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- /*
- * 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;
- }
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