An introductory example
Consider the ordinary differential equation for a solution evolving in time:
This ordinary differential equation (ODE) needs two initial conditions at, say, time . Denote the initial conditions by and where and are some parameters. The following argument shows that the isochrons for this system are here the straight lines .
The general solution of the above ODE is
Now, as time increases, , the exponential terms decays very quickly to zero (exponential decay). Thus all solutions of the ODE quickly approach . That is, all solutions with the same have the same long term evolution. The exponential decay of the 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 .
At the initial time we have and . Algebraically eliminate the immaterial constant from these two equations to deduce that all initial conditions have the same , 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
to new variables transforms the dynamics to the separated form
Hence, near the origin, decays to zero exponentially quickly as its equation is . So the long term evolution is determined solely by : the equation is the model.
Let us use the equation to predict the future. Given some initial values of the original variables: what initial value should we use for ? Answer: the that has the same long term evolution. In the normal form above, evolves independently of . So all initial conditions with the same , but different , have the same long term evolution. Fix and vary gives the curving isochrons in the plane. For example, very near the origin the isochrons of the above system are approximately the lines . Find which isochron the initial values lie on: that isochron is characterised by some ; the initial condition that gives the correct forecast from the model for all time is then .
You may find such normal form transformations for relatively simple systems of ordinary differential equations, both deterministic and stochastic, via an interactive web site.
- J. Guckenheimer, Isochrons and phaseless sets, J. Math. Biol., 1:259–273 (1975)
- S.M. Cox and A.J. Roberts, Initial conditions for models of dynamical systems, Physica D, 85:126–141 (1995)
- 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)