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.
[1][2] Consider the ordinary differential equation for a solution
evolving in time: This ordinary differential equation (ODE) needs two initial conditions at, say, time
Denote the initial conditions by
The following argument shows that the isochrons for this system are here the straight lines
The general solution of the above ODE is Now, as time increases,
, the exponential terms decays very quickly to zero (exponential decay).
Thus all solutions of the ODE quickly approach
have the same long term evolution.
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
Algebraically eliminate the immaterial constant
from these two equations to deduce that all initial conditions
, 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 A marvellous mathematical trick is the normal form (mathematics) transformation.
[3] Here the coordinate transformation near the origin to new variables
transforms the dynamics to the separated form Hence, near the origin,
decays to zero exponentially quickly as its equation is
So the long term evolution is determined solely by
equation to predict the future.
of the original variables: what initial value should we use for
that has the same long term evolution.
, have the same long term evolution.
For example, very near the origin the isochrons of the above system are approximately the lines
Find which isochron the initial values
lie on: that isochron is characterised by some
; the initial condition that gives the correct forecast from the model for all time is then
You may find such normal form transformations for relatively simple systems of ordinary differential equations, both deterministic and stochastic, via an interactive web site.