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{{Distinguish|Isochrone (disambiguation){{!}}Isochrone}}
{{for|the drug with trade name Isochron|Isosorbide dinitrate }}
{{for|the radiometric dating technique|Isochron dating}}

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.<ref>J. Guckenheimer, Isochrons and phaseless sets, J. Math. Biol., 1:259&ndash;273 (1975)</ref><ref>S.M. Cox and A.J. Roberts, Initial conditions for models of dynamical systems, Physica D, 85:126&ndash;141 (1995)</ref>

==Mathematical isochron==
===An introductory example===

Consider the [[ordinary differential equation]] for a solution <math>y(t)</math> evolving in time:

:<math> \frac{d^2y}{dt^2} + \frac{dy}{dt} = 1</math>

This [[ordinary differential equation]] (ODE) needs two [[initial conditions]] at, say, time <math>t=0</math>.  Denote the [[initial conditions]] by <math>y(0)=y_0</math> and <math>dy/dt(0)=y'_0</math>  where <math>y_0</math> and <math>y'_0</math> are some parameters.  The following argument shows that the isochrons for this system are here the straight lines <math>y_0+y'_0=\mbox{constant}</math>.

The general solution of the above ODE is

:<math>y=t+A+B\exp(-t) </math>

Now, as time increases, <math>t\to\infty</math>, the exponential terms decays very quickly to zero ([[exponential decay]]). Thus ''all'' solutions of the ODE quickly approach <math>y\to t+A</math>.  That is, ''all'' solutions with the same <math>A</math> have the same long term evolution.  The [[exponential decay]] of the <math>B\exp(-t)</math> 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 <math>A</math>.

At the initial time <math>t=0</math> we have <math>y_0=A+B</math> and <math>y'_0=1-B</math>.  Algebraically eliminate the immaterial constant <math>B</math> from these two equations to deduce that all initial conditions <math>y_0+y'_0=1+A</math> have the same <math>A</math>, 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]]

:<math> \frac{dx}{dt} = -xy \text{ and } \frac{dy}{dt} = -y+x^2 - 2y^2</math>

A marvellous mathematical trick is the [[normal form (mathematics)]] transformation.<ref>A.J. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, ''Physica A: Statistical Mechanics and its Applications'' 387:12&ndash;38 (2008)</ref>  Here the coordinate transformation near the origin

:<math> x=X+XY+\cdots \text{ and } y=Y+2Y^2+X^2+\cdots</math>

to new variables <math>(X,Y)</math> transforms the dynamics to the separated form

:<math> \frac{dX}{dt} = -X^3+ \cdots \text{ and } \frac{dY}{dt} = (-1-2X^2+\cdots)Y</math>

Hence, near the origin, <math>Y</math> decays to zero exponentially quickly as its equation is <math>dY/dt= (\text{negative})Y</math>.  So the long term evolution is determined solely by <math>X</math>: the <math>X</math> equation is the model.  

Let us use the <math>X</math> equation to predict the future.  Given some initial values <math>(x_0,y_0)</math> of the original variables: what initial value should we use for <math>X(0)</math>?  Answer: the <math>X_0</math> that has the same long term evolution.  In the normal form above, <math>X</math> evolves independently of <math>Y</math>.  So all initial conditions with the same <math>X</math>, but different <math>Y</math>, have the same long term evolution.  Fix <math>X</math> and vary <math>Y</math> gives the curving isochrons in the <math>(x,y)</math> plane.  For example, very near the origin the isochrons of the above system are approximately the lines <math>x-Xy=X-X^3</math>.  Find which isochron the initial values <math>(x_0,y_0)</math> lie on: that isochron is characterised by some  <math>X_0</math>; the initial condition that gives the correct forecast from the model for all time is then  <math>X(0)=X_0</math>.

You may find such normal form transformations for relatively simple systems of ordinary differential equations, both deterministic and stochastic, via an interactive web site.[http://www.maths.adelaide.edu.au/anthony.roberts/sdenf.html]

==References==
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[[Category:Dynamical systems]]