When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

Abstract

In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using time delay embedding techniques. The MGGP algorithm tries to search the functional space of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of the evolved attractors. To demonstrate the potential of the proposed methodology, we report over one hundred new chaotic attractor structures along with their parameters, which are evolved from just the Lorenz system alone. Introduction:An important research theme in non-linear dynamics is to identify sets of differential equations along with their parameters which give rise to chaos. Starting from the advent of Lorenz attractors in three dimensional nonlinear differential equations [1][2], its several other family of attractors have been developed like Rossler, Rucklidge, Chen, Lu, Liu, Sprott, Genesio-Tesi, Shimizu-Morioka etc. [3]. Extension of the basic attractors to four or even higher dimensional systems has resulted in a similar family of hyper-chaotic systems [4]. In addition, several other structures like multi-scroll [5] and multi-wing [6] versions of the Lorenz family of attractors have also been developed by increasing the number of equilibrium points. Development of new chaotic attractors has huge application especially in data encryption, secure communication [7] etc. and in understanding the dynamics of many real world systems whose governing equations match with the template of these chaotic systems [8]. There has been several research reports on the application of master-slave chaos synchronization in secure communication [9], where the fresh set of chaotic attractors can play a big role due to their rich phase space dynamics. Here we explore the potential of automatic generation of chaotic attractors which is developed from the basic three dimensional structure of the Lorenz system.We use the Lyapunov exponent -the most popular signature of chaos, to judge whether a computer generated arbitrary nonlinear structure of third order differential equation along with some chosen parameters exhibits chaotic motion in the phase space. Since there could be chaotic behaviour or complex limit cycles or even stable/unstable motions in the phase space, for unknown mathematical expressions of similar third order dynamical system, the computationally tractable way for the 2. Genetic programming to evolve new chaotic attractors 2.

Evolving chaos: Identifying new attractors of the generalised Lorenz family

“…With such an aim, even sub-optimal intermediate solutions with slightly positive Lyapunov exponents may show rich phase space dynamics as the complexity does not necessarily always correlate with high LLE. This is the reason why we here report even the intermediate search results with LLE > 0, similar to the earlier explorations in [17], during the evolutionary search process and not only at the final converged results of GP algorithm, which has been misinterpreted in Gao et al [31]. Also, in this paper, we have listed the newly found chaotic system as the GP search progressed and not using the calculated LLE.…”

“…Recently Gao et al [31] claimed to improve over the GP based chaos evolution results in [17] for designing chaotic system corresponding to the global optima. However, it is worth noting that even local minima found by the evolutionary search with different expressions and dynamical behaviours can indeed be useful to discover new gallery of chaotic systems.…”

“…In the GP algorithm, the ratio of mathematical operators to the numeric values are kept as 0.5 as also reported in [17]. This enables the GP algorithm to explore new mathematical expressions, rather than searching different constants within the same structure which could have been easily done with simpler versions like genetic algorithms and other evolutionary optimizers for fixed structure parameter optimization problems.…”

“…Single and multigene GP have been used to search for the chaotic system expressions as reported in our previous exploration [17]. The single gene GP helps evolving a single state equation while keeping the other two as that of the Lorenz attractor, whereas the multigene GP simultaneously evolves two or all the three state equations to find a completely new chaotic system compared to the structures of the classical Lorenz system [3].…”

“…Our previous study in [17] report more than 100 chaotic attractors having at least a single sinusoidal nonlinearity in one of the state equations. It was also argued in [17] that the rich phase space dynamics may be an effect of increased number of equilibrium points due to the transcendental terms e.g. sinusoidal function of the state variables.…”