Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 16, 795-800 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.51097 Method of Generation of Chaos Map in the Centre Manifold Evgeny V. Nikulchev Moscow Technological Institute Moscow, Russia 119334 Alexander P. Kondratov Moscow State University of Printing Arts Moscow, Russia 127550 Copyright c 2015 Evgeny V. Nikulchev and Alexander P. Kondratov. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This work has proposed a method to robust chaos generation in the invariant centre manifoldas. It consists of central manifold theory combined with the polynomial chaos approach and symmetry theory. Keywords: Chaos, Robust Chaos, Centre Manifold, Chaos Map. 1 Introduction and preliminaries The modeling of chaotic dynamics currently has a variety of practical applications: data encryption and then comparing the code on the basis of the attractors in different various processes, control systems sampling, simulation in various engineering applications. There are two types of attractors in dynamical systems [1]: the first, when the chaos disappears with little change in the parameters of the system of second type differential equations, then so-called robust chaos, a strange attractor
796 Evgeny V. Nikulchev and Alexander P. Kondratov when you change a range of settings. The methods of robust chaos generation are especially important for technical systems (simulation and identification). In general the accumulated error modeling approximations and minor changes in external factors should not lead to a different system of qualitative states. For example, taking readings of sensors to build an adequate model in the form of differential equations or discrete maps. This model is of fragile chaos type, that is, when you input let s say 0,001 instead of strange chaotic attractor in the phase, portrait will be observed within limited cycles, the value of mathematical model for engineering applications will not be great. In many technical systems it is simply impossible to ensure the accuracy of this parameter, moreover, observational error is often higher than the range of chaotic regimes in such systems [2, 3]. In [4, 5] there have been shown that there may be chaos in smooth systems that served as the creation of a new direction a polynomial chaos. To solve the problems of identification [6] method of robust chaos modeling was developed, which is based in the central building of reduced invariant manifold. Subsequently, there was a theory of polynomial chaos in the central invariant manifold [7, 8]. Invariant manifold is an effective tool that helps to reduce the dynamic system into the Hopf bifurcation point [9]. This approach is based on the idea that all dynamic characteristics near the point of dynamics equilibrium are regulated by the central invariant manifold, when some eigenvalues have zero real part and all other eigenvalues have negative real parts. 2 Main results Dynamical systems are described in the following n- dimensional differential equations ẋ(t) = f(x(t), ε), (1) where x(t) R n is defined as the state vector and ε R control parameter. It is assumed the vector field f should be smooth and the origin point of balance (1). In addition, there is only polynomial nonlinearity considered. Method invariant manifold uses the basic idea, which is important characteristic of a nonlinear dynamical system in the neighborhood of the equilibrium point is determined by the center Variety associated with part of the original system is characterized by the eigenvalues with zero real part to a Hopf bifurcation neighbourhood [10]. Since the system (1) is a polynomial, it can be expressed as follows: ẋ = A(ε)x + Φ 1 (x, ε) +... + Φ k (x, ε), (2)
Generation of chaos in the centre manifold 797 where A(ε) is n n matrix; Φ k is a vector of degree k polynomial functions of x and ε. By means of a linear basis transformation x = T y, the system (2) can be put in a canonical form (3) at the Hopf bifurcation point ε 0. The linear basis transformation is given by the n n matrix T = [ ] T 1,..., T nc, T nc+1,..., T n where T 1,..., T nc and T nc+1,..., T n are the generalized eigenvectors corresponding respectively to the n c eigenvalues λ i (i = 1, n) of A(ε 0 ) with zero real parts and the (n n c ) eigenvalues λ i (i = n c + 1, n) of A(ε 0 ) with nonzero real parts: { ẏc = A c (ε 0 ) + Φ c (y c, y s, ε 0 ), ẏ s = A s (ε 0 ) + Φ s (y c, y s, ε 0 ), (3) where y = [y c y s ] T, A c = λ 1 0 0....., 0... λ nc λ nc+1 0 0 A s =....., 0... λ n [ ] Φc (y c, y s, ε 0 ) = T 1 (Φ Φ s (y c, y s, ε 0 ) 1 (T y, ε 0 ) +... + Φ k (T y, ε 0 )), Φ c (0, 0, ε o ) = 0, Φ s (0, 0, ε o ) = 0 and the jacobian matrice DΦ c (0, 0, ε o ), DΦ s (0, 0, ε o ) are matrices with zero entries In the neighbourhood of the Hopf bifurcation, the system (3) may be defined by the following augmented dynamics: ẏ c = A c ( ε 0 ) + Φ c (y c, y s, ε), ẏ s = A s ( ε 0 ) + Φ s (y c, y s, ε), ε = 0, where ε(1 + δ)ε 0, δ 1. With the centre manifold theorem [6], [10] it is demonstrated that for small y c and ε there is a local centre manifold which helps to express the stable variables y s as a function of the centre variables (y c, ε) such that: y s = h(y c, ε) where h is a function verifying h(0, 0) = 0 and Dh(0, 0) is a matrice with zero entries. Consequently, a reduced order system can be obtained from the system (4) as follows:
798 Evgeny V. Nikulchev and Alexander P. Kondratov { ẏc = A c ( ε 0 ) + Φ c (y c, h(y c ε), ε), ε = 0, A method for constructing central invariant manifolds is based on symmetric properties [11]. The method consists in finding admissible transformations in the reconstructed attractor heuristic methods [12]. It is known that in systems with chaotic dynamics Slobo symmetry breaking occurs. Thus a model of robust chaos may be a system that consists of a regular oscillation pattern and regular departures. The first is built on the basis of symmetric properties found in the geometrical method in the attractor. This is an essence of the developed modeling approach, based on the finding of weak symmetry breaking. In input time series, in output model in the form of finite differential equations: Calculated by numerical methods necessary conditions for the existence of chaos the largest Lyapunov exponent (chaotic dynamics must be greater than zero). Reconstruction attractor. The assumption of chaotic dynamics. Searched the symmetry group (in a weak symmetry violation) [11]. According to the adopted symmetries Hausdorff formula, we obtain the form of equations in a minimal invariant manifold [6]. Structure equations parametrically identified [11]. In accordance with the developed method were obtained chaotic models for a variety of practical applications. For example, it is known that the traffic network oblataet chaotic dynamics. On the basis of the chaos generator was obtained the following: x(t + 1) = Ax(t) + Ψ o (t), y = Cx,, A = 0.9413 0.1805 0.1164 0.0295 0.0545 0.8226 0.1622 0.1056 0.0014 0.0105 0.4455 0.8471 0.0062 0.0341 0.8860 0.5404
Generation of chaos in the centre manifold 799 Ψ 0 = 0.0399 0.0463 0.4848 0.1851 ( exp(t 0.0001 )sin(t 0.4) C = 10 4 [2.1037 0.01240.1202 0.0302]. This mapping is allowed to simulate the traffic management system and build channels of communication to ensure quality of service. 3 Conclusions This paper has proposed a new method to simplify the uncertainty propagation problem in nonlinear dynamic systems. It consists of central manifold theory combined with the polynomial chaos approach and symmetry theory. The first method helps to reduce a parameter dependent system in a Hopf bifurcation neighbourhood while the another allows in the analysis of nonlinear systems to be taken into account. Acknowledgements. This work was funded by RFBR, grant 15-08-08935. References [1] Z. Elhadj, J.C. Sprott, On the robustness of chaos in dynamical systems: Theories and applications, Frontiers of Physics in China, 3 (2008), 195-204. http://dx.doi.org/10.1007/s11467-008-0017-z [2] A. Arneodo, P.H. Coullet, E. A. Spiegel, The dynamics of triple convection, Geophysical and Astrophysical Fluid Dynamics, 31 (1985), 1-48. http://dx.doi.org/10.1080/03091928508219264 [3] M. Drutarovsk, P. Galajda, A robust chaos-based true random number generator embedded in reconfigurable switched-capacitor hardware, 17th International Conference Radioelektronika, (2007), 1-6. http://dx.doi.org/10.1109/radioelek.2007.371423 [4] M. Andrecut, M.K. Ali, Robust chaos in smooth unimodal maps, Physical Review E: statistical, nonlinear, and soft matter physics, 64 (2001), 025203. http://dx.doi.org/10.1103/physreve.64.025203 [5] M. Andrecut, M.K. Ali, Example of robust chaos in a smooth map, Europhys. Lett., 54 (2003), 300-305. http://dx.doi.org/10.1209/epl/i2001-00241-3
800 Evgeny V. Nikulchev and Alexander P. Kondratov [6] E.V. Nikulchev, Modeling technology of complex and chaotic processes permitting symmetry groups, Avtomatizatsiya i Sovremennye Tekhnologii, 11 (2004), 29-33. [7] J.M. Aguirregabiri, Robust chaos with variable Lyapunov exponent in smooth one-dimensional maps, Chaos, Solitons and Fractals, 42 (2009), 2531-2539. http://dx.doi.org/10.1016/j.chaos.2009.03.196 [8] L. Nechak, S. Berger, E. Aubry, Robust Analysis of Uncertain Dynamic Systems: Combination of the Centre Manifold and Polynomial Chaos theories, WSEAS Transactions on Systems, 9 (2010), 386-395. [9] H.W. Knoblock, Construction of Center Manifolds, Journal of Applied Mathematics and Mechanics, 70 (1990), 215-233. http://dx.doi.org/10.1002/zamm.19900700702 [10] J.J. Sinou, L. Jezequel, F. Thouverez, Methods to reduce Non-Linear Mechanical Systems for instability Computation, Archive of Computational Methods in Enginnering, 11 (2004), 257-344. http://dx.doi.org/10.1007/bf02736228 [11] E. Nikulchev, Robust chaos generation on the basis of symmetry violations in attractors, 2nd International Conference on Emission Electronics (ICEE), (2014), 59-61. http://dx.doi.org/10.1109/emission.2014.6893972 [12] E.V. Nikulchev, Simulation of robust chaotic signal with given properties, Advanced Studies in Theoretical Physics, 8 (2014), 939-944. http://dx.doi.org/10.12988/astp.2014.48106 Received: October 15, 2015; Published: December 2, 2015