Frequency Locking and Bifurcation Analysis in Asymmetrically Forced Van der Pol-DuffingModel of Glacial Cycles

Abstract

This paper examines the application of self-sustained oscillator systems, particularly the van der Pol-Duffing oscillator, to understand the complex dynamics of Pleistocene glacial cycles. We investigate how asymmetric forcing (β ̸= 0) influences frequency locking and bifurcation behavior through geometric singular perturbation theory (GSPT) analysis and Poincar ́e return map construction. Our numerical results demonstrate that the van der Pol-Duffing oscillator possesses substantially larger regions of stable periodic behavior in parameter space compared to standard van der Pol oscillators. As asymmetry increases from β = 0.25 to β = 1.2, we observed progressive narrowing of Arnold tongue structures with most frequency locking regions requiring stronger forcing amplitudes (a ≥ 1.5) to initiate synchronization. However, remarkably resilient 2:1 frequency locking regions persist across all asymmetry levels. This provides a mathematical framework for explaining dominant frequency transitions observed in paleoclimate records, particularly the Mid-Pleistocene Transition from 41 kyr to 100 kyr glacial cycles (4-significance)