{'fraction_non_alphanumeric': 0.04238440616500453, 'fraction_numerical': 0.009066183136899365, 'mean_word_length': 4.304086538461538, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Nonlinear system synchronization to sum signals of multiple chaotic systems

Coupling of chaotic oscillators has evidenced conditions where synchronization is possible, therefore 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 is possible to induce it to follow a \textit{linear} superposition of signals from multiple equivalent systems, using only partial information from them, through one- or more variable-signal. Moreover, we show that the larger the number of trajectories added to the input signal, the better the convergence of the system trajectory to the sum input.


INTRODUCTION
Nonlinear systems, such as Lorenz [1], Rössler [2], or Gauthier-Bienfang [3] systems, have been numerically shown to converge to a particular solution by adding into the system information on a particular trajectory, either through substitution of variables [4] for the synchronization 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 trajectory by driving it with a linear combination of signals generated from N independent equivalent systems. Moreover, the system trajectory better mimics that of the sum signal when the number of added trajectories increases.
In order to investigate this behavior of synchronization of a nonlinear system with a sum signal obtained from two or more drives, we prepare N similar systems with slightly mismatched parameters, oscillating independently from one another. Each drive X n is a particular solution of the free-running system equatioṅ where X T n = (x n , y n , z n ) and F is the vector field describing the flux of the system. Information from the drives trajectories is sent to the nonlinear receiving oscillator by way of a sum signal, as described by the modified equation, where X represents the receiving oscillator in 3 , with X T = (x, y, z). K is a m × m matrix allowing for the coupling of the X components and c n represents the strength of the coupling for each solution, X n . Here we numerically show that 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, 2, 3, ...) of Eq. (1) starting from arbitrary initial conditions for each solution and allowing small mismatches between 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 are then linearly combined and introduced in Eq. (2). We establish which variables are coupled through the choice of K components. We applied this technique to a few nonlinear systems such as Lorenz, Rössler, 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 where , and α, A, B, C, and D are positive constants in [3]. We initially analyzed the system with two drives, with X 1 and X 2 their respective trajectories, solutions of equation (1). For the sake of simplicity we consider c 1 = c 2 = 1.0.
We solved numerically equation (2) for the particular flow F described in eq. (4), using K 11 = K 22 = 1.0, and K ij = 0, for all the other components.
To analyze the synchronization, the components (x, y, z) of the response solution are compared to each variable (x n , y n , z n ), with n = 1, 2, of the individual drive oscillators (see and X s is displayed in all the components synchronization curves, X × X s , as well as in the temporal series X(t) and X s (t). Comparing the solution variables X n (t) (n = 1, 2) to those of the oscillator, X(t), we do not observe any regular relation of phase between the solution and individual variables (Figs. 1-3 (a,b)), but only with the sum of the solutions used to define X s (Figs. 1-3 (c)). convergence of X toward X s improves as the number of solutions in X s increases. We have likewise applied this technique to Lorenz and Rössler systems. Both also converge to the solution-sum, yet Rössler shows a slower convergence for the z-variable with the number of solutions. 5