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Orbit quantization in a retarded harmonic oscillator

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  • López, Álvaro G.

Abstract

We study the dynamics of a damped harmonic oscillator in the presence of a retarded potential with state-dependent time-delayed feedback. In the limit of small time-delays, we show that the oscillator is equivalent to a Liénard system. This allows us to analytically predict the value of the first Hopf bifurcation, unleashing a self-oscillatory motion. We compute bifurcation diagrams for several model parameter values and analyze multistable domains in detail. Using the Lyapunov energy function, two well-resolved energy levels represented by two coexisting stable limit cycles are discerned. Further exploration of the parameter space reveals the existence of a superposition limit cycle, encompassing two degenerate coexisting limit cycles at the fundamental energy level. When the system is driven very far from equilibrium, a multiscale strange attractor displaying intrinsic and robust intermittency is uncovered.

Suggested Citation

  • López, Álvaro G., 2023. "Orbit quantization in a retarded harmonic oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    • Handle: RePEc:eee:chsofr:v:170:y:2023:i:c:s0960077923003132
    DOI: 10.1016/j.chaos.2023.113412
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    References listed on IDEAS

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    1. Hansen, Matheus & Protachevicz, Paulo R. & Iarosz, Kelly C. & Caldas, Iberê L. & Batista, Antonio M. & Macau, Elbert E.N., 2022. "The effect of time delay for synchronisation suppression in neuronal networks," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    2. Jade Martínez-Llinàs & Xavier Porte & Miguel C. Soriano & Pere Colet & Ingo Fischer, 2015. "Dynamical properties induced by state-dependent delays in photonic systems," Nature Communications, Nature, vol. 6(1), pages 1-9, November.
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