Dynamics of a Modified Third–Order Phase–Locked Loops (PLL): Melnikov Approach, Simulations

Kyurkchiev, Nikolay; Zaevski, Tsvetelin; Iliev, Anton; Kyurkchiev, Vesselin; Rahnev, Asen

doi:10.3390/math14122071

Open AccessFeature PaperArticle

Dynamics of a Modified Third–Order Phase–Locked Loops (PLL): Melnikov Approach, Simulations

by

Nikolay Kyurkchiev

^1,2,3

,

Tsvetelin Zaevski

^1,3,4

,

Anton Iliev

^2,3,*

,

Vesselin Kyurkchiev

²

and

Asen Rahnev

^2,3

¹

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria

²

Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24, Tzar Asen Str., 4000 Plovdiv, Bulgaria

³

Centre of Excellence in Informatics and Information and Communication Technologies, 1113 Sofia, Bulgaria

⁴

Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5, James Bourchier Blvd., 1164 Sofia, Bulgaria

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(12), 2071; https://doi.org/10.3390/math14122071

Submission received: 9 May 2026 / Revised: 30 May 2026 / Accepted: 9 June 2026 / Published: 10 June 2026

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions, 3rd Edition)

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Abstract

In this article, we investigate the dynamics of new modified third-order Phase-Locked Loops (PLLs). Our goal here is to investigate the effect of the new factor

\sum_{j = 1}^{N} a_{j} \sin (j ω t)

on the dynamics of the proposed model. Using perturbation techniques based on Andronov–Melnikov concepts, we demonstrate that horseshoe chaos exists in three-dimensional nonautonomous systems. Several simulations are performed. Additionally, we present a few specific modules for examining the dynamics of the hypothetical oscillator circuit under consideration. This will be a crucial component of a much broader web-based scientific computing application. We will explicitly note that the proposed model is hypothetical and specialists working in this scientific field have a say. We will consider a numerical example of the possible application of the Melnikov function in the modeling of the radiation Melnikov antenna diagram. In addition, we examine a generalization based on probability distributions.

Keywords: modified perturbed third-order Phase-Locked Loops (PLLs); horseshoe chaos; Melnikov integral; Melnikov criterion in determining the occurrence of potential chaos; Melnikov antenna diagram

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MDPI and ACS Style

Kyurkchiev, N.; Zaevski, T.; Iliev, A.; Kyurkchiev, V.; Rahnev, A. Dynamics of a Modified Third–Order Phase–Locked Loops (PLL): Melnikov Approach, Simulations. Mathematics 2026, 14, 2071. https://doi.org/10.3390/math14122071

AMA Style

Kyurkchiev N, Zaevski T, Iliev A, Kyurkchiev V, Rahnev A. Dynamics of a Modified Third–Order Phase–Locked Loops (PLL): Melnikov Approach, Simulations. Mathematics. 2026; 14(12):2071. https://doi.org/10.3390/math14122071

Chicago/Turabian Style

Kyurkchiev, Nikolay, Tsvetelin Zaevski, Anton Iliev, Vesselin Kyurkchiev, and Asen Rahnev. 2026. "Dynamics of a Modified Third–Order Phase–Locked Loops (PLL): Melnikov Approach, Simulations" Mathematics 14, no. 12: 2071. https://doi.org/10.3390/math14122071

APA Style

Kyurkchiev, N., Zaevski, T., Iliev, A., Kyurkchiev, V., & Rahnev, A. (2026). Dynamics of a Modified Third–Order Phase–Locked Loops (PLL): Melnikov Approach, Simulations. Mathematics, 14(12), 2071. https://doi.org/10.3390/math14122071

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Dynamics of a Modified Third–Order Phase–Locked Loops (PLL): Melnikov Approach, Simulations

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