In the mathematical theory of dynamical systems, an isochron is a set of initial conditions for the system that all lead to the same long-term behaviour.
Consider the ordinary differential equation for a solution
y
(
t
)
evolving in time:
d
2
y
d
t
2
+
d
y
d
t
=
1
This ordinary differential equation (ODE) needs two initial conditions at, say, time
t
=
0
. Denote the initial conditions by
y
(
0
)
=
y
0
and
d
y
/
d
t
(
0
)
=
y
0
′
where
y
0
and
y
0
′
are some parameters. The following argument shows that the isochrons for this system are here the straight lines
y
0
+
y
0
′
=
constant
.
The general solution of the above ODE is
y
=
t
+
A
+
B
exp
(
−
t
)
Now, as time increases,
t
→
∞
, the exponential terms decays very quickly to zero (exponential decay). Thus all solutions of the ODE quickly approach
y
→
t
+
A
. That is, all solutions with the same
A
have the same long term evolution. The exponential decay of the
B
exp
(
−
t
)
term brings together a host of solutions to share the same long term evolution. Find the isochrons by answering which initial conditions have the same
A
.
At the initial time
t
=
0
we have
y
0
=
A
+
B
and
y
0
′
=
1
−
B
. Algebraically eliminate the immaterial constant
B
from these two equations to deduce that all initial conditions
y
0
+
y
0
′
=
1
+
A
have the same
A
, hence the same long term evolution, and hence form an isochron.
Let's turn to a more interesting application of the notion of isochrons. Isochrons arise when trying to forecast predictions from models of dynamical systems. Consider the toy system of two coupled ordinary differential equations
d
x
d
t
=
−
x
y
and
d
y
d
t
=
−
y
+
x
2
−
2
y
2
A marvellous mathematical trick is the normal form (mathematics) transformation. Here the coordinate transformation near the origin
x
=
X
+
X
Y
+
⋯
and
y
=
Y
+
2
Y
2
+
X
2
+
⋯
to new variables
(
X
,
Y
)
transforms the dynamics to the separated form
d
X
d
t
=
−
X
3
+
⋯
and
d
Y
d
t
=
(
−
1
−
2
X
2
+
⋯
)
Y
Hence, near the origin,
Y
decays to zero exponentially quickly as its equation is
d
Y
/
d
t
=
(
negative
)
Y
. So the long term evolution is determined solely by
X
: the
X
equation is the model.
Let us use the
X
equation to predict the future. Given some initial values
(
x
0
,
y
0
)
of the original variables: what initial value should we use for
X
(
0
)
? Answer: the
X
0
that has the same long term evolution. In the normal form above,
X
evolves independently of
Y
. So all initial conditions with the same
X
, but different
Y
, have the same long term evolution. Fix
X
and vary
Y
gives the curving isochrons in the
(
x
,
y
)
plane. For example, very near the origin the isochrons of the above system are approximately the lines
x
−
X
y
=
X
−
X
3
. Find which isochron the initial values
(
x
0
,
y
0
)
lie on: that isochron is characterised by some
X
0
; the initial condition that gives the correct forecast from the model for all time is then
X
(
0
)
=
X
0
.
You may find such normal form transformations for relatively simple systems of ordinary differential equations, both deterministic and stochastic, via an interactive web site.[1]
