[Paper Review] Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited

M. I. Gluzman, Richard H. Rand|arXiv (Cornell University)|May 8, 2017

Nonlinear Dynamics and Pattern Formation2 references3 citations

TL;DR

This paper revisits the dynamics of two delay-coupled van der Pol oscillators by retaining delay terms in the slow flow, contrasting with prior work that approximated delays as non-delayed terms. It derives a series solution for the transcendental characteristic equation of the DDE slow flow, showing that neglecting delays leads to 3–15% error in critical delay for Hopf bifurcation, while a 3-term truncation reduces error to under 1%.

ABSTRACT

The problem of two van der Pol oscillators coupled by velocity delay terms was studied by Wirkus and Rand in 2002. The small-epsilon analysis resulted in a slow flow which contained delay terms. To simplify the analysis, Wirkus and Rand followed a common procedure of replacing the delay terms by non-delayed terms, a step said to be valid for small epsilon, resulting in a slow flow which was an ODE rather than a DDE (delay-differential equation). In the present paper we consider the same problem but leave the delay terms in the slow flow, thereby offering an evaluation of the approximate simplification made previously.

Motivation & Objective

• To re-express the slow flow of two delay-coupled van der Pol oscillators with delay terms retained, rather than approximated as non-delayed terms.
• To evaluate the accuracy of the common simplification in prior literature that replaces delayed variables with non-delayed ones in the slow flow.
• To derive and analyze a series solution for the transcendental characteristic equation arising from the DDE slow flow.
• To quantify the error introduced by neglecting delay terms in the slow flow, particularly in predicting the critical delay for Hopf bifurcation.
• To compare analytical approximations (1-, 2-, and 3-term truncations) with numerical integration results for the DDE system.

Proposed method

• Uses two-variable perturbation method to derive a slow flow in terms of stretched time τ, retaining delay terms.
• Applies Lindstedt’s method to obtain a periodic approximation of the in-phase mode, with amplitude and frequency corrected for ε and delay T.
• Linearizes the system around the in-phase mode to derive a DDE for deviations, leading to a DDE slow flow with delayed variables.
• Derives a transcendental characteristic equation from the DDE slow flow, which is solved via a series expansion (eq. 41, 42) for small ε.
• Compares analytical approximations (1-, 2-, and 3-term truncations of the series) with numerical integration of the DDEs.
• Evaluates errors in predicting the critical delay for Hopf bifurcation using absolute, relative, and percent error metrics.

Experimental results

Research questions

• RQ1 How does retaining delay terms in the slow flow affect the prediction of Hopf bifurcation in two delay-coupled van der Pol oscillators?
• RQ2 What is the magnitude of error introduced by approximating delayed variables with non-delayed ones in the slow flow?
• RQ3 How accurate are analytical series approximations of the DDE slow flow compared to numerical integration?
• RQ4 At what coupling strength α and delay T does the 3-term series approximation achieve less than 1% error in critical delay prediction?
• RQ5 Where does the maximum error in the 1-term approximation occur, and what is its magnitude for ε = 0.5?

Key findings

• The 1-term truncation of the series solution (equivalent to replacing delays with non-delayed terms) results in a 3–15% error in predicting the critical delay for Hopf bifurcation, depending on ε.
• For ε = 0.5, the maximum absolute error in the 1-term approximation reaches 0.2218, with a 13.11% percent error, primarily due to neglecting delay effects.
• The 3-term truncation reduces the maximum percent error to 0.71% for ε = 0.5, indicating high accuracy when higher-order delay effects are included.
• The maximum error in all error metrics (absolute, relative, percent) for the 1-, 2-, and 3-term approximations occurs at α = 1 across all ε values tested.
• Numerical integration of the DDE slow flow (26)–(27) provides the reference for error evaluation, showing that the 3-term series is highly accurate compared to this benchmark.
• The dashed curve in Figure 1 (1-term approximation) significantly departs from the numerical results (plotted + signs), especially at higher α, confirming the inaccuracy of the non-delayed simplification.