Bifurcations

  • 1D flows seem to have one of two behaviour

    • Settle down to an equilibrium point
    • Fly away to infinity

  • Apart from this we may have a behaviour where

    • Fixed points are created/destroyed
    • Stability of these points are changed

We will study here the bifurcations of flows on a 1D line Let's consider the first example: $\dot{x} = r + x^2$

In [1]:
import numpy as np;
import matplotlib.pyplot as plt;
plt.rcParams.update({"text.usetex":True});
%config InlineBackend.figure_format = "svg"
from ipywidgets import interactive
from scipy.optimize import fsolve
In [2]:
def see_fixed_pts(r=1):
    x = np.linspace(-2, 2);
    xd = r + x**2;
    plt.plot(x, xd);
    plt.plot(x, 0*x, '-k');

w = interactive(see_fixed_pts, r = (-2, 2, 0.2));
w

We can then plot the equilibrium points as the parameter r is varied. Along with that we can also plot whether that particular point is a stable or unstable point. This gives a clear identification of the nature of the roots.

In [3]:
def f(x, r):
    return r + x**2;

def dfdx(x, r):
    return 2*x;

x = np.linspace(-2, 2);
r_a = np.linspace(-2, 2);

for r in r_a:
    sol = fsolve(f, [-2, 2], args=(r), full_output=True);
    if sol[2] == 1: # Solution has convered
        for i in np.arange(0, np.size(sol[0])):
            root = sol[0][i];
            slope_at_root = dfdx(root, r)
            if slope_at_root > 0:
                plt.plot(r, root, 'bx')
            else:
                plt.plot(r, root, 'ro')
ax = plt.gca(); ax.set_aspect(1);
plt.xlim(np.min(r_a), np.max(r_a))
plt.axhline(0); plt.axvline(0)
plt.xlabel("r");
plt.ylabel("x*");
plt.title("Saddle node bifurcation")
Out[3]:
Text(0.5, 1.0, 'Saddle node bifurcation')

The above diagram clearly elucidates how increasing the parameter r beyond 0 yields no equilibrium point while for $r<0$ we have one stable and one unstable branch. This is an example of a saddle node bifurcation.

In [4]:
slope_at_root = dfdx(sol[0][0], r); print(slope_at_root)

0.00018740347684758447

In the example below, we show the same saddle node bifurcation in a slightly distored form. The normal form of the equation below, i.e. $\dot{x} = r - x - \exp{-x}$ is of the same form as $\dot{x} = r + x^2$

In [5]:
def f(x, r):
    return r - x - np.exp(-x);

def dfdx(x, r):
    return -1 + np.exp(-x);

x = np.linspace(-2, 2);
r_a = np.linspace(-2, 2, 200);

for r in r_a:
    sol = fsolve(f, [-2, 2], args=(r), full_output=True);
    if sol[2] == 1: # Solution has convered
        for i in np.arange(0, np.size(sol[0])):
            root = sol[0][i];
            slope_at_root = dfdx(root, r)
            if slope_at_root > 0:
                plt.plot(r, root, 'bx')
            else:
                plt.plot(r, root, 'ro')
ax = plt.gca(); ax.set_aspect(1);
plt.xlim(np.min(r_a), np.max(r_a))
plt.axhline(0); plt.axvline(0)
plt.xlabel("r");
plt.ylabel("x*");
plt.title("Saddle node bifurcation")
plt.text(0.5, -1.2, "Unstable branch")
plt.text(0.5, 1.2, "Stable branch")
plt.axvline(1)
Out[5]:
<matplotlib.lines.Line2D at 0x1aabb5586a0>
In [ ]: