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