# 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.[1][2]

## Mathematical isochron

### An introductory example

Consider the ordinary differential equation for a solution ${\displaystyle y(t)}$  evolving in time:

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+{\frac {dy}{dt}}=1}$

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

The general solution of the above ODE is

${\displaystyle y=t+A+B\exp(-t)}$

Now, as time increases, ${\displaystyle t\to \infty }$ , the exponential terms decays very quickly to zero (exponential decay). Thus all solutions of the ODE quickly approach ${\displaystyle y\to t+A}$ . That is, all solutions with the same ${\displaystyle A}$  have the same long term evolution. The exponential decay of the ${\displaystyle B\exp(-t)}$  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 ${\displaystyle A}$ .

At the initial time ${\displaystyle t=0}$  we have ${\displaystyle y_{0}=A+B}$  and ${\displaystyle y'_{0}=1-B}$ . Algebraically eliminate the immaterial constant ${\displaystyle B}$  from these two equations to deduce that all initial conditions ${\displaystyle y_{0}+y'_{0}=1+A}$  have the same ${\displaystyle A}$ , 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

${\displaystyle {\frac {dx}{dt}}=-xy{\text{ and }}{\frac {dy}{dt}}=-y+x^{2}-2y^{2}}$

A marvellous mathematical trick is the normal form (mathematics) transformation.[3] Here the coordinate transformation near the origin

${\displaystyle x=X+XY+\cdots {\text{ and }}y=Y+2Y^{2}+X^{2}+\cdots }$

to new variables ${\displaystyle (X,Y)}$  transforms the dynamics to the separated form

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

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

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

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]

## References

1. ^ J. Guckenheimer, Isochrons and phaseless sets, J. Math. Biol., 1:259–273 (1975)
2. ^ S.M. Cox and A.J. Roberts, Initial conditions for models of dynamical systems, Physica D, 85:126–141 (1995)
3. ^ A.J. Roberts, Normal form transforms separate slow and fast modes in stochastic dynamical systems, Physica A: Statistical Mechanics and its Applications 387:12–38 (2008)