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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_numeric_odeint.tutorial.chaotic_systems_and_lyapunov_exponents"></a><a class="link" href="chaotic_systems_and_lyapunov_exponents.html" title="Chaotic systems and Lyapunov exponents">Chaotic |
| systems and Lyapunov exponents</a> |
| </h3></div></div></div> |
| <p> |
| In this example we present application of odeint to investigation of the |
| properties of chaotic deterministic systems. In mathematical terms chaotic |
| refers to an exponential growth of perturbations <span class="emphasis"><em>δ x</em></span>. |
| In order to observe this exponential growth one usually solves the equations |
| for the tangential dynamics which is again an ordinary differential equation. |
| These equations are linear but time dependent and can be obtained via |
| </p> |
| <p> |
| <span class="emphasis"><em>d δ x / dt = J(x) δ x</em></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>J</em></span> is the Jacobian of the system under consideration. |
| <span class="emphasis"><em>δ x</em></span> can also be interpreted as a perturbation of the original |
| system. In principle <span class="emphasis"><em>n</em></span> of these perturbations exist, |
| they form a hypercube and evolve in the time. The Lyapunov exponents are |
| then defined as logarithmic growth rates of the perturbations. If one Lyapunov |
| exponent is larger then zero the nearby trajectories diverge exponentially |
| hence they are chaotic. If the largest Lyapunov exponent is zero one is usually |
| faced with periodic motion. In the case of a largest Lyapunov exponent smaller |
| then zero convergence to a fixed point is expected. More information's about |
| Lyapunov exponents and nonlinear dynamical systems can be found in many textbooks, |
| see for example: E. Ott "Chaos is Dynamical Systems", Cambridge. |
| </p> |
| <p> |
| To calculate the Lyapunov exponents numerically one usually solves the equations |
| of motion for <span class="emphasis"><em>n</em></span> perturbations and orthonormalizes them |
| every <span class="emphasis"><em>k</em></span> steps. The Lyapunov exponent is the average |
| of the logarithm of the stretching factor of each perturbation. |
| </p> |
| <p> |
| To demonstrate how one can use odeint to determine the Lyapunov exponents |
| we choose the Lorenz system. It is one of the most studied dynamical systems |
| in the nonlinear dynamics community. For the standard parameters it possesses |
| a strange attractor with non-integer dimension. The Lyapunov exponents take |
| values of approximately 0.9, 0 and -12. |
| </p> |
| <p> |
| The implementation of the Lorenz system is |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">10.0</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">R</span> <span class="special">=</span> <span class="number">28.0</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">8.0</span> <span class="special">/</span> <span class="number">3.0</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="number">3</span> <span class="special">></span> <span class="identifier">lorenz_state_type</span><span class="special">;</span> |
| |
| <span class="keyword">void</span> <span class="identifier">lorenz</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">lorenz_state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">lorenz_state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">);</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">];</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| We need also to integrate the set of the perturbations. This is done in parallel |
| to the original system, hence within one system function. Of course, we want |
| to use the above definition of the Lorenz system, hence the definition of |
| the system function including the Lorenz system itself and the perturbation |
| could look like: |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">num_of_lyap</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">N</span> <span class="special">=</span> <span class="identifier">n</span> <span class="special">+</span> <span class="identifier">n</span><span class="special">*</span><span class="identifier">num_of_lyap</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">num_of_lyap</span> <span class="special">></span> <span class="identifier">lyap_type</span><span class="special">;</span> |
| |
| <span class="keyword">void</span> <span class="identifier">lorenz_with_lyap</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">lorenz</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">);</span> |
| |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">l</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">l</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">l</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">pert</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">3</span> <span class="special">+</span> <span class="identifier">l</span> <span class="special">*</span> <span class="number">3</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="special">*</span><span class="identifier">dpert</span> <span class="special">=</span> <span class="identifier">dxdt</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">3</span> <span class="special">+</span> <span class="identifier">l</span> <span class="special">*</span> <span class="number">3</span><span class="special">;</span> |
| <span class="identifier">dpert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">+</span> <span class="number">10.0</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">1</span><span class="special">];</span> |
| <span class="identifier">dpert</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="special">(</span> <span class="identifier">R</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">2</span><span class="special">];</span> |
| <span class="identifier">dpert</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">b</span> <span class="special">*</span> <span class="identifier">pert</span><span class="special">[</span><span class="number">2</span><span class="special">];</span> |
| <span class="special">}</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| The perturbations are stored linearly in the <code class="computeroutput"><span class="identifier">state_type</span></code> |
| behind the state of the Lorenz system. The problem of lorenz() and lorenz_with_lyap() having different |
| state types may be solved putting the Lorenz system inside a functor with |
| templatized arguments: |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span> |
| <span class="special">{</span> |
| <span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">StateIn</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">StateOut</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Value</span> <span class="special">></span> |
| <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">StateIn</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">StateOut</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">Value</span> <span class="identifier">t</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">);</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">];</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span> |
| <span class="special">}</span> |
| <span class="special">};</span> |
| |
| <span class="keyword">void</span> <span class="identifier">lorenz_with_lyap</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">lorenz</span><span class="special">()(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">);</span> |
| <span class="special">...</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| This works fine and <code class="computeroutput"><span class="identifier">lorenz_with_lyap</span></code> |
| can be used for example via |
| </p> |
| <pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">;</span> |
| <span class="comment">// initialize x..</span> |
| |
| <span class="identifier">explicit_rk4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk4</span><span class="special">;</span> |
| <span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz_with_lyap</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span> <span class="number">1000</span> <span class="special">);</span> |
| </pre> |
| <p> |
| This code snippet performs 1000 steps with constant step size 0.01. |
| </p> |
| <p> |
| A real world use case for the calculation of the Lyapunov exponents of Lorenz |
| system would always include some transient steps, just to ensure that the |
| current state lies on the attractor, hence it would look like |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">;</span> |
| <span class="comment">// initialize x</span> |
| <span class="identifier">explicit_rk4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="identifier">rk4</span><span class="special">;</span> |
| <span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">0.01</span> <span class="special">,</span> <span class="number">1000</span> <span class="special">);</span> |
| </pre> |
| <p> |
| The problem is now, that <code class="computeroutput"><span class="identifier">x</span></code> |
| is the full state containing also the perturbations and <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> |
| does not know that it should only use 3 elements. In detail, odeint and its |
| steppers determine the length of the system under consideration by determining |
| the length of the state. In the classical solvers, e.g. from Numerical Recipes, |
| the problem was solved by pointer to the state and an appropriate length, |
| something similar to |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">lorenz</span><span class="special">(</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">*</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span><span class="special">,</span> <span class="keyword">void</span><span class="special">*</span> <span class="identifier">params</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="special">...</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">int</span> <span class="identifier">system_length</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span> |
| <span class="identifier">rk4</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">system_length</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">lorenz</span> <span class="special">);</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| But odeint supports a similar and much more sophisticated concept: <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a>. |
| To make the steppers and the system ready to work with <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a> |
| the system has to be changed: |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">lorenz</span> |
| <span class="special">{</span> |
| <span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Deriv</span> <span class="special">></span> |
| <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x_</span> <span class="special">,</span> <span class="identifier">Deriv</span> <span class="special">&</span><span class="identifier">dxdt_</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span> |
| <span class="special">{</span> |
| <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">x_</span> <span class="special">);</span> |
| <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">range_iterator</span><span class="special"><</span> <span class="identifier">Deriv</span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">dxdt</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">begin</span><span class="special">(</span> <span class="identifier">dxdt_</span> <span class="special">);</span> |
| |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">);</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">];</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span> |
| <span class="special">}</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| This is in principle all. Now, we only have to call <code class="computeroutput"><span class="identifier">integrate_n_steps</span></code> |
| with a range including only the first 3 components of <span class="emphasis"><em>x</em></span>: |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="comment">// explicitly choose range_algebra to override default choice of array_algebra</span> |
| <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">range_algebra</span> <span class="special">></span> <span class="identifier">rk4</span><span class="special">;</span> |
| |
| <span class="comment">// perform 10000 transient steps</span> |
| <span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="identifier">n</span> <span class="special">)</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="number">10000</span> <span class="special">);</span> |
| </pre> |
| <p> |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| Note that when using <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a>, |
| we have to explicitly configure the stepper to use the <code class="computeroutput"><span class="identifier">range_algebra</span></code> |
| as otherwise odeint would automatically chose the <code class="computeroutput"><span class="identifier">array_algebra</span></code>, |
| which is incompatible with the usage of <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a>, |
| because the original state_type is an <code class="computeroutput"><span class="identifier">array</span></code>. |
| </p></td></tr> |
| </table></div> |
| <p> |
| Having integrated a sufficient number of transients steps we are now able |
| to calculate the Lyapunov exponents: |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| Initialize the perturbations. They are stored linearly behind the state |
| of the Lorenz system. The perturbations are initialized such that <span class="emphasis"><em>p |
| <sub>​ij</sub> = δ <sub>​ij</sub></em></span>, where <span class="emphasis"><em>p <sub>​ij</sub></em></span> is the <span class="emphasis"><em>j</em></span>-component |
| of the <span class="emphasis"><em>i</em></span>.-th perturbation and <span class="emphasis"><em>δ <sub>​ij</sub></em></span> |
| is the Kronecker symbol. |
| </li> |
| <li class="listitem"> |
| Integrate 100 steps of the full system with perturbations |
| </li> |
| <li class="listitem"> |
| Orthonormalize the perturbation using Gram-Schmidt orthonormalization |
| algorithm. |
| </li> |
| <li class="listitem"> |
| Repeat step 2 and 3. Every 10000 steps write the current Lyapunov exponent. |
| </li> |
| </ol></div> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="identifier">fill</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()+</span><span class="identifier">n</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span> |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">n</span><span class="special">+</span><span class="identifier">n</span><span class="special">*</span><span class="identifier">i</span><span class="special">+</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> |
| <span class="identifier">fill</span><span class="special">(</span> <span class="identifier">lyap</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">lyap</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span> |
| |
| <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> |
| <span class="identifier">size_t</span> <span class="identifier">count</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> |
| <span class="keyword">while</span><span class="special">(</span> <span class="keyword">true</span> <span class="special">)</span> |
| <span class="special">{</span> |
| |
| <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">integrate_n_steps</span><span class="special">(</span> <span class="identifier">rk4</span> <span class="special">,</span> <span class="identifier">lorenz_with_lyap</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="number">100</span> <span class="special">);</span> |
| <span class="identifier">gram_schmidt</span><span class="special"><</span> <span class="identifier">num_of_lyap</span> <span class="special">>(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">lyap</span> <span class="special">,</span> <span class="identifier">n</span> <span class="special">);</span> |
| <span class="special">++</span><span class="identifier">count</span><span class="special">;</span> |
| |
| <span class="keyword">if</span><span class="special">(</span> <span class="special">!(</span><span class="identifier">count</span> <span class="special">%</span> <span class="number">100000</span><span class="special">)</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">t</span><span class="special">;</span> |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">num_of_lyap</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">lyap</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="identifier">t</span> <span class="special">;</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| The full code can be found here: <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/chaotic_system.cpp" target="_top">chaotic_system.cpp</a> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2009-2012 Karsten |
| Ahnert and Mario Mulansky<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
| </tr></table> |
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