Abstract
In a transmission line oscillator (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. Recent experiments on a TLO driven by a negative resistor demonstrated rich delay-induced dynamics and high-frequency chaotic behaviour. Furthermore, good agreement was found with a neutral delay differential equation (NDDE) model.
In this paper we perform a numerical bifurcation analysis of the NDDE model of the TLO. Our main focus is on homoclinic orbits, which give rise to complicated dynamics and bifurcations. For small time delay there is a homoclinic orbit to a steady-state. However, past 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 snaking branches of periodic solutions. We summarise our results in a two-parameter bifurcation diagram in the plane of resistance against time delay.
Our study demonstrates that the theory of homoclinic bifurcations in ordinary differential equations largely carries over to NDDEs. However, we find that the neutral delay nature of the problem influences some bifurcations, especially convergence rates of homoclinic snaking.
Original language | English |
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Publication status | Published - 2006 |
Bibliographical note
Sponsorship: David Barton is supported by EPSRC DTG/EMAT.SB1349.6525Research Groups and Themes
- Engineering Mathematics Research Group
Keywords
- NDDE
- homoclinic
- delay
- neutral
- transmission line
- numerical continuation