How to plot bifurcation diagram in python

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How to plot bifurcation diagram in python

In [ ]: def bifurcation_plot(f,f_x,r,x,rlabel='r'): """ produce a bifurcation diagram for a function f(r,x) given f and its partial derivative f_x(r,x) over a domain given by numpy arrays r and x f(r,x) : RHS function of autonomous ode dx/dt = f(r,x) f_x(r,x): partial derivative of f with respect to x r : numpy array giving r coordinates of domain x : numpy array giving x coordinates of domain rlabel : string for x axis

parameter label """ # set up a mesh grid and extract the 0 level set of f R,X = numpy.meshgrid(r,x) plt.figure() CS = plt.contour(R,X,f(R,X),[0],colors='k') plt.clf() c0 = CS.collections[0] # for each path in the contour extract vertices and mask by the sign of df/dx for path in c0.get_paths(): vertices = path.vertices vr = vertices[:,0] vx = vertices[:,1] mask = numpy.sign(f_x(vr,vx)) stable = mask < 0. unstable =

mask > 0. # plot the stable and unstable branches for each path plt.plot(vr[stable],vx[stable],'b') plt.hold(True) plt.plot(vr[unstable],vx[unstable],'b--') plt.xlabel('parameter {0}'.format(rlabel)) plt.ylabel('x') plt.legend(('stable','unstable'),loc='best') plt.xlim(r[0],r[-1]) plt.ylim(x[0],x[-1]) Next: Three Dimensional Plotting Tool Up: Running AUTO using Python Previous: The .autorc File Contents The two dimensional

plotting tool can be run by using the command plot() to plot the files fort.7 and fort.8 after a calculation has been run, or using the command plot('foo') to plote the data in the files s.foo and b.foo. The menu bar provides two buttons. The File button brings up a menu which allows the user to save the current plot as a Postscript file or to quit the plotter. The Options button allows the plotter configuration

options to be modified. The available options are decribed in Table 4.7. In addition, the options can be set from within the CLUI. For example, the set of commands in Figure 4.16 shows how to create a plotter and change its background color to black. The demo script auto/07p/demo/python/plotter.py contains several examples of changing options in plotters. Pressing the right mouse button in the plotting

window brings up a menu of buttons which control several aspects of the plotting window. The top two toggle buttons control what function the left button performs. The print value button causes the left button to print out the numerical value underneath the pointer when it is clicked. When zoom button is checked the left mouse button may be held down to create a box in the plot. When the left button is

released the plot will zoom to the selected portion of the diagram. The unzoom button returns the diagram to the default zoom. The Postscript button allows the user to save the plot as a Postscript file. The Configure... button brings up the dialog for setting configuration options. The options for the AUTO CLUI two dimensional plotting window. Query string Meaning background The background color of the

plot. bifurcation_column_defaults A set of bifurcation columns the user is likely to use. bifurcation_diagram A parsed bifurcation diagram file to plot. bifurcation_diagram_filename The filename of the bifurcation diagram to plot. bifurcation_symbol The symbol to use for bifurcation points. bifurcation_x The column to plot along the X-axis for bifurcation diagrams. bifurcation_y The column to plot along the Yaxis for bifurcation diagrams. color_list A list of colors to use for multiple plots. decorations Turn on or off the axis, tick marks, etc. error_symbol The symbol to use for error points. foreground The background color of the plot. grid Turn on or off the grid. hopf_symbol The symbol to use for Hopf bifurcation points. index An array of indicies to plot. label An array of labels to plot. label_defaults A set of labels

that the user is likely to use. limit_point_symbol The symbol to use for limit points. mark_t The t value to marker with a small ball. maxx The upper bound for the x-axis of the plot. maxy The upper bound for the y-axis of the plot. minx The lower bound for the x-axis of the plot. miny The lower bound for the y-axis of the plot. period_doubling_symbol The symbol to use for period doubling bifurcation points.

ps_colormode The PostScript output mode: 'color', 'gray' or 'monochrome'. runner The runner object from which to get data. special_point_colors An array of colors used to mark special points. special_point_radius The radius of the spheres used to mark special points. solution A parsed solution file to plot. solution_column_defaults A set of solution columns the user is likely to use. solution_filename The

filename of the solution to plot. solution_x The column to plot along the X-axis for solutions. solution_y The column to plot along the Y-axis for solutions. symbol_font The font to use for marker symbols. symbol_color The color to use for the marker symbols. tick_label_template A string which defines the format of the tick labels. tick_length The length of the tick marks. torus_symbol The symbol to use for

torus bifurcation points. type The type of the plot, either ``solution'' or ``bifurcation''. use_labels Whether or not to display label numbers in the graph. user_point_symbol The symbol to use for user defined output points. xlabel The label for the x-axis. xmargin The margin between the graph and the right and left edges. xticks The number of ticks on the x-axis. ylabel The label for the y-axis. ymargin The

margin between the graph and the top and bottom edges. yticks The number of ticks on the y-axis. Next: Three Dimensional Plotting Tool Up: Running AUTO using Python Previous: The .autorc File Contents Gabriel Lord 2007-11-19 This is one of the 100+ free recipes of the IPython Cookbook, Second Edition, by Cyrille Rossant, a guide to numerical computing and data science in the Jupyter

Notebook. The ebook and printed book are available for purchase at Packt Publishing. ? Text on GitHub with a CC-BY-NC-ND license ? Code on GitHub with a MIT license ? Go to Chapter 12 : Deterministic Dynamical Systems ? Get the Jupyter notebook A chaotic dynamical system is highly sensitive to initial conditions; small perturbations at any given time yield completely different trajectories. The

trajectories of a chaotic system tend to have complex and unpredictable behaviors. Many real-world phenomena are chaotic, particularly those that involve nonlinear interactions among many agents (complex systems). Examples can be found in meteorology, economics, biology, and other disciplines. In this recipe, we will simulate a famous chaotic system: the logistic map. This is an archetypal example of

how chaos can arise from a very simple nonlinear equation. The logistic map models the evolution of a population, taking into account both reproduction and density-dependent mortality (starvation). We will draw the system's bifurcation diagram, which shows the possible long-term behaviors (equilibria, fixed points, periodic orbits, and chaotic trajectories) as a function of the system's parameter. We will

also compute an approximation of the system's Lyapunov exponent, characterizing the model's sensitivity to initial conditions. How to do it... 1. Let's import NumPy and matplotlib: import numpy as np import matplotlib.pyplot as plt %matplotlib inline 2. We define the logistic function by: Here is the implementation of this function in Python: def logistic(r, x): return r * x * (1 - x) 3. Here is a graphic

representation of this function x = np.linspace(0, 1) fig, ax = plt.subplots(1, 1) ax.plot(x, logistic(2, x), 'k') 4. Our discrete dynamical system is defined by the recursive application of the logistic function: $$x_{n+1}^{(r)} = f_r(x_n^{(r)}) = rx_n^{(r)}(1-x_n^{(r)})$$ Let's simulate a few iterations of this system with two different values of \(r\): def plot_system(r, x0, n, ax=None): # Plot the function and the # y=x

diagonal line. t = np.linspace(0, 1) ax.plot(t, logistic(r, t), 'k', lw=2) ax.plot([0, 1], [0, 1], 'k', lw=2) # Recursively apply y=f(x) and plot two lines: # (x, x) -> (x, y) # (x, y) -> (y, y) x = x0 for i in range(n): y = logistic(r, x) # Plot the two lines. ax.plot([x, x], [x, y], 'k', lw=1) ax.plot([x, y], [y, y], 'k', lw=1) # Plot the positions with increasing # opacity. ax.plot([x], [y], 'ok', ms=10, alpha=(i + 1) / n) x = y ax.set_xlim(0, 1)

ax.set_ylim(0, 1) ax.set_title(f"$r={r:.1f}, \, x_0={x0:.1f}$") fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 6), sharey=True) plot_system(2.5, .1, 10, ax=ax1) plot_system(3.5, .1, 10, ax=ax2) On the left panel, we can see that our system converges to the intersection point of the curve and the diagonal line (fixed point). On the right panel however, using a different value for \(r\), we observe a seemingly chaotic

behavior of the system. 5. Now, we simulate this system for 10000 values of \(r\) linearly spaced between 2.5 and 4, and vectorize the simulation with NumPy by considering a vector of independent systems (one dynamical system per parameter value): n = 10000 r = np.linspace(2.5, 4.0, n) 6. We use 1000 iterations of the logistic map and keep the last 100 iterations to display the bifurcation diagram:

iterations = 1000 last = 100 7. We initialize our system with the same initial condition \(x_0 = 0.00001\): 8. We also compute an approximation of the Lyapunov exponent for every value of \(r\). The Lyapunov exponent is defined by: $$\lambda(r) = \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} \log\left| \frac{df_r}{dx}\left(x_i^{(r)}\right) \right|$$ We first initialize the lyapunov vector: 9. Now, we simulate the

system and plot the bifurcation diagram. The simulation only involves the iterative evaluation of the logistic() function on our vector x. Then, to display the bifurcation diagram, we draw one pixel per point \(x_n^{(r)}\) during the last 100 iterations: fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 9), sharex=True) for i in range(iterations): x = logistic(r, x) # We compute the partial sum of the # Lyapunov exponent.

lyapunov += np.log(abs(r - 2 * r * x)) # We display the bifurcation diagram. if i >= (iterations - last): ax1.plot(r, x, ',k', alpha=.25) ax1.set_xlim(2.5, 4) ax1.set_title("Bifurcation diagram") # We display the Lyapunov exponent. # Horizontal line. ax2.axhline(0, color='k', lw=.5, alpha=.5) # Negative Lyapunov exponent. ax2.plot(r[lyapunov < 0], lyapunov[lyapunov < 0] / iterations, '.k', alpha=.5, ms=.5) # Positive

Lyapunov exponent. ax2.plot(r[lyapunov >= 0], lyapunov[lyapunov >= 0] / iterations, '.r', alpha=.5, ms=.5) ax2.set_xlim(2.5, 4) ax2.set_ylim(-2, 1) ax2.set_title("Lyapunov exponent") plt.tight_layout() The bifurcation diagram brings out the existence of a fixed point for \(r

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