Bistability and Synchronization in Coupled Maps Models
BBistability and Synchronization in Coupled Maps Models
Ricardo L´opez-Ruiz
University of [email protected]
Several coupled maps models are sketched and reviewed in this short communication.First, a discrete logistic type model that was proposed for the symbiotic interaction of twospecies [1, 2]. The coupling between the species depends on the size of the other species ( x n , y n )at each time n and of a constant λ that it was called the mutual benefit : x n +1 = λ (3 y n + 1) x n (1 − x n ) (1) y n +1 = λ (3 x n + 1) y n (1 − y n ) . If the species are isolated, we recover the logistic model for each of them. For λ values whereone isolated species would extinguish, the coexistence with the other species allows them to havean alternative state to survive, supporting the idea that the symbiosis between both species makespossible the bistability (Fig. 1). For increasing values of λ different dynamical regimes are obtained,from periodic oscillations of the populations, through quasiperiodicity up to a chaotic regime beforethe final collapse. Figure 1: Bistability between the fixed popula-tions P and P and their basins of attraction for themap (1) when λ = 0 . . Second, if many of these symbiotic species, i =1 , · · · , N , evolve together ( x in , y in ) under the coupling ac-tion of a global mean field, X Nn and Y Nn , with strength (cid:15) , x in +1 = f λ,N ( x in , y in ) (2) y in +1 = f λ,N ( y in , x in )with f λ,N ( x in , y in ) = λ (cid:34) (cid:32) y in + (cid:15)Y Nn (cid:15) (cid:33) + 1 (cid:35) x in (1 − x in )and X Nn = N (cid:80) Ni =1 x in , Y Nn = N (cid:80) Ni =1 y in , then thesynchronization phenomenon of many oscillators appears,not only under the effect of a strong coupling (cid:15) if not alsounder the effect of an increasing size N of the system. This is one of the first models appeared inthe literature where the synchronization of many chaotic oscillators was reported [1]. Thus, it canbe easily seen that when (cid:15) = 0 the species (maps) are uncoupled and they evolve freely but when (cid:15) → ∞ the maps are totally synchronized and they evolve with the same dynamics all of them, ina periodic, quasiperiodic or chaotic state depending on the mutual benefit λ . But it is still moresurprising the effect provoked by the size of the system consisting in that when N increases thereis an induced synchronization in this many agents model, such as it was also reported in Ref. [1]and it is sketched in Fig. 2. a r X i v : . [ n li n . AO ] A ug hird, if the species x i interact in a symbiotic way with the local mean field X in coming fromthe nearest neighboring species then we can implement this model in different kind of networks.This implementation was done in Ref. [3] interpreting this model in a neuronal context as a naivebrain model where the bistability of the global system mimics the sleeping and the awaking statesof a primitive brain. These are the two global synchronized states, the turned off ( x θ ) and theturned on ( x + ) states of the network in question, that in the context of species would mean theextinction or the coexistence of the species in the ecosystem. This model is x in +1 = ¯ p i x in (1 − x in ) (3)where ¯ p i is the symbiotic parameter that depends on the local mean value, X in , generated by theneighboring species, ¯ p i = p (3 X in + 1), with X in = N i (cid:80) N i j =1 x jn . N i is the number of neighbors ofthe ith species (node i of the network), and p is the actual version of the mutual benefit λ ofthe model (1). The synchronized states of system (3) can be analytically found. The solutionsare x θ = 0 and x + (cid:54) = 0. The first state x θ is stable for 0 < p < x + is stablefor p > .
75. Therefore the bistability between both states, ( x in = x θ , x in = x + ) , ∀ i , is possiblefor p > p = 0 .
75 in the case of many interacting species on a network. Let us observe at thispoint that the non-null state x + appears in the system as a kind of explosive synchronization mediated by a global saddle-node bifurcation in the network that takes place exactly at p = p ,but in this case the appearance of the new synchronized state x + coexists with another non-noisysynchronized state x θ . Also, different routes to enhance or switch on these both states and othersimilar multi-dimensional models where reported at that time in Refs. [3, 4].Finally, some new results concerning this last model embedded in different topologies will bepresented [5]. Figure 2: ( a − c ) Iterates and Fourier spectraof the globally coupled map (2) for λ = 1 . when (cid:15) = 0 . and N = 100 : (a) Iterates of a single ele-ment (the exterior attractor) and of the average (thecentral spot), (b) temporal Fourier spectrum of a sin-gle unit (arbitrary units), (c) temporal Fourier spec-trum for the average (mean field). ( d − f ) The sameas (a-c) when N = 1000 . The synchronization effectmediated by the increasing size N is observed with theappearance of a period- oscillation (at w = 1 / ) inthe mean field, and by extension in the single element. References [1] R. L´opez-Ruiz, C. P´erez-Garc´ıa, Dynamics of maps with aglobal multiplicative coupling, Chaos, Solitons and Fractals 1(1991) 511–528.[2] R. L´opez-Ruiz, D. Fournier-Prunaret , Complex behaviour ina discrete logistic model for the symbiotic interaction of twospecies, Math. Biosci. Eng. 1 (2004) 307–324.[3] R. L´opez-Ruiz, Y. Moreno, A.P. Fern´andez, S. Boccaletti,D.U. Hwang, Awaking and sleeping of a complex network,arXi:nlin/0406053 (2004); Neural Networks 20 (2007) 102–108.[4] R. L´opez-Ruiz, D. Fournier-Prunaret , The bistable brain:A neuronal model with symbiotic interactions, Ch. 10 in