external pinning approach. Here the weakly chaotic LCML and a periodic

204

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

Fig. 1. Uniform pinning in LCML and ECML: Bifurcation plot for a chosen lattice site, i = 30, as a function of pinning strength p of the—(a) LCML (r = 3:6, = 0:3); and (b) ECML (r = 2:5, = 0:3). Data for n = 100 consecutive time steps are superimposed after eliminating the transients. Initial conditions are randomly chosen from (0,1) for i = 1; 2; : : : ; 60.

ECML are uniformly pinned for a wide range of positive and negative pinning strength p. The bifurcation diagrams in Figs. 1a and b depict the typical local response of a chosen site (L=2), as all sites in the lattice exhibit similar behaviour under uniform pinning. Pinning LCML. Fig. 1a clearly shows that, under the in,uence of increased negative pinning, the local weakly chaotic dynamics at a site in the logistic-CML exhibits period reversals leading to periodic and 3xed-point dynamics. Conversely, increased chaotic behaviour is observed with increasing strength of positive p. Analysis of the local bifurcation plots for di-erent values of r can help in determining the strength and sign of pinning required for attaining the desired state. Thus by an appropriate choice of the strength and sign of pinning or perturbation, it is possible to target the LCML to any desired spatiotemporal dynamical state. Pinning ECML. The situation is di-erent in the case of ECML. Fig. 1b shows that the bifurcation diagram with local periodic exponential-map exhibits period-reversals to periodic and 3xed-point dynamics for both large positive and negative pinning. Though small values of negative pinning induce chaos in the periodic dynamics, increasing the strength of negative pinning further, period-reversal behaviour is observed. In this case, at very high pinning strengths, the dynamics is independent of the sign of pinning and always results in control (i.e., suppression of chaos). It may also be noted that in the case of positive pinning, chaos in the lattice is suppressed with very small p, through controlling it to equilibrium state requires a high pinning strength. This di-erence in the response to perturbation of the two one-dimensional single hump map functions, belonging to the same universality class, results from the di-erence in the exact nature of the functional form f [23,24,29].

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

205

3.2. (B) Controlling CML with coupled discrete maps—the host–parasitoid system The host–parasitoid (HP) system [30] in ecology is described by two coupled discrete equations H (n + 1) ≡ f1 = rH (n)[1 − H (n)] exp[ − P(n)] ; P(n + 1) ≡ f2 = H (n)[1 − exp(−P(n))] ;

(3)

where the variables H (n) and P(n) are the densities of the host and parasite (or parasitoid) populations at the nth generation. The growth of the host (e.g. insect) population follows the logistic growth function which is modulated by parasitism (exp[ − P(n)]). The parasite grows only by infecting the hosts. The parameters r and  represent the intrinsic growth rate of the host and the searching eKciency of the parasitoid, and the populations exhibit quasi-periodic dynamics for r = 4 and  = 3:5. The metapopulation (many subpopulations connected through migration) of the host–parasitoid system is modeled on a one-dimensional lattice where both the hosts and parasitoids di-use to the nearest-neighbouring sites with = 0:2. This system (HP-CML) exhibits a variety of complex spatiotemporal patterns [30,31]. Here we show the response of this system when pinning is applied to either the host (H), the parasitoid (P), or to both the populations simultaneously. In all the cases studied, we start with host–parasitoid populations exhibiting quasi-periodic dynamics. 3.2.1. Uniform pinning In Fig. 2 we compare the bifurcation plots of a single site in the HP metapopulation showing the e-ect of pinning H and P individually, and both simultaneously. Pinning on H. Fig. 2a depicts the long-term dynamics of the host and parasitoid populations at a representative site in HP-CML, when pinning is applied to the hosts only in all the subpopulations. It is clear from the 3gure that positive pinning suppresses the quasi-periodic dynamics while negative pinning increases the complexity in dynamics as shown by the increased phase-space spanned by both the host and parasitoid variables. It may be noted that very small pinning strength is required to control the HP dynamics to equilibrium state in this case. Increasing strength of negative pinning induces higher amplitude complex dynamics around the same unstable steady state. Thus the sign of pinning plays an important role while pinning the host and may lead to suppression or enhancement of the complexity in the dynamics on appropriately choosing the sign of pinning. Pinning on P. Fig. 2b shows the e-ect of pinning the parasitoids only. The result is quite similar to the behaviour observed in the case of ECML (shown in Fig. 1b). Here also, the dynamics is independent of the sign of pinning, and the complexity in the dynamics is suppressed in both host and parasitoid for large values of both positive and negative pinning strength. For small positive pinning the system shows anti-control, but this enhanced complex dynamics is suppressed by p ¿ 0:05. In contrast, smaller negative pinning (p 6 − 0:025) suppresses the dynamics and H and P assume increasingly higher and lower equilibrium states, respectively.

206

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

Fig. 2. Uniform pinning in HP-CML: Bifurcation plots for host (H in black) and parasitoid (P in grey) at the lattice site i = 30 as a function of pinning strength, p on—(a) pinning the host; (b) pinning the parasitoid; (c) pinning both host and parasitoid. Data for n = 100 consecutive time steps are superimposed after eliminating the transients.

Pinning on both H and P. Fig. 2c shows that pinning both H and P simultaneously has an overall system response that is similar to Fig. 2b, i.e., the complexity in the dynamics is suppressed for large values of both positive and negative p, though the pinning strengths required for control is quite di-erent here. When compared to Fig. 2a, small positive p is suKcient in this case also for controlling the dynamics. It also shows enhancement of complexity at smaller negative p, but the dynamics is suppressed at larger negative p as is the case in Fig. 2b. 3.2.2. Non-uniform pinning Having shown the eKcacy of the method for controlling the dynamics by uniformly pinning H with opposite signs of p, we next consider the situation when all the sites are not pinned. In Fig. 3a is shown the quasi-periodic dynamics of the host in the HP-CML for p=0. Negatively pinning (p=−0:05) the host at every alternate lattice site (ip =2) increases the amplitude of the quasi-periodic oscillations leading to anti-control (Fig. 3b). On positively pinning (p = 0:04) the host at every alternate site, the system exhibits spatially periodic but temporally equilibrium dynamics in the lattice (Fig. 3c). In Fig. 3d, even on reducing the density of pinning to every 6th site, it is possible to control the HP-CML to stable periodic state. However, in this case stronger pinning strengths (p = 0:15) are required for control. As expected from Fig. 2b, non-uniform pinning of the parasitoid also leads to global control in the CML for both positive and negative p (results not shown).

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

207

Fig. 3. Non-uniform pinning of the host in HP-CML: (a) unpinned quasi-periodic dynamics of the host; (b) anti-control by negative pinning at ip = 2; (c) control by positive pinning at ip = 2; and (d) control by positive pinning at ip = 6. The pinned and unpinned sites are denoted by ‘×’ and ‘•’ respectively.

Fig. 4. Local pinning of H in HP-CML: The lattice sites 1–20 are pinned negatively with p=−0:04, and sites 41– 60 pinned positively with p = 0:04. The central 20 sites are unpinned showing quasi-periodic dynamics.

3.2.3. Local pinning The e-ect of pinning the host in spatially localized regions in the HP-CML on their long term dynamics is shown in the space-time-amplitude plot (Fig. 4). Here di-erent regions of the HP-CML (L = 60) are pinned di-erentially—the sites 1–20 are pinned negatively, while the sites 41– 60 are pinned positively. The central region (sites 21– 40) are not pinned and exhibit the normal quasi-periodic dynamics. Fig. 4 shows that these di-erent regions of the HP-CML settle in three di-erent dynamical states: the left region showing higher amplitude oscillations due to positive p, and, the right region controlled to an equilibrium state due to negative p. This result shows

208

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

that di-erent regions in the same lattice can be held in di-erent dynamical states by localized pinning. The boundary sites show small oscillations due to coupling. 3.3. (C) Controlling CML with coupled di:erential equations—the Lorenz system The Lorenz system [32] is described by three coupled di-erential equations: x˙ ≡ f1 = (y − x) ; y˙ ≡ f2 = −xz + rx − y ; z˙ ≡ f3 = xy − bz :

(4)

This system shows low-dimensional chaos for the parameter values r = 28, b = 2:67, and  = 10. We use this continuous dynamical system on a spatially discrete one-dimensional lattice (Lorenz-CML) with all three variables di-using to the nearest neighbours with the same strength ( = 0:3). Similar to the host–parasitoid system, here also pinning is applied to the three variables x, y and z, individually and to all three simultaneously, and the dynamic response of the CML in each case is shown for the z variable. 3.3.1. Uniform pinning The bifurcation diagrams in Fig. 5 show the e-ect of increasing strength of pinning (for both positive and negative values) on the dynamics of z, at a chosen site (i = 10) in the Lorenz-CML, when pinning is applied to the variables x, y and z individually. Fig. 6 shows the bifurcation diagram of z when all the three variables are pinned simultaneously. Pinning on x. Fig. 5a depicts the dynamic response of z on pinning the x variable. It may be noted that for small magnitudes of both positive and negative pinning, the dynamics becomes more chaotic, exhibiting larger amplitude oscillations. On further increasing the magnitude of p, the oscillations are gradually suppressed, eventually leading to equilibrium dynamics for higher values of both positive and negative p. Pinning on y. Similar behaviour is observed on pinning the y variable (Fig. 5b). However, the strength of pinning required for control in this case is much smaller (| ∼ 0:04|) compared to that required on pinning x (| ∼ 0:15|). Also, in this case, the z value at the equilibrium state gradually increases with increasing pinning strengths, compared to that in Fig. 5a, which remains nearly constant. Pinning on z. Very di-erent behaviour is observed on pinning the z variable. As shown in Fig. 5c, positive pinning results in the suppression of chaos in the dynamics, while negative pinning enhances it. For control of the Lorenz-CML dynamics, in this case too, very small pinning strengths are required (| ∼ 0:03|). On comparing the response of pinning the three variables individually, it may be noted that complete controllability (i.e., control and “anti-control”) is attainable in the Lorenz-CML only on pinning the z variable, as pinning the other two variables leads to control for both positive and negative values. Pinning on x; y; z. In Fig. 6 we summarize the dynamical behaviour of the LorenzCML on pinning all the three variables simultaneously. The overall behaviour is similar

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

209

Fig. 5. Uniform pinning in Lorenz-CML (L = 20): Bifurcation plots for variable z at site i = 10 as a function of pinning strength p on pinning the—(a) x-variable, (b) y-variable; and (c) z-variable. Data for 2000 time points are superimposed after eliminating the transients.

Fig. 6. Bifurcation plot for variable z on uniformly pinning all the three variables, x, y, and z in Lorenz-CML.

to that of pinning z alone, that is, positive pinning suppresses while negative pinning enhances chaos in the dynamics. However, on a closer look it may be observed that control is achieved for even smaller values of p (compared to Fig. 5c). And, for negative p, the unstable steady state does not remain the same but changes with increasing

210

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

Fig. 7. Non-uniform pinning (ip = 2) of z in Lorenz-CML (L = 20): Central panel—normal chaotic dynamics (p = 0:0); left panel—increasing chaos on negative pinning (p = −0:3); and right panel—controlled to a spatiotemporally regular state on positive pinning (p = 0:3). The pinned and unpinned sites are denoted by ‘×’ and ‘•’ respectively.

magnitude of p. Furthermore, the amplitude of oscillations for larger values of negative p are reduced, probably because of the simultaneous pinning of the x and y variables. 3.3.2. Non-uniform pinning In Fig. 7 we show the space-amplitude plot for z on non-uniform pinning of only the z variable in the Lorenz-CML since it allows controllability by changing the sign of p (c.f., Fig. 5c). The central panel shows the normal chaotic dynamics in all lattice sites for p = 0. On negatively pinning every alternate site (ip = 2) of the CML, the complexity in dynamics in the entire lattice is increased, with larger ,uctuations on the pinned site (left panel). Positive pinning of every alternate site results in a lattice controlled to a regular state (right panel). It may be noted that the pinned and unpinned sites settle into di-erent stable states resulting in a spatially periodic and temporally equilibrium state. 3.3.3. Local pinning To show localised control and anti-control, we consider the Lorenz-CML with L=60 and pinning is applied to the z-variable. In this lattice, sites 11–20 are pinned negatively with p = −0:1 and the sites 41–50 are pinned positively with p = 0:1 respectively, leaving the rest of the system unpinned. It is clear from the space–amplitude plots in Fig. 8, that the local region in the lattice with positive p is controlled to a 3xed-point state, while the region with negative p exhibits larger amplitude oscillations indicating a local increase in the chaotic oscillations. The remaining lattice sites continue to exhibit the uncontrolled chaotic dynamics. Thus, we show that di-erent dynamical states can be induced in coupled, continuous dynamical systems also by localized pinning of di-erent sign and strength.

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

211

Fig. 8. Local control in Lorenz-CML (L = 60) by pinning the z-variable: Space–amplitude plots with the lattice sites from 21–30 pinned positively and the lattice sites 41–50 pinned negatively.

4. Conclusion Many physical and biological systems exhibit a variety of spatiotemporal dynamics—from stable to chaotic—that can change under pathological conditions and impair their normal functions. Thus being able to control the altered dynamics for improved functioning has potential for wide ranging applications in real and arti3cial systems. In this paper, we have assessed the eKcacy of complete controllability of the pinning approach in spatially extended systems whose local dynamics is regulated by coupled multi-variable processes, by pinning each variable separately and together. Our results clearly show that though control/suppression of unstable dynamics is easily achieved by the pinning approach, not all spatiotemporal systems lend themselves to anti-control by simply changing the sign and strength of the pinning signal. This implies that the nonlinearity and feedback processes connecting these variables can lead to quite di-erent response to perturbation of each of these variables. Therefore, the control eKciency of this method crucially depends on the variable perturbed. In situations where a tractable mathematical model for the system is available, linear stability analysis can be used to obtain the type of response, and a rough estimate of the strength and sign of pinning required for control. Alternatively, a few short test experiments on the local functional form can help assess the controllability. Our study is important in the context of many real situations. In Ecology, onedimensional maps are used to model discrete population growth both of single and coupled populations of di-erent species. A metapopulation of such species may respond di-erently to pinning (dispersal or harvesting) depending on the growth function followed by the species. Also, our results on the host-parasitoid system imply that, in adopting control measures in a diseased population, killing the parasites and removing the hosts (quarantine) may have opposite e-ects. This is a non-intuitive result.

212

N. Parekh, S. Sinha / Physica A 318 (2003) 200 – 212

Thus, the same vaccination strategy, harvesting policy, or conservation measures of removal/introduction of individuals in a population may not yield the same results. Same is true for continuous dynamical systems that are regulated by coupled multi-variable processes. Acknowledgements The idea was conceived while SS was a Fellow of the Wissenschaftskolleg zu Berlin. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

N. Parekh, V. Ravi Kumar, B.D. Kulkarni, Phys. Rev. E 52 (1998) 5100. M.S. Spach, Int. J. Bifurc. Chaos 6 (1996) 1637. K.S. Thornburg, M. Moller, R. Roy, T.W. Carr, R.D. Li, T. Erneux, Phys. Rev. E 55 (1997) 3865. V. Petrov, V. Gaspar, J. Masere, K. Showalter, Nature 361 (1993) 240. N. Parekh, V. Ravi Kumar, B.D. Kulkarni, Physica A 224 (1996) 369. I. Hanski, P. Turchin, E. Korplmaki, H. Henttonen, Nature 364 (1993) 232. D.J.D. Earn, P. Rohani, B. Grenfell, Proc. R. Soc. London B 265 (1998) 7. S.J. Schi-, K. Jerger, D.H. Duong, T. Chang, M.L. Spano, W.L. Ditto, Nature 370 (1994) 615. J.B Jack, D. Noble, R.W. Tsien, Electric Current Flow in Excitable Cells, Clarendon Press, Oxford University Press, Oxford, 1983. J.D. Murray, Mathematical Biology, Springer, New York, 1989. M. Santillan, M.C. Mackey, Proc. Natl. Acad. Sci. USA 98 (2001) 1364. A.V. Pan3lov, S.C. Mueller, V.S. Zykov, J.P. Keener, Phys. Rev. E 61 (2000) 4644. T. Shinbrot, Adv. Phys. 44 (1995) 73. M. Ding, W. Yang, V. In, W.L. Ditto, M.L. Spano, B. Gluckman, Phys. Rev. E 53 (1996) 4334. N. Parekh, V. Ravi Kumar, B.D. Kulkarni, Chaos 8 (1998) 300. A. Gar3nkel, M.L. Spano, W.L. Ditto, J.N. Weiss, Science 257 (1992) 1230. K. Hall, D.J. Christini, M. Tremblay, J.J. Collins, L. Glass, J. Billette, Phys. Rev. Lett. 78 (1997) 4518. W.L. Ditto, S.N. Rauseo, M.L. Spano, Phys. Rev. Lett. 65 (1990) 3211. N.J. Corron, S.D. Pethel, B.A. Hopper, Phys. Rev. Lett. 84 (2000) 3835. N. Parekh, S. Parthasarathy, S. Sinha, Phys. Rev. Lett. 81 (1998) 1401. N. Parekh, S. Sinha, in: P.E. Rapp, N. Pradhan, R. Sreenivasan (Eds.), Nonlinear Dynamics and Brain Functioning, Nova Science Publishers, Inc., New York, 1999. N. Parekh, S. Sinha, Phys. Rev. E 65 (2002) 036227. S. Parthasarathy, S. Sinha, Phys. Rev. E 51 (1995) 6239. S. Sinha, S. Parthasarathy, Proc. Natl. Acad. Sci. 93 (1996) 1540. P.K. Das, S. Sinha, in: S.K. Malik, M.K. Chandrashekaran, N. Pradhan (Eds.), Nonlinear Phenomena in Biological and Physical Sciences, Indian Natl. Sci. Acad., New Delhi, 2000, p. 195. R.M. May, Nature 261 (1976) 459. K. Kaneko, Physica D 34 (1989) 2. T. Elbert, W.J. Ray, Z.J. Kowalik, J.E., Skinner, K.E. Graf, N. Birbaumer, Physiol. Rev. 74 (1994) 1. L. Stone, Nature 365 (1993) 617. R.V. Sole, J. Valls, J. Bascompte, Phys. Lett. A 166 (1992) 123. P. Rohani, O. Miramontes, Proc. R. Soc. London B 260 (1995) 335. E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130.