Phase-plane analysis
dx
d
or  , , etc.)
dt
dt
System state is the coordinate on its phase plane.
Phase plane: ( x ,
dx
dt
System trajectory
On phase-space
x
A system may traverse any one of the (potentially) infinite
trajectories. Initial conditions decide which way it’d
evolve.
A
dx
dt
B
x
A system is at a stable state if all its trajectories converge at
that state (equilibrium) state. This means all trajectories
form a closed bundle for all neighborhood points.
dx
 0 . On the
dt
phase-space below for a system, its equilibrium points are
shown.
An equilibrium state is that state where its
dx
dt
Equilibrium points
of a system
x
Equilibrium points are:
Stable equilibrium: System tends to come back here after
a small perturbation.
Unstable equilibrium: System tends to diverge away from
this point after a small perturbation.
Saddle-point: Exhibits both stable and unstable
characteristics.
Reality. Everything should be stable to qualify as an
observable.
Consider the meta-logistic system with two equilibrium
points.
dp
p
p
 a(  1)(1  ) Equilibrium points are:
dt
k
m
p  k , p  m . Assume k  m
Its phase diagram shows the stable and unstable
equilibrium points.
k
m
unstable
At a stable equilibrium:
stable
d2p
dt
At an unstable equilibrium:
At a saddle-point:
d2p
dt
2
negative
2
d2p
dt
2
positive
is zero
Using Taylor’s series, indicate the rationale.
Another case.
Examples of phase-space around unstable equilibriums.
Another case.
When neither attracted nor repelled, the equilibrium points
exhibit limit cycles (oscillatory systems around the
equilibrium points)
Another interesting case is the saddle point states. Here
trajectories both enter and leave an equilibrium point.
Examples.
The system is given by
d2y
dt 2

dy
 y  y2  0
dt
Let x1  y and x2 
dx2
 x12  x1  x2
dt
dy dx1
This implies

dt
dt
dx2 x12  x1  x2
and

dx1
x2
Note that the system has two equilibrium points on the
phase plane:
x1  0
x2  0
and
x1  1
x2  0
dx2
  , the trajectories are there
dx1
perpendicular to x1 axis. On the phase plane, the radius
vector r is given by
At x2  0 ,
r 2  x12  x2 2
Now,
d 2
dx
dx
r  2 x1 1  2 x2 2  2 x1 x2  2 x2 ( x12  x1  x2 )
dt
dt
dt
 2 x2 ( x12  x2 )
If this is positive, r is increasing i.e. trajectories are
diverging. If this negative, the trajectories are converging.
Therefore, if x2  0 and x12  x2  convergent case
If x2  0 , the system is always stable.
The phase plane portrait in this case appears as