isochron

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

:$\backslash frac\{d^2y\}\{dt^2\}\; +\; \backslash frac\{dy\}\{dt\}\; =\; 1$

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

The general solution of the above ODE is

:$y=t+A+B\backslash exp(-t)$

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

At the initial time $t=0$ we have $y\_0=A+B$ and $y\text{'}\_0=1-B$. Algebraically eliminate the immaterial constant $B$ from these two equations to deduce that all initial conditions $y\_0+y\text{'}\_0=1+A$ have the same $A$, hence the same long term evolution, and hence form an isochron.

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

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

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

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

to new variables $(X,Y)$ transforms the dynamics to the separated form

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

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

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.[http://www.maths.adelaide.edu.au/anthony.roberts/sdenf.html]

Category:Dynamical systems