No abstract

We demonstrate by means of several examples that an easily calculable measure of algorithmic complexity c which has been introduced by Lempel and Ziv 25 (1976)] is extremely useful for characterizing spatiotemporal patterns in high-dimensionality nonlinear systems. It is shown that, for time series, c can be a finer measure for order than the Liapunov exponent. We find that, for simple cellular automata, pattern formation can be clearly separated from a mere reduction of the source entropy and dift'erent types of automata can be distinguished. For a chain of coupled logistic maps, c signals pattern formation which cannot be seen in the spatial correlation function alone.

No abstract

939In biological oscillator communities the transmission of information between two oscillators needs a finite time r. We investigate the influence of this effect on the mutual entrainment of two limit cycle oscillators with different frequencies by coupling them with a time delay r. It is shown that for a finite delay time there exists a multitude of synchronized solutions in contrast to the situation without delay where one has at most one solution. § 1. IntroductionInteracting nonlinear oscillators with different individual frequencies can spontaneously synchronize themselves to a common frequency-if the coupling strength exceeds a certain threshold value. This phenomenon is of much relevance to the understanding of biological oscillators such as, e.g., coupled heart pacemaker cells. Inspired by Winfree's earlier ideas l ) there has been a great deal of theoretical work on self-synchronization especially by Kuramoto and his coworkers. 2H3 )In this article we want to take into account the fact that in biological oscillator communities the information transmission between two oscillators needs a finite time r. 14 ),15) Important examples would be neural assemblies which could be modelled by oscillator communities where the delay is caused by the finite velocity of the signal along the axons. 16H8 ) There the crucial question occurs whether synchronization is still possible if the inverse delay time becomes of the same order of magnitude as the oscillator frequencies. In order to investigate the effect of this time delay on the synchronization of nonlinear oscillators we study the following model equations for two limit cycle oscillators:Here if;1, if;2 are the phases of the oscillators with individual frequencies WI, W2, and the interaction terms K sin( if;1,2( t) -if;2,1 (t -r» with coupling strength K tend to synchronize both oscillators. Apart from the delay time r which takes into account that different oscillators know from each others phases only after a retardation time, these equations are (for r=O) just the model equations which have been extensively investigated by Kuramoto.

Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the method for a 1D lattice is also presented.[ S0031-9007(97) The present Letter represents an effort to develop a general control algorithm for spatiotemporally chaotic systems using the methodology of linear control theory, which already proved to be fruitful [6]. Clarifying a number of issues will have direct bearing on this. For instance, it is not clear how many parameters are required for successful control. If the control is applied locally, what is the minimal density of controllers and how should they be arranged to obtain optimal performance? What are the limitations of the linear control scheme and how can they be overcome?Consider the coupled map lattice (CML), originally introduced by Kaneko [7], in an alternative form:with i 1, 2, · · · , L and periodic boundary conditions (BC), z t i1L z t i imposed. We also assume that the local map f͑z, a͒ is a nonlinear function with parameter a, such that f͑z ‫ء‬ , a͒ z ‫ء‬ .To be specific, we choosebut emphasize that all the major results hold independent of this choice. This CML has a homogeneous steady state z ‫ء‬ 1 2 1͞a, which is unstable for a . 3.0. Our goal is to stabilize it using a minimal number of controllers.The first attempt in this direction was undertaken by Hu and Qu [8]. The authors tried to stabilize the homogeneous state by controlling an array of M periodically placed pinning sites ͕i 1 , · · · , i M ͖ with appropriately cho-This however required a very dense array with distance between controllers L p L͞M # 3 in the physically interesting interval of parameters 3.57 , a , 4.0.The reason for this is the spatial periodicity of the pinnings. Since the system is spatially uniform, its eigenmodes are just Fourier modes and the control does not affect the modes whose nodes happen to lie at the pinnings, i.e., modes with periods equal to 2L p , 2L p ͞2, 2L p ͞3, etc., provided those are integer. The control scheme worked only when all such modes were stable.It is however not necessary to destroy the periodicity completely to achieve control: that would complicate the analysis unnecessarily. Instead we add one more pinning site between each of the existing ones. Not all positions are good, but some do solve the problem-previously uncontrollable modes become controllable.In order to understand how the pinnings should be placed and see whether we achieve improved performance by introducing additional controllers, we have to use a few results of the linear control t...

We analyze the dynamic behavior of large two-dimensional systems of limit-cycle oscillators with random intrinsic frequencies that interact via time-delayed nearest-neighbor coupling. We find that even small delay times lead to a novel form of frequency depression where the system decays to stable states which oscillate at a delay and interaction-dependent reduced collective frequency. For greater delay or tighter coupling between oscillators we find metastable synchronized states that we describe analytically and numerically.PACS numbers: 05.45. +b, 87 .I 0. +e Arrays of coupled limit-cycle oscillators are of fundamental interest in physics [I-3], biology [4][5][6][7][8], and engineering [9]. In each of these cases, finite transmission velocities or discrete events, such as the propagation of information through a network node or "synapse," introduce delays in the system that are not commonly incorporated into the picture of interacting oscillators. The recognition that delay increases the dimensionality, and hence the complexity, of the system has focused efforts on those domains where delay is not a major factor, or where the system becomes chaotic [I 0-141. The ubiquitous nature of time delay leads naturally to the subject of this Letter: an exploration of the dynamic behavior and metastable states of coupled time-delayed oscillators.We study a system of N coupled oscillators with phases ¢'; E [0,2Jr]. The Langevin dynamics of the system are described by N differential-difference equations:where K is the coupling constant, r is the delay, and wo is the intrinsic frequency of the oscillators. The sum runs over all nearest neighbors of oscillator i. The temperature T is incorporated in the usual way by the Gaussian noise term 1J;(t):We simulated this system for N=I6384 (128xi28) oscillators on the CM-2 Connection Machine, using a two-dimensional square lattice with periodic boundary conditions. We found the surprising result that even at low temperatures (T«w0 ) the system exhibits a strong "frequency suppression": With increasing coupling and delay the mean frequency, 0 = (I IN) '1:,; (~;) [the angular brackets denote the average over the random noise 1]; (t) ], of the system is drastically reduced as shown in Fig. I (as a function of r ). We obtain a similar frequency depression if instead of a single frequency w0 the oscillators have different intrinsic frequencies with a distribution that has a mean wo and a width comparable to wo (data not shown).In order to understand this effect we consider a situation where all angles change with the same frequency 0 without fluctuation (i.e., ¢'; = 0 t +a) and obtain for T=O from Eq. (I) the resultwhere n is the number of neighbors (four in the case of a square lattice with nearest-neighbor interaction). This equation has multiple solutions. In Fig. I we plot the lowest stable frequency Omin = wo/(1 + Knr) and obtain excellent agreement with our simulations.It is to be expected that the approximation of common frequency and phase of all oscillators is less justified...

No abstract

SUMMARYThe Prisoner's Dilemma has become a paradigm for the evolution of altruistic behaviour. Here we present results of numerical simulations of the infinitely iterated stochastic simultaneous Prisoner's Dilemma considering players with longer memory, encounters of more than two players as well as different pay-off values. This provides us with a better foundation to compare theoretical results to experimental data. We show that the success of the strategy Pavlov, regardless of its simplicity, is far more general by having an outstanding role in the iterated N -player N -memory Prisoner's Dilemma. Besides, we study influences of increased memory sizes in the iterated two-player Prisoner's Dilemma, and present comparisons to results of experiments with first-year students.