/* * chaotic_system.cpp * * This example demonstrates how one can use odeint to determine the Lyapunov * exponents of a chaotic system namely the well known Lorenz system. Furthermore, * it shows how odeint interacts with boost.range. * * Copyright 2011-2012 Karsten Ahnert * Copyright 2011-2013 Mario Mulansky * * Distributed under the Boost Software License, Version 1.0. * (See accompanying file LICENSE_1_0.txt or * copy at http://www.boost.org/LICENSE_1_0.txt) */ #include <iostream> #include <boost/array.hpp> #include <boost/numeric/odeint.hpp> #include "gram_schmidt.hpp" using namespace std; using namespace boost::numeric::odeint; const double sigma = 10.0; const double R = 28.0; const double b = 8.0 / 3.0; //[ system_function_without_perturbations struct lorenz { template< class State , class Deriv > void operator()( const State &x_ , Deriv &dxdt_ , double t ) const { typename boost::range_iterator< const State >::type x = boost::begin( x_ ); typename boost::range_iterator< Deriv >::type dxdt = boost::begin( dxdt_ ); dxdt[0] = sigma * ( x[1] - x[0] ); dxdt[1] = R * x[0] - x[1] - x[0] * x[2]; dxdt[2] = -b * x[2] + x[0] * x[1]; } }; //] //[ system_function_with_perturbations const size_t n = 3; const size_t num_of_lyap = 3; const size_t N = n + n*num_of_lyap; typedef boost::array< double , N > state_type; typedef boost::array< double , num_of_lyap > lyap_type; void lorenz_with_lyap( const state_type &x , state_type &dxdt , double t ) { lorenz()( x , dxdt , t ); for( size_t l=0 ; l<num_of_lyap ; ++l ) { const double *pert = x.begin() + 3 + l * 3; double *dpert = dxdt.begin() + 3 + l * 3; dpert[0] = - sigma * pert[0] + 10.0 * pert[1]; dpert[1] = ( R - x[2] ) * pert[0] - pert[1] - x[0] * pert[2]; dpert[2] = x[1] * pert[0] + x[0] * pert[1] - b * pert[2]; } } //] int main( int argc , char **argv ) { state_type x; lyap_type lyap; fill( x.begin() , x.end() , 0.0 ); x[0] = 10.0 ; x[1] = 10.0 ; x[2] = 5.0; const double dt = 0.01; //[ integrate_transients_with_range // explicitly choose range_algebra to override default choice of array_algebra runge_kutta4< state_type , double , state_type , double , range_algebra > rk4; // perform 10000 transient steps integrate_n_steps( rk4 , lorenz() , std::make_pair( x.begin() , x.begin() + n ) , 0.0 , dt , 10000 ); //] //[ lyapunov_full_code fill( x.begin()+n , x.end() , 0.0 ); for( size_t i=0 ; i<num_of_lyap ; ++i ) x[n+n*i+i] = 1.0; fill( lyap.begin() , lyap.end() , 0.0 ); double t = 0.0; size_t count = 0; while( true ) { t = integrate_n_steps( rk4 , lorenz_with_lyap , x , t , dt , 100 ); gram_schmidt< num_of_lyap >( x , lyap , n ); ++count; if( !(count % 100000) ) { cout << t; for( size_t i=0 ; i<num_of_lyap ; ++i ) cout << "\t" << lyap[i] / t ; cout << endl; } } //] return 0; }