The use of the Euler method in identification of multiple bifurcations and chaotic behavior in numerical approximation of delay-differential equations

A discrete model is proposed to delve into the rich dynamics of nonlinear delayed systems under Euler discretization, such as multiple bifurcations, stable limit cycles (periodic or quasiperiodic solutions), and chaotic behavior. A method of using a finite-dimensional discrete dynamical system to approximate an infinite-dimensional dynamical system is developed here. We find that the effect of breaking the symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include complex periodic oscillations, quasiperiodicity and chaos. A route from complex periodic/quasiperiodic oscillations to chaos and then to quasiperiodic oscillations can be observed. The delay model also gives a family of examples for chaotic behavior usable to demonstrate analyzing, controlling and anti-controlling schemes.