# Isochron

Template:Distinguish {{#invoke:Hatnote|hatnote}}

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

${\frac {d^{2}y}{dt^{2}}}+{\frac {dy}{dt}}=1$ The general solution of the above ODE is

$y=t+A+B\exp(-t)\,$ Now, as time increases, $t\to \infty$ , the exponential terms decays very quickly to zero (exponential decay). Thus all solutions of the ODE quickly approach $y\to t+A$ . 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 $A$ .

### 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

${\frac {dx}{dt}}=-xy{\text{ and }}{\frac {dy}{dt}}=-y+x^{2}-2y^{2}$ A marvellous mathematical trick is the normal form (mathematics) transformation. Here the coordinate transformation near the origin

$x=X+XY+\cdots {\text{ and }}y=Y+2Y^{2}+X^{2}+\cdots$ to new variables $(X,Y)$ transforms the dynamics to the separated form

${\frac {dX}{dt}}=-X^{3}+\cdots {\text{ and }}{\frac {dY}{dt}}=(-1-2X^{2}+\cdots )Y$ 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 $x-Xy=X-X^{3}$ . 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.