Homoclinic bifurcations in a neutral delay model of a transmission line oscillator.

*(English)*Zbl 1129.34049The authors make use of a simulation tool that they have extended in order to investigate the complicated behavior of a transmission line oscillator (TLO). In this TLO, a linear wave travels along a piece of cable, the transmission line, and interacts with terminating electrical components. A fixed time delay arises due to the transmission time through the transmission line. The authors experiments on the neutral delay differential equation (NDDE) that models the TLO driven by a negative resistor, demonstrate rich delay-induced dynamics and high-frequency chaotic behaviour. The main focus is on homoclinic orbits. For small time delay there is a homoclinic orbit to a steady-state. However, after a codimension-two Shil’nikov-Hopf bifurcation, the homoclinic orbit connects to a saddle-type periodic solution, which exists in a region bounded by homoclinic tangencies. Both types of homoclinic bifurcations are associated with accumulating branches of periodic solutions. The authors summarize their results in a two-parameter bifurcation diagram in the plane of resistance against time delay. This study demonstrates that the theory of homoclinic bifurcations in ordinary differential equations largely carries over to NDDEs. However, the neutral delay nature of the problem influences some bifurcations, especially the convergence rates of folds associated with the homoclinic tangencies.

Reviewer: H. B. Bouzahir (Agadir)

##### MSC:

34K18 | Bifurcation theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

34K40 | Neutral functional-differential equations |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |