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 [math]\displaystyle{ y(t) }[/math] evolving in time:

[math]\displaystyle{ \frac{d^2y}{dt^2} + \frac{dy}{dt} = 1 }[/math]

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

The general solution of the above ODE is

[math]\displaystyle{ y=t+A+B\exp(-t) }[/math]

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

At the initial time [math]\displaystyle{ t=0 }[/math] we have [math]\displaystyle{ y_0=A+B }[/math] and [math]\displaystyle{ y'_0=1-B }[/math]. Algebraically eliminate the immaterial constant [math]\displaystyle{ B }[/math] from these two equations to deduce that all initial conditions [math]\displaystyle{ y_0+y'_0=1+A }[/math] have the same [math]\displaystyle{ 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]\displaystyle{ \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. Here the coordinate transformation near the origin

[math]\displaystyle{ x=X+XY+\cdots \text{ and } y=Y+2Y^2+X^2+\cdots }[/math]

to new variables [math]\displaystyle{ (X,Y) }[/math] transforms the dynamics to the separated form

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

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

Let us use the [math]\displaystyle{ X }[/math] equation to predict the future. Given some initial values [math]\displaystyle{ (x_0,y_0) }[/math] of the original variables: what initial value should we use for [math]\displaystyle{ X(0) }[/math]? Answer: the [math]\displaystyle{ X_0 }[/math] that has the same long term evolution. In the normal form above, [math]\displaystyle{ X }[/math] evolves independently of [math]\displaystyle{ Y }[/math]. So all initial conditions with the same [math]\displaystyle{ X }[/math], but different [math]\displaystyle{ Y }[/math], have the same long term evolution. Fix [math]\displaystyle{ X }[/math] and vary [math]\displaystyle{ Y }[/math] gives the curving isochrons in the [math]\displaystyle{ (x,y) }[/math] plane. For example, very near the origin the isochrons of the above system are approximately the lines [math]\displaystyle{ x-Xy=X-X^3 }[/math]. Find which isochron the initial values [math]\displaystyle{ (x_0,y_0) }[/math] lie on: that isochron is characterised by some [math]\displaystyle{ X_0 }[/math]; the initial condition that gives the correct forecast from the model for all time is then [math]\displaystyle{ 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.[1]