Bifurcations in First-Order ODEs

ODE

saddle

transcritical

supercritical pitchfork

subcritical pitchfork

parameter r

-1

t

max

5

initial states

7

A first-order autonomous ordinary differential equation (ODE) with a parameter

r

has the general form

dx/dt=f(x,r)

. The fixed points are the values of

x

for which

f(x,r)=0

. A bifurcation occurs when the number or the stability of the fixed points changes as system parameters change. The classical types of bifurcations that occur in nonlinear dynamical systems are produced from the following prototypical differential equations:

saddle:

dx/dt=r+

2

x

transcritical:

dx/dt=rx-

2

x

supercritical pitchfork:

dx/dt=rx-

3

x

subcritical pitchfork:

dx/dt=rx+

3

x

This Demonstration shows bifurcations of these nonlinear first-order ODEs as you vary the parameter

r

.

The top figure shows the phase portrait,

dx/dt

versus

x

, with stable fixed points indicated by solid disks and unstable fixed points as open circles. The bottom figure shows the solutions,

x(t)

versus

t

, starting from a number of initial states with stable fixed points indicated by solid lines and unstable fixed points as dashed lines. You can vary both the number of initial states and time duration,

t

max

. Note how the fixed points and solutions change as bifurcations occur as you vary the parameter

r

.