# Weakly Coupled Distributed Calculation of Lyapunov Exponents for Non-Linear Dynamical Systems
## 1 Introduction  
Although the study of non-linear differential equations began with the rise of differential equations themselves, formally, the birth of the modern field of non-linear dynamical systems began in 1962 when [PERSON], a MIT meteorologist, computationally simulated a set of differential equations for fluid convection in the atmosphere. In such simulations, he noticed a complicated behaviour that seemed to depend sensitively on initial conditions, and here he found the \"Lorenz\" strange attractor. The field of non-linear dynamical systems, particularly chaos theory, is a very active research field today, and its applications cover huge areas of science as well as other disciplines.  
Nowadays, it is widely accepted that if a system of differential equations possesses significant dependence on initial conditions, then its behaviour is catalogued as chaotic [1]. The most popular tools to measure sensitivity dependence on initial conditions are the Lyapunov exponents (LE), named after the Russian mathematician [PERSON] (1857-1918) [2]. As Lyapunov exponents give a measure of the separation of closely adjacent solutions (in initial conditions) to the set of differential equations when the system has evolved into a steady state (after a very long time), their numerical calculation has always led to high processing times. Moreover, the smaller the step in which possible initial conditionsare swept and the smaller the time step of the simulation as well as the larger the final simulation time, the better the obtained approach to the theoretical Lyapunov exponent.  
The numerical processing time problem is aggravated when the set of ODEs (Ordinary Differential Equations) is large, because for each degree of freedom, a Lyapunov exponent ought to be calculated. Historically, in order to alleviate computational load, different approximation techniques have been developed. The most popular is the calculation of the maximal Lyapunov exponent, which, if negative, guarantees that the system is not chaotic. Several methods to estimate the maximal Lyapunov exponent have been developed [3; 4; 5; 6].  
Nevertheless, the calculation of the maximal Lyapunov exponent only serves as a guide to detect chaos--or not--in a system. In order to fully exploit the richness of the non-linear dynamical behaviour embodied in the set of ODEs, the full Lyapunov spectrum is required. This way, the calculation of Lyapunov exponents is a high-computational-load problem that is benefited by parallelisation techniques. Notwithstanding, the parallelisation might be tricky because the dependence of the adjacent initial conditions solutions of the system in order to calculate each exponent of the spectra. Thus, in this work we present the weakly coupled distributed calculation of Lyapunov exponents in a high latency cluster. This paper is organized as follows: In Section 2 we present the theory of Lyapunov exponents as well as the system of differential equations to be tackled in this paper. In Section 3 we describe the design of the application, focusing in the MPI-distributed implementation for a high-latency cluster. In Section 4 we describe the performance experiments along with their results. Finally, in Section 6 we present some interesting final remarks.