- Email: [email protected]
Adaptive dynamics on circle maps
3 April 1995
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 199 ( 1995) 365-374
Adaptive dynamics on circle maps Sudeshna Sinha Indian
Received 24 October
1994;
Institute
of Astrophysics.
Koramangala,
Bangalore
560 034, India
revised manuscriptreceived 13 December 1994; accepted for publication 18 December 1994 Communicated
by C.R. Doering
Abstract We have studied a model of Sinha and Biswas of adaptive dynamics on a lattice of circle maps. The model reveals two remarkable features: First, even when the individual local elements are regular, the adaptive mechanisms can given rise to chaotic lattice dynamics. Secondly, the adaptive relaxation time between chaotic updates determines the nature of the power spectrum. In the limit of small relaxation times (when the tails of the adaptive processes interfere) we obtain 1lfcharacteristics.
1. Introduction
The dynamics of networks of chaotic elements is important not only as a model for complex nonlinear systems with many degrees of freedom, but also from the viewpoint of possible engineering applications [ 11. Here we analyse an extension of a model of adaptive dynamics on a lattice of chaotic elements, proposed very recently [ 2,3]. The system is spatially extended, with local nonlinear dynamics along with a self-regulatory process incorporated as threshold dynamics. Such systems are relevant in the context of a variety of physical and biological phenomena (and even in social sciences such as economics). In this study, the local on-site dynamics of the lattice is given by circle maps, which are important as prototypes of chaos in a wide range of systems involving nonlinear oscillator-like behaviour, such as Josephson junctions, nonlinear optics1 &vices, and neutral activity in biology [ 1] . We investigate phenomenologically the wealth of spatiotemporal structures, “phases” and pattern dynamics that emerge from the adaptive mechanism operating on such a lattice. 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00075-5
The other motivation for this work is to study the effect of the variation of adaptive relaxation times A, from the adiabatic limit where the chaotic updates take place only after the system has relaxed completely (i.e., A large), to the limit of a “coupled map”-like situation where adaptive dynamics and chaotic updates take place almost simultaneously (i.e., A very small). We investigate the time evolution of various dynamical quantities over a range of A in order to ascertain the presence of low frequency, or 1 lf noise. (There is a lot of evidence to indicate that llf noise is ubiquitous in nature, occurring widely as it does in a host of composite processes, as diverse as star-flicker in a astronomy, flow of sand in an hour glass, to traffic movement and stock market fluctuations [4] ! It is therefore of immense interest to construct paradigmatic models of dynamic processes which are capable of yielding llf spectral characteristics.) 1.1. Model We first recall the one-dimensional unidirectional model [ 2,3]. Here time is discrete, labelled by n, space is discrete, labelled by i, i= l-N, where N is system
366
S. Sinhu / Physics Letters A 199 (1995) 365-374
size, and the state variable x,(i) (which in physical systems could be quantities like energy or pressure) is continuous. Each individual site in the lattice evolves under a suitable nonlinear mapf(x) (In Ref. [ 21 the local map f(x) was chosen to be the logistic map: f(x) = 1 --ax’, XE [ - 1.0, 1.01, with the nonlinearity parametera in the chaotic regime-u = 2.0 in all numerical experiments.) On such a lattice a self-regulatory threshold dynamics is incorporated. The adaptive mechanism ’ is triggered when a site in the lattice exceeds the critical value x,, i.e., when a certain site x,(i) > x,. The super critical site then relaxes (or “topples”) by transporting its excess 6x= x,(i) -x, to its neighbour as follows,
x,(i+
1) +x,(i+
1) +6x.
t
(1)
This algorithm thus induces a unidirectional nonlinear transport down the array (by initiating a domino effect). The boundary is open so that the “excess” may be transported out of the system. This kind of threshold mechanism imposed on local chaos, makes the above scenario especially relevant for certain mechanical systems like chains of nonlinear springs, as also for some biological systems, such as synapses of nerve tissue (note that individual neurons display complex chaotic behaviour and have step function-like responses to stimuli). Note that in Ref. [ 21 the system was allowed to relax complete before each chaotic update (the local chaos in the maps can be thought of as an “intrinsic” or “internal” perturbation - in which case, the model can be considered as operating in the “dilute” perturbation limit). One self-regulatory update of all existing supercritical sites in the lattice, according to Eq. ( 1) (which may create some other supercritical sites) constitutes one relaxation time step. After A such relaxation steps, the system undergoes the next chaotic update. In some sense then, time n associated with the chaotic dynamics is measured in units of A. When A is large enough (A > O(N) ) as in Ref. [ 21, the complete dynamical picture is as follows: after every time step in the chaotic evolution, IZ, the system undergoes a self-regulatory ’ The dynamics is adaptive or self-regulatory as it incorporates a feedback mechanism which resets (or “adapts”) the state of the system to subcritical when it finds ti to be super-critical. See analogy with adaptive control in Ref. [ 3) and references therein.
04
(b) i-
02
-0.2
-
-
I -
A
t
034
036
030
I
04
-Kc
Fig. I. Diagram of the curvesf,(x,) lying belowfo(xC) =I,, representative case of ~=4.0, for (a) X,E 10.0. 0.5). and blown up section wherex, E [0.34,0.4). Note the “star”-like tures (around which there is effective chaos) and windows, support low order cycles.
for the (b) a strucwhich
relaxation leading to stable “undercritical” configurations (x,,(i) ), after which the next chaotic iteration of the lattice takes place, governed by the nonlinear evolution mapping: x, + , (i) =f(xn( i) ) . The nonlinear threshold adaptive dynamics is reminiscent of the Bak-Tang-Wiesenfeld algorithm [ 5 J , or
S. Sinha /Physics
Letters A 199 (1995) 365-374
the “sandpile” model, which gives rise to self-organized criticality (SOC) . This model is, however, significantly different, the most important difference being that the self-regulatory mechanism now occurs on a chaotic “substrate”, i.e., there is an “intrinsic” or “internal” deterministic dynamics at each site. Further, the state variable analogous to the integer “height” variable z in the sandpile model is continuous here. All this accounts for enhanced complexity and this system is thus capable of exhibiting a wider repertoire of dynamics. So, unlike lD-SOC, the one-dimensional model here is not trivial and can give rise to many interesting features, including a host of dynamical phases. In this paper we use the circle map in the local dynamics, &,I
=f(x,)
=x, f Fr sin( 27rxJ .
(2)
The nonlinearity parameter here is K and it determines the dynamics of the individual elements. For instance, for ~=4.4 the local map is chaotic, and for K= 4.0 it supports a regular cycle. The relevant parameters in the model then are: the local nonlinearity K, the critical x,, the strength of perturbation, u and the relaxation time A. Two of the most interesting features to be examined are (i) the temporal evolution of local quantities, such as the individual sites x,(i), and (ii) the temporal evolution of globally defined quantities, such as “avalanches”, which are defined as the total number of “active” sites, i.e., sites that have “toppled” during the adaptive relaxation 2, denoted by s. Note that x, is the most significant parameter in the system. For the case of the local logistic map [2], numerical simulations showed the presence of the many “phases” in x, space. For example, for x, < 0.5 the dynamics went to a fixed point. When 0.5
for the
361
parameter) provides a wide repertoire of dynamics for lattice dynamical systems, just at the logistic map does for one-dimensional systems (by variation of the nonlinearity parameter). Thus it may be used as a paradigm for complex behaviour in extended systems. In Ref. [ 31 a detailed and rigorous analytical picture of the above dynamical phases was obtained. Ref. [2] also introduced the bidirectional model. This is given by the following algorithm, if x,( i) > x,, the supercritical site then relaxes to x, by transporting the excess (x,,(i) -x,) equally to its two neighbours, x,(i) -f& 3
n,(i-l)+x,(i-1)+6x,
(3)
where 8x= [x,,(i) -x,] 12. Now disturbances in the course of self-regulation, can spread like “ripples” and “refract’ ’ . When f(x) was the logistic map [2], the dynamics of the bidirectional model looked, by and large, like a “fuzzier” or “noisier” version of its unidirectional counterpart, and for certain large x, values gave rise to 1lfspectrum. Note now, that this adaptive dynamics depends on the algorithm for autonomously updating each site and propagating threshold coupling between sites. In Ref. [ 21 these two evolutionary steps were carried out separately. The adaptive dynamics began after each step in the site dynamics and continued till the system reached a steady state where all sites were less than critical, i.e., all x( i) d xc, and the system was stationary, after which the next step in the site dynamics took place. So the time scales of the two dynamics, the intrinsic chaotic dynamics of each lattice site and the adaptive relaxation, were adiabatically separable. The relaxation mechanism was much faster than the chaotic evolution, and this enabled the system to relax completely before the next chaotic iteration. A random driving force can also be introduced in the model. Under this, the system is perturbed at some site j in the lattice: x,,(j) + a(j) + u where g is the strength of the perturbation and j is chosen at random. The random driving force likewise, operative at time scales comparable to the chaotic dynamics, was much slower than the adaptive dynamics in Ref. [ 21. The scenario there was similar to the SOC algorithm, where the driving force (perturbation) is very dilute, and the system is allowed to relax completely before the next perturbation and
S. Sinha /Physics
Letters A 199 (1995) 36S-374
I (b)
10 t
r Fig. 2. Order k of the cycle versus x,, for x, E [O.O. 0.5) with k going up to 20, for the cases of K= 3.6 (a), 3.8 (b). 4.0 (c), 4.2 (d), and 4.4 (e). A section of the plot. x, E [ 0.26.0.27 1,is blown up for the case of ~=4.0, in (f).
time is usually measured in units of the perturbing force (for example, in units of grains of “sand” added in the sandpile model of SOC) and the configurations studied are the relaxed configurations after each perturbation step. This is quite different from the coupled map lattice (CML) dynamics [6], where the coupling is incorporated in the map evolution step.
Here we will vary the time allowed between updates
(A), from the large A limit (when relaxation is complete as in Refs. [ 21 and [3]) to the limit of A = 1 which is the CML version of the adaptive model.
S. Sinha /Physics
Letters A 199 (I 995) 365-374
369
Fig. 2. (continued.)
2. Results 2. I. Cycles and effective chaos First, we investigate the behaviour of a single element x (the case of N = 1) as a function of time n, as it sheds light on the basic structure of the dynamical phases in x, space. (Note that there is no additional random driving force here, i.e. (T= 0.) The first step in the analysis is to study the curves of the various iterates of the map. Letf,(x,) denote the curve in X, space of the nth iterate of the map with initial condition X=-Q. That is, (1) n=O; fo(xC) =x,; (2) It= 1; f,(x,) = 1-2x:; (3) n=2; f*(x,)=l-2(1-2x,2)*=-1--8x:+ 8x:, and so on. In general, U-G)
=Pfn-
1(XC) =P.P
. . .f(&) .
The important thing is the intersection of thef, curves witif,, i.e. the 45” line. Whenever thef, curve crosses above the f0 line we have an n-cycle (see Fig. 1) , as this implies that the nth iterate exceeds the critical value X, and therefore is adapted back to x, ( =fo, which is the first point of the cycle). Thus the threshold mechanism forces a regular “cyclic” evolution, whose period depends on x0 varying which we get periods of
all orders. So even completely chaotic single elements can now yield a wide variety of regular dynamical behaviour determined by the critical x,. Figs. 2a-2e give the order of the cycles thus obtained, with respect to x,, for different values of the map nonlinearity parameter K. (We report results ’ for K = 3.6,3.8,4.0, 4.2,4.4). Now we consider the scenario where N> 1 and A is large (i.e., complete relaxation, as in Refs. [ 21 and [ 31). Here the unidirectional adaptive dynamics without external noise yields exact cycles (say in the temporal evolution of quantities such as avalanche size s), for lattices of any arbitrary size, with the order of the cycle given precisely by the above diagrams (see Ref. [ 31 for rigorous arguments regarding why this should hold true). With the addition of noise, the cycles persist up to some noise strength a,, and then get “fuzzy”, (mostly with a 1lfO-like very low frequency shoulder followed by 1 /f-like slope). The value of oC is proportional to the width of the window supporting the cycle in X, space and the distance of x, from the edge of the window, and is inversely proportional to the order of 3Note that we have not plotted in Figs. 1 and 2 the fixed point region. which occurs whenf, lies above the& curve. This happens for values of x,: 0.5 bx,. Here we have a single point attractor x* =x, for all times.
370
S. Sinha / Physics Letters A I99 (1995) 365-374
I
(6
the cyclic patterns in the bidirectional model, where the dynamics is a noisier version of the above. Now note the other significant fact -just as chaotic elements have periodicity enforced on them by the adaptive dynamics, completely regular elements, with low order cycles, can also have “quasi-chaotic” 4 be&Sour as a result of the threshold coupling. For example note the very high order cycles (that is, effective chaos), obtained for certain x, values, for the case of K= 4.0 (Fig. 2c), where the single circle map supports an exact cycle. This is also clearly evident from Fig. 3a which shows the regular spectrum of a single circle map with ~=4.0, and Fig. 3b which shows the noisy spectrum of a single site in a lattice of circle maps with K = 4.0. A small hand waving argument to rationalise this is given below. The local nonlinear map (without adaptive dynamics) operating in the regular K regime, will for generic initial conditions reach its natural stable cycle. But the important thing to note is that, when it starts its dynamics from a certain initial condition, it can take a very large number of transient time steps (though in principle finite) to reach the cycle. Now the adaptive mechanism “resets” the initial conditions again and again to xC, after some number of time steps (this number is the order of the adaptive cycle, see Fig. 2). So the local map most often does not reach its asymptotic cycle. If transience is long and the system is close to “star”like structures (see Fig. 1) it can effectively look “chaotic” in the finite precision picture. 2.2. Variation of relaxation times
Fig. 3. Power spectrum of (a) a single circle map at K = 4.0. and (b ) a single site in a lattice of circle maps, evolving under unidirectional adaptive dynamics, with ~=4.0. I,= 0.4, o=O.O and N= 20 (A large). We have averaged over eight time runs of 2048 each. The x axis has frequency@ [0,0.51, and they axis has In SfJ), whereSV) is the power.
the cycle. For instance, for the case of K =4.5, for x, = 0.1 and for X, = 0.35 we obtain exact two-cycles even when ~7is as high as 0.2, as these X, values lie in the middle of wide windows of low periodicities. On the other hand, X, = 0.2, which supports an exact fivecycle when CJ= 0, yields a noisy cycle for c as low as 0.01. The unidirectional case also gives the skeleton of
We now investigate the effect of relaxation time on dynamical characteristics. Numerical simulations on a variety of systems (with differingx,, land K) showed that the variation of A affects the dynamics crucially. The reason for this stems from the fact that varying A
aThe
dynamics is “quasi-chaotic” in the sense that it “appears” chaotic in thefinite time eoolution orfinite precision picture. So when one considers. say a finite time series (which may be very long), various computed quantities (for example power spectrum, or phase portrait) are quite indistinguishable from the ones characteristic of chaos. So though these systems may not be chaotic in the strict sense (in fact, the reported cases are in principle not chaotic, but have extremely long period), the dynamics is still very closely mimicked by chaos. This is relevant, as in laboratory experiments and computer simulations, we are always in the finite precision regime, and work with finite data sets.
S. Sinha /Physics
Letters A 199 (1995) 365-374
371
In the unidirectional case, in the presence of weak noise, most often we encounter either of the two qualitative scenarios described below: (a) Complete relaxation (A large) yields a slightly flat shoulder in the very low frequency region ( - 1/f O), followed by 1 /f-like behaviour, while small A leads to flatter low frequency portion of the spectrum - i.e., the 1 /f ’ behaviour persists to higher frequencies. For A = 1, which is the situation analogous to CML, the spectrum is almost completely fat. (See Figs. 4a and 4b.) (b) Complete relaxation (A large) yields exact cycles, while the very low A yield a pronounced 1 lfspectrum. (See Fig. 5.) For the bidirectional case, the basic trend is as follows: for small A we have a distinct 1 lfspectrum, while for large A we get flatter shoulders at the low frequency end (for certain parameters we may also get exact cycles) ‘. Figs. 6 and 7 show some examples. For instance, Fig. 6 gives representative cases where complete relaxation (A large) gives rise to a flatter noisy spectrum ( w 1lf ’ at the lowest frequencies) while the intermediate ranges, A N N/4 to N/2, yield interesting spectra with 1/f-like behaviour persisting to very low frequencies (of the order of 10-3). The particular
Fig. 4. Power spectrum of avalanches in a unidirectional adaptive model on a lattice of circle maps, with ~=4.4, x,=0.3, u=O.Ol, N = 20. and A = 1 (a) and 20 (b) (when the system is completely relaxed). We have averaged over eight time runs of 2048 each. The x axis has lnf, wherefis the frequency cf~ (0.0.5] ), and the .v axis has In Sy), where Sy) is the power.
essentially allows us to move from a picture of adiabatically separated avalanches to one where avalanches overlap, and disturbances do not die before the subsequent chaotic update, i.e. the tails of the adaptiveprocesses carry over to the next event and there are “interference” effects. We will present evidence for this primarily pictorially, for a few representativecases.
Fig. 5. Power spectrum of avalanches in a unidirectional adaptive model on a lattice of circle maps, with K= 3.6, x,=0.285, (r=O.Ol, N= 20, and A = 1. (When A =4 we have an exact cycle.) Here we average over eight time runs of 2048 each. The x axis has Inf, where f is the frequency (f~ (0,051). and the y axis has In SU,, where Sy) is the power.
372
S. Sinha /Physics
10
Letters A 199 (I 995) 365-374
/
Fig. 6. Power spectrum of avalanches in a bidirectional adaptive model on a lattice of gives rise to a flatter noisy spectrum ( - 1lf a at the lowest frequencies) whereas spectrum. The case shown are (a) ~=3.8, x,=0.28, ~=0.01, with (i) A= 12 and A = 5 and (ii) A = 100 (N = 20 in all cases) Here we average over eight time runs (f~ (0,0.5] ), and the y axis has In Sy). where S(f, is the power.
’ The CML analogue of the model, i.e. A close to 1, gives rise to two kinds of behaviour: (a) Somewhat flat very low frequency spectra. which fall thereafter as Ilf 4 +- 1,and (b) fixed point dynamics, where the avalanches are all of the same size for all time, s = AN, i.e., all sites move at all relaxation time steps. This implies that the entire lattice is super-critical, always, as none of the sites have enough time to relax to undercriticality. (Also, for A just below -N/4 we sometimes get flatter I/,f” like spectra which for slightly larger values of A then become 1If-like.)
circle maps, for cases where complete relaxation (A large) the intermediate range of A yields an interesting 1/f-like w=O.O with (i) (ii) A= 100, and (b) ~=4.0, x,=0.25, of 2048 each. The x axis has Inf, where f is the frequency
examples shown are (a) ~=3.8,&=0.28 (u=O.Ol), and (b) ~=4.0,x,=0.25 (u=O.O). Fig. 7 shows cases where large A ( > N) yields exact cycles, for instance: (a) X, = 0.285, K= 3.6, (b) x,=0.16, ~=4.4, and (C) x,=0.3, ~~4.4. Interestingly again, in these systems, for intermediate values of A, we get 1lfspectrum. This is clearly evident from
S. Sinha /Physics
Letters A 199 (1995) 365-374
313
Fig. 7. Power spectrum of avalanches in a bidirectional adaptive model on a lattice of circle maps, for (a) x, = 0.285, K = 3.6, (b) x, = 0.16, u=4.4, and (c) x,=0.3, ~=4.4. Here N=20, with a=O.Ol, A=N/2 for cases (a) and (b), and a=O.l, A=N/4 for case (c). For all these cases complete relaxation (i.e., A large) yields exact cycles. We have averaged over eight time runs of 2048 each. The x axis has Inf, wheref is the frequency cf~ (0,0.51), and they axis has In Sy), where S(f) is the power.
7a-7c which give the spectra at an intermediate valueofd (N/4gA
adaptive processes, and so the 1 if characteristicspersist up to very large values of A. See Fig. 8 for an example: K = 4.0, X, = 0.3 and U= 0.1. 3. Summary In this paper we have presented the rich and extensive phenomenology arising from adaptive dynamics
374
S. Sinha / Physics Letters A 199 (1995) 365-374
( 1) Complete relaxation (A large) yields a slightly flat shoulder in the very low frequency region ( N l/ f ‘), followed by 1/f-like behaviour, while small A leads to a flatter low frequency portion of the spectrum - i.e., the 1lf ” behaviour persists to higher frequencies (Fig. 4). (2) Complete relaxation (A large) yields exact cycles, while the very low A yield a pronounced 1If spectrum (Fig. 5).
Fig. 8. Power spectrum of avalanches in a bidirectional adaptive model onalatticeofcircle maps, forI,= 0.3, K=~.O,N= 20, (T= 0.1, and A = 100. We have averaged over eight time runs of 2048 each. The x axis has Inf, wherefis the frequency (fE (0,O.S 1). and the .I axis has In Sy). where.S(fl is the power. Note the clear 1Ifspectrum even though A is large (A B- N).
on a lattice of circle maps. We summarize now the various observations reported in detail above: 3.1. Single circle map with adaptive dynamics Here we find that the threshold mechanism is capable of imposing a dynamics quite different from that which occurs naturally. So, even completely chaotic maps can now yield a wide variety of regular dynamical behaviour, such as cycles whose period is determined by the critical X, (Figs. 1 and 2). Further, more counter-intuitively, even the reverse is possible, that is, regular elements with low order cycles, can have effectively chaotic behaviour as a result of the threshold coupling (Fig. 3). 3.2. Lattice of circle maps with adaptive dynamics
3.2.2. Bidirectional transport Here, there are three distinct situations arising from different large A (A > N) behaviour, while for intermediate A (N/4 < A < N/2) we have uniform trend, namely the presence of well pronounced l/f noise in all cases ‘. Now we list the observed cases below: ( 1) Large A gives rise to a flatter noisy spectrum ( N 1lf ” at the lowest frequencies) while intermediate A yields distinct 1lfspectra (Fig. 6). (2) Large A yields exact cycles, while for intermediate values of A, again, we get a 1/fspectrum (Fig. 7). (3) The time taken for complete relaxation tends to be infinitely large, so here the 1/fcharacteristics persist up to very large values of A (Fig. 8). In conclusion, the dynamics of this extended adaptive nonlinear system of circle maps is capable of yielding a wide range of behaviours, including low frequency noise, and this makes it of considerable interest as a model for complex nonlinear phenomena. References 11 I 1. Crutchfield and K. Kaneko, in: Directions in chaos, ed. Hao Bai-Lin (World Scientific, Singapore, 1987). and references therein; K. Kaneko, Physica D 54 ( 1994) 5, and references therein. [ 2 I S.Sinha and D. Biswas. Phys. Rev. Lett. 71 (1993) 2010. 13 I S. Sinha, Phys. Rev. E (June 1994). 141 W.H. Press, Comm. Mod. Phys. C 7 (1978) 103. 151P. Bak, C. Tang and K. Weisenfeld, Phys. Rev. Lett. 59 ( 1987) 381. Phys. Rev. A 38 (1988) 364. 161 R. Kapral, J. Math. Chem. 6 (1991) 113, andreferences therein.
We report results from models incorporating unidirectional (Eq. ( 1) ) and bidirectional transport (Eq. (3)). 3.2.1. Unidirectional transport: Here there are two cases:
’ While the bidirectional model displays the uniform trend of 1/f spectra for low A, the unidirectional model displays two kinds of behaviour: in the cases where large A yields exact cycles, low A yields I lfspectra, as in the bidirectional case; but in all other cases, spectra for low A appear noisier and flatter than those for high A. Work towards understanding this phenomenon is in progress.
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 199 ( 1995) 365-374
Adaptive dynamics on circle maps Sudeshna Sinha Indian
Received 24 October
1994;
Institute
of Astrophysics.
Koramangala,
Bangalore
560 034, India
revised manuscriptreceived 13 December 1994; accepted for publication 18 December 1994 Communicated
by C.R. Doering
Abstract We have studied a model of Sinha and Biswas of adaptive dynamics on a lattice of circle maps. The model reveals two remarkable features: First, even when the individual local elements are regular, the adaptive mechanisms can given rise to chaotic lattice dynamics. Secondly, the adaptive relaxation time between chaotic updates determines the nature of the power spectrum. In the limit of small relaxation times (when the tails of the adaptive processes interfere) we obtain 1lfcharacteristics.
1. Introduction
The dynamics of networks of chaotic elements is important not only as a model for complex nonlinear systems with many degrees of freedom, but also from the viewpoint of possible engineering applications [ 11. Here we analyse an extension of a model of adaptive dynamics on a lattice of chaotic elements, proposed very recently [ 2,3]. The system is spatially extended, with local nonlinear dynamics along with a self-regulatory process incorporated as threshold dynamics. Such systems are relevant in the context of a variety of physical and biological phenomena (and even in social sciences such as economics). In this study, the local on-site dynamics of the lattice is given by circle maps, which are important as prototypes of chaos in a wide range of systems involving nonlinear oscillator-like behaviour, such as Josephson junctions, nonlinear optics1 &vices, and neutral activity in biology [ 1] . We investigate phenomenologically the wealth of spatiotemporal structures, “phases” and pattern dynamics that emerge from the adaptive mechanism operating on such a lattice. 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00075-5
The other motivation for this work is to study the effect of the variation of adaptive relaxation times A, from the adiabatic limit where the chaotic updates take place only after the system has relaxed completely (i.e., A large), to the limit of a “coupled map”-like situation where adaptive dynamics and chaotic updates take place almost simultaneously (i.e., A very small). We investigate the time evolution of various dynamical quantities over a range of A in order to ascertain the presence of low frequency, or 1 lf noise. (There is a lot of evidence to indicate that llf noise is ubiquitous in nature, occurring widely as it does in a host of composite processes, as diverse as star-flicker in a astronomy, flow of sand in an hour glass, to traffic movement and stock market fluctuations [4] ! It is therefore of immense interest to construct paradigmatic models of dynamic processes which are capable of yielding llf spectral characteristics.) 1.1. Model We first recall the one-dimensional unidirectional model [ 2,3]. Here time is discrete, labelled by n, space is discrete, labelled by i, i= l-N, where N is system
366
S. Sinhu / Physics Letters A 199 (1995) 365-374
size, and the state variable x,(i) (which in physical systems could be quantities like energy or pressure) is continuous. Each individual site in the lattice evolves under a suitable nonlinear mapf(x) (In Ref. [ 21 the local map f(x) was chosen to be the logistic map: f(x) = 1 --ax’, XE [ - 1.0, 1.01, with the nonlinearity parametera in the chaotic regime-u = 2.0 in all numerical experiments.) On such a lattice a self-regulatory threshold dynamics is incorporated. The adaptive mechanism ’ is triggered when a site in the lattice exceeds the critical value x,, i.e., when a certain site x,(i) > x,. The super critical site then relaxes (or “topples”) by transporting its excess 6x= x,(i) -x, to its neighbour as follows,
x,(i+
1) +x,(i+
1) +6x.
t
(1)
This algorithm thus induces a unidirectional nonlinear transport down the array (by initiating a domino effect). The boundary is open so that the “excess” may be transported out of the system. This kind of threshold mechanism imposed on local chaos, makes the above scenario especially relevant for certain mechanical systems like chains of nonlinear springs, as also for some biological systems, such as synapses of nerve tissue (note that individual neurons display complex chaotic behaviour and have step function-like responses to stimuli). Note that in Ref. [ 21 the system was allowed to relax complete before each chaotic update (the local chaos in the maps can be thought of as an “intrinsic” or “internal” perturbation - in which case, the model can be considered as operating in the “dilute” perturbation limit). One self-regulatory update of all existing supercritical sites in the lattice, according to Eq. ( 1) (which may create some other supercritical sites) constitutes one relaxation time step. After A such relaxation steps, the system undergoes the next chaotic update. In some sense then, time n associated with the chaotic dynamics is measured in units of A. When A is large enough (A > O(N) ) as in Ref. [ 21, the complete dynamical picture is as follows: after every time step in the chaotic evolution, IZ, the system undergoes a self-regulatory ’ The dynamics is adaptive or self-regulatory as it incorporates a feedback mechanism which resets (or “adapts”) the state of the system to subcritical when it finds ti to be super-critical. See analogy with adaptive control in Ref. [ 3) and references therein.
04
(b) i-
02
-0.2
-
-
I -
A
t
034
036
030
I
04
-Kc
Fig. I. Diagram of the curvesf,(x,) lying belowfo(xC) =I,, representative case of ~=4.0, for (a) X,E 10.0. 0.5). and blown up section wherex, E [0.34,0.4). Note the “star”-like tures (around which there is effective chaos) and windows, support low order cycles.
for the (b) a strucwhich
relaxation leading to stable “undercritical” configurations (x,,(i) ), after which the next chaotic iteration of the lattice takes place, governed by the nonlinear evolution mapping: x, + , (i) =f(xn( i) ) . The nonlinear threshold adaptive dynamics is reminiscent of the Bak-Tang-Wiesenfeld algorithm [ 5 J , or
S. Sinha /Physics
Letters A 199 (1995) 365-374
the “sandpile” model, which gives rise to self-organized criticality (SOC) . This model is, however, significantly different, the most important difference being that the self-regulatory mechanism now occurs on a chaotic “substrate”, i.e., there is an “intrinsic” or “internal” deterministic dynamics at each site. Further, the state variable analogous to the integer “height” variable z in the sandpile model is continuous here. All this accounts for enhanced complexity and this system is thus capable of exhibiting a wider repertoire of dynamics. So, unlike lD-SOC, the one-dimensional model here is not trivial and can give rise to many interesting features, including a host of dynamical phases. In this paper we use the circle map in the local dynamics, &,I
=f(x,)
=x, f Fr sin( 27rxJ .
(2)
The nonlinearity parameter here is K and it determines the dynamics of the individual elements. For instance, for ~=4.4 the local map is chaotic, and for K= 4.0 it supports a regular cycle. The relevant parameters in the model then are: the local nonlinearity K, the critical x,, the strength of perturbation, u and the relaxation time A. Two of the most interesting features to be examined are (i) the temporal evolution of local quantities, such as the individual sites x,(i), and (ii) the temporal evolution of globally defined quantities, such as “avalanches”, which are defined as the total number of “active” sites, i.e., sites that have “toppled” during the adaptive relaxation 2, denoted by s. Note that x, is the most significant parameter in the system. For the case of the local logistic map [2], numerical simulations showed the presence of the many “phases” in x, space. For example, for x, < 0.5 the dynamics went to a fixed point. When 0.5
for the
361
parameter) provides a wide repertoire of dynamics for lattice dynamical systems, just at the logistic map does for one-dimensional systems (by variation of the nonlinearity parameter). Thus it may be used as a paradigm for complex behaviour in extended systems. In Ref. [ 31 a detailed and rigorous analytical picture of the above dynamical phases was obtained. Ref. [2] also introduced the bidirectional model. This is given by the following algorithm, if x,( i) > x,, the supercritical site then relaxes to x, by transporting the excess (x,,(i) -x,) equally to its two neighbours, x,(i) -f& 3
n,(i-l)+x,(i-1)+6x,
(3)
where 8x= [x,,(i) -x,] 12. Now disturbances in the course of self-regulation, can spread like “ripples” and “refract’ ’ . When f(x) was the logistic map [2], the dynamics of the bidirectional model looked, by and large, like a “fuzzier” or “noisier” version of its unidirectional counterpart, and for certain large x, values gave rise to 1lfspectrum. Note now, that this adaptive dynamics depends on the algorithm for autonomously updating each site and propagating threshold coupling between sites. In Ref. [ 21 these two evolutionary steps were carried out separately. The adaptive dynamics began after each step in the site dynamics and continued till the system reached a steady state where all sites were less than critical, i.e., all x( i) d xc, and the system was stationary, after which the next step in the site dynamics took place. So the time scales of the two dynamics, the intrinsic chaotic dynamics of each lattice site and the adaptive relaxation, were adiabatically separable. The relaxation mechanism was much faster than the chaotic evolution, and this enabled the system to relax completely before the next chaotic iteration. A random driving force can also be introduced in the model. Under this, the system is perturbed at some site j in the lattice: x,,(j) + a(j) + u where g is the strength of the perturbation and j is chosen at random. The random driving force likewise, operative at time scales comparable to the chaotic dynamics, was much slower than the adaptive dynamics in Ref. [ 21. The scenario there was similar to the SOC algorithm, where the driving force (perturbation) is very dilute, and the system is allowed to relax completely before the next perturbation and
S. Sinha /Physics
Letters A 199 (1995) 36S-374
I (b)
10 t
r Fig. 2. Order k of the cycle versus x,, for x, E [O.O. 0.5) with k going up to 20, for the cases of K= 3.6 (a), 3.8 (b). 4.0 (c), 4.2 (d), and 4.4 (e). A section of the plot. x, E [ 0.26.0.27 1,is blown up for the case of ~=4.0, in (f).
time is usually measured in units of the perturbing force (for example, in units of grains of “sand” added in the sandpile model of SOC) and the configurations studied are the relaxed configurations after each perturbation step. This is quite different from the coupled map lattice (CML) dynamics [6], where the coupling is incorporated in the map evolution step.
Here we will vary the time allowed between updates
(A), from the large A limit (when relaxation is complete as in Refs. [ 21 and [3]) to the limit of A = 1 which is the CML version of the adaptive model.
S. Sinha /Physics
Letters A 199 (I 995) 365-374
369
Fig. 2. (continued.)
2. Results 2. I. Cycles and effective chaos First, we investigate the behaviour of a single element x (the case of N = 1) as a function of time n, as it sheds light on the basic structure of the dynamical phases in x, space. (Note that there is no additional random driving force here, i.e. (T= 0.) The first step in the analysis is to study the curves of the various iterates of the map. Letf,(x,) denote the curve in X, space of the nth iterate of the map with initial condition X=-Q. That is, (1) n=O; fo(xC) =x,; (2) It= 1; f,(x,) = 1-2x:; (3) n=2; f*(x,)=l-2(1-2x,2)*=-1--8x:+ 8x:, and so on. In general, U-G)
=Pfn-
1(XC) =P.P
. . .f(&) .
The important thing is the intersection of thef, curves witif,, i.e. the 45” line. Whenever thef, curve crosses above the f0 line we have an n-cycle (see Fig. 1) , as this implies that the nth iterate exceeds the critical value X, and therefore is adapted back to x, ( =fo, which is the first point of the cycle). Thus the threshold mechanism forces a regular “cyclic” evolution, whose period depends on x0 varying which we get periods of
all orders. So even completely chaotic single elements can now yield a wide variety of regular dynamical behaviour determined by the critical x,. Figs. 2a-2e give the order of the cycles thus obtained, with respect to x,, for different values of the map nonlinearity parameter K. (We report results ’ for K = 3.6,3.8,4.0, 4.2,4.4). Now we consider the scenario where N> 1 and A is large (i.e., complete relaxation, as in Refs. [ 21 and [ 31). Here the unidirectional adaptive dynamics without external noise yields exact cycles (say in the temporal evolution of quantities such as avalanche size s), for lattices of any arbitrary size, with the order of the cycle given precisely by the above diagrams (see Ref. [ 31 for rigorous arguments regarding why this should hold true). With the addition of noise, the cycles persist up to some noise strength a,, and then get “fuzzy”, (mostly with a 1lfO-like very low frequency shoulder followed by 1 /f-like slope). The value of oC is proportional to the width of the window supporting the cycle in X, space and the distance of x, from the edge of the window, and is inversely proportional to the order of 3Note that we have not plotted in Figs. 1 and 2 the fixed point region. which occurs whenf, lies above the& curve. This happens for values of x,: 0.5 bx,. Here we have a single point attractor x* =x, for all times.
370
S. Sinha / Physics Letters A I99 (1995) 365-374
I
(6
the cyclic patterns in the bidirectional model, where the dynamics is a noisier version of the above. Now note the other significant fact -just as chaotic elements have periodicity enforced on them by the adaptive dynamics, completely regular elements, with low order cycles, can also have “quasi-chaotic” 4 be&Sour as a result of the threshold coupling. For example note the very high order cycles (that is, effective chaos), obtained for certain x, values, for the case of K= 4.0 (Fig. 2c), where the single circle map supports an exact cycle. This is also clearly evident from Fig. 3a which shows the regular spectrum of a single circle map with ~=4.0, and Fig. 3b which shows the noisy spectrum of a single site in a lattice of circle maps with K = 4.0. A small hand waving argument to rationalise this is given below. The local nonlinear map (without adaptive dynamics) operating in the regular K regime, will for generic initial conditions reach its natural stable cycle. But the important thing to note is that, when it starts its dynamics from a certain initial condition, it can take a very large number of transient time steps (though in principle finite) to reach the cycle. Now the adaptive mechanism “resets” the initial conditions again and again to xC, after some number of time steps (this number is the order of the adaptive cycle, see Fig. 2). So the local map most often does not reach its asymptotic cycle. If transience is long and the system is close to “star”like structures (see Fig. 1) it can effectively look “chaotic” in the finite precision picture. 2.2. Variation of relaxation times
Fig. 3. Power spectrum of (a) a single circle map at K = 4.0. and (b ) a single site in a lattice of circle maps, evolving under unidirectional adaptive dynamics, with ~=4.0. I,= 0.4, o=O.O and N= 20 (A large). We have averaged over eight time runs of 2048 each. The x axis has frequency@ [0,0.51, and they axis has In SfJ), whereSV) is the power.
the cycle. For instance, for the case of K =4.5, for x, = 0.1 and for X, = 0.35 we obtain exact two-cycles even when ~7is as high as 0.2, as these X, values lie in the middle of wide windows of low periodicities. On the other hand, X, = 0.2, which supports an exact fivecycle when CJ= 0, yields a noisy cycle for c as low as 0.01. The unidirectional case also gives the skeleton of
We now investigate the effect of relaxation time on dynamical characteristics. Numerical simulations on a variety of systems (with differingx,, land K) showed that the variation of A affects the dynamics crucially. The reason for this stems from the fact that varying A
aThe
dynamics is “quasi-chaotic” in the sense that it “appears” chaotic in thefinite time eoolution orfinite precision picture. So when one considers. say a finite time series (which may be very long), various computed quantities (for example power spectrum, or phase portrait) are quite indistinguishable from the ones characteristic of chaos. So though these systems may not be chaotic in the strict sense (in fact, the reported cases are in principle not chaotic, but have extremely long period), the dynamics is still very closely mimicked by chaos. This is relevant, as in laboratory experiments and computer simulations, we are always in the finite precision regime, and work with finite data sets.
S. Sinha /Physics
Letters A 199 (1995) 365-374
371
In the unidirectional case, in the presence of weak noise, most often we encounter either of the two qualitative scenarios described below: (a) Complete relaxation (A large) yields a slightly flat shoulder in the very low frequency region ( - 1/f O), followed by 1 /f-like behaviour, while small A leads to flatter low frequency portion of the spectrum - i.e., the 1 /f ’ behaviour persists to higher frequencies. For A = 1, which is the situation analogous to CML, the spectrum is almost completely fat. (See Figs. 4a and 4b.) (b) Complete relaxation (A large) yields exact cycles, while the very low A yield a pronounced 1 lfspectrum. (See Fig. 5.) For the bidirectional case, the basic trend is as follows: for small A we have a distinct 1 lfspectrum, while for large A we get flatter shoulders at the low frequency end (for certain parameters we may also get exact cycles) ‘. Figs. 6 and 7 show some examples. For instance, Fig. 6 gives representative cases where complete relaxation (A large) gives rise to a flatter noisy spectrum ( w 1lf ’ at the lowest frequencies) while the intermediate ranges, A N N/4 to N/2, yield interesting spectra with 1/f-like behaviour persisting to very low frequencies (of the order of 10-3). The particular
Fig. 4. Power spectrum of avalanches in a unidirectional adaptive model on a lattice of circle maps, with ~=4.4, x,=0.3, u=O.Ol, N = 20. and A = 1 (a) and 20 (b) (when the system is completely relaxed). We have averaged over eight time runs of 2048 each. The x axis has lnf, wherefis the frequency cf~ (0.0.5] ), and the .v axis has In Sy), where Sy) is the power.
essentially allows us to move from a picture of adiabatically separated avalanches to one where avalanches overlap, and disturbances do not die before the subsequent chaotic update, i.e. the tails of the adaptiveprocesses carry over to the next event and there are “interference” effects. We will present evidence for this primarily pictorially, for a few representativecases.
Fig. 5. Power spectrum of avalanches in a unidirectional adaptive model on a lattice of circle maps, with K= 3.6, x,=0.285, (r=O.Ol, N= 20, and A = 1. (When A =4 we have an exact cycle.) Here we average over eight time runs of 2048 each. The x axis has Inf, where f is the frequency (f~ (0,051). and the y axis has In SU,, where Sy) is the power.
372
S. Sinha /Physics
10
Letters A 199 (I 995) 365-374
/
Fig. 6. Power spectrum of avalanches in a bidirectional adaptive model on a lattice of gives rise to a flatter noisy spectrum ( - 1lf a at the lowest frequencies) whereas spectrum. The case shown are (a) ~=3.8, x,=0.28, ~=0.01, with (i) A= 12 and A = 5 and (ii) A = 100 (N = 20 in all cases) Here we average over eight time runs (f~ (0,0.5] ), and the y axis has In Sy). where S(f, is the power.
’ The CML analogue of the model, i.e. A close to 1, gives rise to two kinds of behaviour: (a) Somewhat flat very low frequency spectra. which fall thereafter as Ilf 4 +- 1,and (b) fixed point dynamics, where the avalanches are all of the same size for all time, s = AN, i.e., all sites move at all relaxation time steps. This implies that the entire lattice is super-critical, always, as none of the sites have enough time to relax to undercriticality. (Also, for A just below -N/4 we sometimes get flatter I/,f” like spectra which for slightly larger values of A then become 1If-like.)
circle maps, for cases where complete relaxation (A large) the intermediate range of A yields an interesting 1/f-like w=O.O with (i) (ii) A= 100, and (b) ~=4.0, x,=0.25, of 2048 each. The x axis has Inf, where f is the frequency
examples shown are (a) ~=3.8,&=0.28 (u=O.Ol), and (b) ~=4.0,x,=0.25 (u=O.O). Fig. 7 shows cases where large A ( > N) yields exact cycles, for instance: (a) X, = 0.285, K= 3.6, (b) x,=0.16, ~=4.4, and (C) x,=0.3, ~~4.4. Interestingly again, in these systems, for intermediate values of A, we get 1lfspectrum. This is clearly evident from
S. Sinha /Physics
Letters A 199 (1995) 365-374
313
Fig. 7. Power spectrum of avalanches in a bidirectional adaptive model on a lattice of circle maps, for (a) x, = 0.285, K = 3.6, (b) x, = 0.16, u=4.4, and (c) x,=0.3, ~=4.4. Here N=20, with a=O.Ol, A=N/2 for cases (a) and (b), and a=O.l, A=N/4 for case (c). For all these cases complete relaxation (i.e., A large) yields exact cycles. We have averaged over eight time runs of 2048 each. The x axis has Inf, wheref is the frequency cf~ (0,0.51), and they axis has In Sy), where S(f) is the power.
7a-7c which give the spectra at an intermediate valueofd (N/4gA
adaptive processes, and so the 1 if characteristicspersist up to very large values of A. See Fig. 8 for an example: K = 4.0, X, = 0.3 and U= 0.1. 3. Summary In this paper we have presented the rich and extensive phenomenology arising from adaptive dynamics
374
S. Sinha / Physics Letters A 199 (1995) 365-374
( 1) Complete relaxation (A large) yields a slightly flat shoulder in the very low frequency region ( N l/ f ‘), followed by 1/f-like behaviour, while small A leads to a flatter low frequency portion of the spectrum - i.e., the 1lf ” behaviour persists to higher frequencies (Fig. 4). (2) Complete relaxation (A large) yields exact cycles, while the very low A yield a pronounced 1If spectrum (Fig. 5).
Fig. 8. Power spectrum of avalanches in a bidirectional adaptive model onalatticeofcircle maps, forI,= 0.3, K=~.O,N= 20, (T= 0.1, and A = 100. We have averaged over eight time runs of 2048 each. The x axis has Inf, wherefis the frequency (fE (0,O.S 1). and the .I axis has In Sy). where.S(fl is the power. Note the clear 1Ifspectrum even though A is large (A B- N).
on a lattice of circle maps. We summarize now the various observations reported in detail above: 3.1. Single circle map with adaptive dynamics Here we find that the threshold mechanism is capable of imposing a dynamics quite different from that which occurs naturally. So, even completely chaotic maps can now yield a wide variety of regular dynamical behaviour, such as cycles whose period is determined by the critical X, (Figs. 1 and 2). Further, more counter-intuitively, even the reverse is possible, that is, regular elements with low order cycles, can have effectively chaotic behaviour as a result of the threshold coupling (Fig. 3). 3.2. Lattice of circle maps with adaptive dynamics
3.2.2. Bidirectional transport Here, there are three distinct situations arising from different large A (A > N) behaviour, while for intermediate A (N/4 < A < N/2) we have uniform trend, namely the presence of well pronounced l/f noise in all cases ‘. Now we list the observed cases below: ( 1) Large A gives rise to a flatter noisy spectrum ( N 1lf ” at the lowest frequencies) while intermediate A yields distinct 1lfspectra (Fig. 6). (2) Large A yields exact cycles, while for intermediate values of A, again, we get a 1/fspectrum (Fig. 7). (3) The time taken for complete relaxation tends to be infinitely large, so here the 1/fcharacteristics persist up to very large values of A (Fig. 8). In conclusion, the dynamics of this extended adaptive nonlinear system of circle maps is capable of yielding a wide range of behaviours, including low frequency noise, and this makes it of considerable interest as a model for complex nonlinear phenomena. References 11 I 1. Crutchfield and K. Kaneko, in: Directions in chaos, ed. Hao Bai-Lin (World Scientific, Singapore, 1987). and references therein; K. Kaneko, Physica D 54 ( 1994) 5, and references therein. [ 2 I S.Sinha and D. Biswas. Phys. Rev. Lett. 71 (1993) 2010. 13 I S. Sinha, Phys. Rev. E (June 1994). 141 W.H. Press, Comm. Mod. Phys. C 7 (1978) 103. 151P. Bak, C. Tang and K. Weisenfeld, Phys. Rev. Lett. 59 ( 1987) 381. Phys. Rev. A 38 (1988) 364. 161 R. Kapral, J. Math. Chem. 6 (1991) 113, andreferences therein.
We report results from models incorporating unidirectional (Eq. ( 1) ) and bidirectional transport (Eq. (3)). 3.2.1. Unidirectional transport: Here there are two cases:
’ While the bidirectional model displays the uniform trend of 1/f spectra for low A, the unidirectional model displays two kinds of behaviour: in the cases where large A yields exact cycles, low A yields I lfspectra, as in the bidirectional case; but in all other cases, spectra for low A appear noisier and flatter than those for high A. Work towards understanding this phenomenon is in progress.