Van der Pol Equation with a Large Feedback Delay

The well-known Van der Pol equation with delayed feedback is considered. It is assumed that the delay factor is large enough. In the study of the dynamics, the critical cases in the problem of the stability of the zero equilibrium state are identified. It is shown that they have infinite dimension. For such critical cases, special local analysis methods have been developed. The main result is the construction of nonlinear evolutionary boundary value problems, which play the role of normal forms. Such boundary value problems can be equations of the Ginzburg-Landau type, as well as equations with delay and special nonlinearity. The nonlocal dynamics of the constructed equations determines the local behavior of the solutions to the original equation. It is shown that similar normalized boundary value problems also arise for the Van der Pol equation with a large coefficient of the delay equation. The important problem of a small perturbation containing a large delay is considered separately. In addition, the Van der Pol equation, in which the cubic nonlinearity contains a large delay, is considered. One of the general conclusions is that the dynamics of the Van der Pol equation in the presence of a large delay is complex and diverse. It fundamentally differs from the dynamics of the classical Van der Pol equation.

Automated summary

This article investigates the well-known Van der Pol equation with delayed feedback. Assuming a sufficiently large delay factor, critical cases in the stability problem of the zero equilibrium state are identified and found to have infinite dimension. Special local analysis methods are developed for these critical cases, resulting in the construction of nonlinear evolutionary boundary value problems that serve as normal forms. These boundary value problems can be equations of the Ginzburg-Landau type, as well as equations with delay and special nonlinearity. The nonlocal dynamics of the constructed equations determine the local behavior of the solutions to the original equation. Similar normalized boundary value problems also arise for the Van der Pol equation with a large coefficient of the delay equation. The important problem of a small perturbation containing a large delay is considered separately, as well as the Van der Pol equation with cubic nonlinearity containing a large delay. In conclusion, the dynamics of the Van der Pol equation with a large delay is complex and diverse, fundamentally differing from the dynamics of the classical Van der Pol equation.