Nonlinear system synchronization to sum signals of multiple chaotic systems
Robson Vieira, Weliton S. Martins, Sergio Barreiro, Rafael A. de Oliveira, Martine Chevrollier, Marcos Oriá
NNonlinear system synchronization to sum signals of multiplechaotic systems
Robson Vieira, Weliton S. Martins, Sergio Barreiro,Rafael A. de Oliveira, Martine Chevrollier, Marcos Ori´a
Universidade Federal Rural de Pernambuco - UACSACabo de Santo Agostinho - PE - Brazil and (Dated: August 18, 2020)
Abstract
Coupling of chaotic oscillators has evidenced conditions where synchronization is possible, there-fore a nonlinear system can be driven to a particular state through input from a similar oscillator.Here we expand this concept of control of the state of a nonlinear system by showing that it ispossible to induce it to follow a linear superposition of signals from multiple equivalent systems,using only partial information from them, through one- or more variable-signal. Moreover, we showthat the larger the number of trajectories added to the input signal, the better the convergence ofthe system trajectory to the sum input. a r X i v : . [ n li n . AO ] A ug NTRODUCTION
Nonlinear systems, such as Lorenz [1], R¨ossler [2], or Gauthier-Bienfang [3] systems, havebeen numerically shown to converge to a particular solution by adding into the systeminformation on a particular trajectory, either through substitution of variables [4] for thesynchronization of chaotic oscillators, or by adding a linear term [3] to the original system.Here we show that it is possible to lead a nonlinear system to a new defined trajectoryby driving it with a linear combination of signals generated from N independent equivalentsystems. Moreover, the system trajectory better mimics that of the sum signal when thenumber of added trajectories increases.In order to investigate this behavior of synchronization of a nonlinear system with asum signal obtained from two or more drives, we prepare N similar systems with slightlymismatched parameters, oscillating independently from one another. Each drive X n is aparticular solution of the free-running system equation ˙X n = F ( X n ) , (1)where X Tn = ( x n , y n , z n ) and F is the vector field describing the flux of the system. Infor-mation from the drives trajectories is sent to the nonlinear receiving oscillator by way of asum signal, as described by the modified equation, ˙X = F ( X ) + K N (cid:88) n =1 c n ( X n − X ) , (2)where X represents the receiving oscillator in (cid:60) , with X T = ( x, y, z ). K is a m × m matrixallowing for the coupling of the X components and c n represents the strength of the couplingfor each solution, X n . Here we numerically show that X −→ X s = (cid:80) Nn =1 c n X n (cid:80) Nn =1 c n , (3)i.e., for large values of N and positive c n , the trajectories of the chaotic response oscillator X converge to X s , with X s T = ( x s , y s , z s ).To demonstrate statement (3) we generated solutions X n ( n = 1 , , , ... ) of Eq. (1)starting from arbitrary initial conditions for each solution and allowing small mismatchesbetween drives, i.e., the parameters defining F may be slightly modified (by a few percent)in order to check for the robustness of the technique. The generated drive solutions are2hen linearly combined and introduced in Eq.(2). We establish which variables are coupledthrough the choice of K components. We applied this technique to a few nonlinear systemssuch as Lorenz, R¨ossler, and Gauthier-Bienfang systems. The results are discussed below. RESULTS
We present here in some detail our syncronization technique applied to a Gauthier-Bienfang system. The flow F in equations (1) and (2) takes the form F = x n − g [ x n − y n ] g [ x n − y n ] − z n y n − Cz n , (4)where g [ χ ] = χ/B + D [exp( αχ ) − exp( − αχ )], and α , A , B , C , and D are positive constantsin (cid:60) [3]. We initially analyzed the system with two drives, with X and X their respectivetrajectories, solutions of equation (1). For the sake of simplicity we consider c = c = 1 . F described in eq. (4), using K = K = 1 .
0, and K ij = 0, for all the other components.To analyze the synchronization, the components ( x, y, z ) of the response solution arecompared to each variable ( x n , y n , z n ), with n = 1 ,
2, of the individual drive oscillators (seeFigures 1-3). We show in Figure 1(a,b) the graphs x × x and x × x , as well as the temporalseries for x , x and x . Clearly, the system does not synchronize with any of the twoindividual components, with both phase and amplitude uncorrelated. These comparativetrajectories and temporal series are shown in Figure 1(c) for the component x s = ( x + x ) / x and x . The component x of the response oscillatorsynchronizes with the component x s of the weighted linear sum signal.Figures 2 and 3 exhibit graphs y × y n and z × z n for n = 1 , X and X s is displayed in all the components synchronization curves, X × X s , as well as in thetemporal series X ( t ) and X s ( t ). Comparing the solution variables X n ( t ) ( n = 1 ,
2) to thoseof the oscillator, X ( t ), we do not observe any regular relation of phase between the solutionand individual variables (Figs. 1-3 (a,b)), but only with the sum of the solutions used todefine X s (Figs. 1-3 (c)). 3 x1 x ( c )( b ) x s x x x x x t ( a ) x2 x t xs x t FIG. 1. Output of the Gauthier-Bienfang nonlinear system with addition of a linear combinationof 2-solution coupling. (a) x × x ; (b) x × x ; (c) x s = ( x + x )2 × x . Left: synchronization analysis;Right: temporal series. We have investigated the behaviour of the solution of equation (2) as a function of thenumber N of solutions of Eq.(1) composing X s . For simplification purposes we consider c n = c ( n = 1 , , ....N ). We define the vector X ⊥ = ( X s − X ) and show in Figure 4 thebehavior of the distance | X ⊥ | [5] as a function of the coupling coefficient c , that evidences theconvergence to synchronization for a large number of added solutions. Figure 4(a) exhibits | X ⊥ | max , the maximum value of | X ⊥ | in the temporal series, which is more sensitive to localinstability, and Figure 4(b) displays | X ⊥ | rms , the average value of | X ⊥ | in the temporalseries, as a measure of the global stability of solution X . A monotonic convergence of | X ⊥ | to zero is observed as the number of solutions added into X s is increased.The decreasing of both | X ⊥ | max and | X ⊥ | rms as N increases confirms the behaviourobserved in the graphs ( X + X ) × X of Figs 1-3(c) in the case of two solutions only: the4 y1 y ( c )( b ) y s y y y y y t ( a ) y2 y t ys y t FIG. 2. Output of the Gauthier-Bienfang nonlinear system with addition of a linear combinationof 2-solution coupling. (a) y × y ; (b) y × y ; (c) y s = ( y + y )2 × y . Left: synchronization analysis;Right: temporal series. convergence of X toward X s improves as the number of solutions in X s increases. We havelikewise applied this technique to Lorenz and R¨ossler systems. Both also converge to thesolution-sum, yet R¨ossler shows a slower convergence for the z -variable with the number ofsolutions. 5 z1 z ( c )( b ) z s z z z z z t ( a ) z2 z t zs z t FIG. 3. Output of the Gauthier-Bienfang nonlinear system with addition of a linear combinationof 2-solution coupling. (a) z × z ; (b) z × z ; (c) z s = ( z + z )2 × z . Left: synchronization analysis;Right: temporal series. CONCLUSION
We showed that is possible to create a state of coherence in a classical, chaotic system.The non-linear system oscillates, following trajectories that are not a simple solution ofthe system equations but a linear combination of solutions. The increased sensitivity ofnonlinear systems to a high number of solutions, even when only a partial informationis transmitted through one or few variables, gives an insight into processes where a largenumber of inputs determine a single output, as occurs in complex networks such as neuralsystems.This work was supported by Universidade Federal Rural de Pernambuco (UFRPE) andthe Brazilian agencies CNPq/Universal, CAPES, and FACEPE.6 . 0 0 . 5 1 . 00246 0 . 0 0 . 5 1 . 0012 ( b ) œ X ^œ max c ( a )
2 4 6 1 0 2 4 6 1 0 œ X ^œ rms c FIG. 4. Measure of the convergence of the oscillator trajectory to a sum signal. (a) | X ⊥ | max and(b) | X ⊥ | rms , for the sum of: 2 (black squares), 4 (red circles), 6 (green up triangle), and 10 (bluedown triangle) solutions.[1] E. N. Lorenz, J. Atmos. Sci. , 130 (1963).[2] O. E. R¨ossler, Physics Letters A , 397 (1976).[3] D. J. Gauthier and J. C. Bienfang, Physical Review Letters , 1751 (1996).[4] L. M. Pecora and T. L. Carroll, Physical Review Letters , 821 (1990).[5] The distance | X ⊥ | is defined as | X ⊥ | ≡ ( | x s − x | + | y s − y | ) + | z s − z | ).).