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Discontinuity, Nonlinearity, and Complexity 12(2) (2023) 313--327 | DOI:10.5890/DNC.2023.06.007

Department of Mathematics, University of Central Florida, Orlando FL32816-8005, USA

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Abstract

The dynamics of a delayed multiparameter nonlinear Mathieu equation: $$ \ddot{x}+(\delta+\epsilon\alpha \cos{t})x+\epsilon\gamma x^3=\epsilon\beta\int_{-\infty}^{t}{x(\tau)\xi e^{-\xi(t-\tau)}}d\tau,$$ is investigated in the neighborhood of $\delta=1/4$. Three different features interact here: a distributed delay, cubic nonlinearity and 2:1 parametric resonance. The averaging method is used to obtain a slow flow that is analyzed for stability and bifurcations, and the resulting predictions are compared against actual system responses. In particular, we find regimes where: i. the slow flow has a zero stable fixed point (implying Amplitude Death), or ii. the slow flow goes to a stable non-zero fixed point (implying periodic solutions), or iii. the slow flow goes to a stable periodic solution at large times (corresponding to a quasiperiodic system response). All of these types of behavior would be very difficult to isolate otherwise, except by intensive numerical searching of the multiparameter space. However, there are also parameter regimes where the slow flow predictions may occasionally disagree with the actual system response $x(t)$ in cases where that has large amplitude or exhibits bounded aperiodicity. The reasons for these discrepancies are also carefully considered.

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