Isochron

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.
Mathematical isochron.
An introductory example.
Consider the ordinary differential equation for a solution formula_1 evolving in time:
This ordinary differential equation (ODE) needs two initial conditions at, say, time formula_3. Denote the initial conditions by formula_4 and formula_5 where formula_6 and formula_7 are some parameters. The following argument shows that the isochrons for this system are here the straight lines formula_8.
The general solution of the above ODE is
Now, as time increases, formula_10, the exponential terms decays very quickly to zero (exponential decay). Thus "all" solutions of the ODE quickly approach formula_11. That is, "all" solutions with the same formula_12 have the same long term evolution. The exponential decay of the formula_13 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 formula_12.
At the initial time formula_3 we have formula_16 and formula_17. Algebraically eliminate the immaterial constant formula_18 from these two equations to deduce that all initial conditions formula_19 have the same formula_12, hence the same long term evolution, and hence form an isochron.
Accurate forecasting requires isochrons.
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. Here the coordinate transformation near the origin
to new variables formula_23 transforms the dynamics to the separated form
Hence, near the origin, formula_25 decays to zero exponentially quickly as its equation is formula_26. So the long term evolution is determined solely by formula_27: the formula_27 equation is the model. 
Let us use the formula_27 equation to predict the future. Given some initial values formula_30 of the original variables: what initial value should we use for formula_31? Answer: the formula_32 that has the same long term evolution. In the normal form above, formula_27 evolves independently of formula_25. So all initial conditions with the same formula_27, but different formula_25, have the same long term evolution. Fix formula_27 and vary formula_25 gives the curving isochrons in the formula_39 plane. For example, very near the origin the isochrons of the above system are approximately the lines formula_40. Find which isochron the initial values formula_30 lie on: that isochron is characterised by some formula_32; the initial condition that gives the correct forecast from the model for all time is then formula_43.
You may find such normal form transformations for relatively simple systems of ordinary differential equations, both deterministic and stochastic, via an interactive web site.