Citation: Machmudah, A.; Dutykh, D.; Parman, S. Coupled and Synchronization Models of Rhythmic Arm Movement in Planar Plane. Bioengineering 2022, 9, 385. https://doi.org/10.3390/ bioengineering9080385 Academic Editor: Chengfei Zhang Received: 31 May 2022 Accepted: 4 August 2022 Published: 12 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). bioengineering Article Coupled and Synchronization Models of Rhythmic Arm Movement in Planar Plane Affiani Machmudah 1,2 , Denys Dutykh 3, * and Setyamartana Parman 4 1 Faculty of Advanced Technology and Multidiscipline, Kampus C Jalan Mulyorejo, Universitas Airlangga, Surabaya 60115, Indonesia 2 Research Center for Hydrodynamics Technology, National Research and Innovation Agency (BRIN), Jl. Hidro Dinamika, Keputih, Sukolilo, Surabaya 60112, Indonesia 3 Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France 4 Fakulti Teknologi Kejuruteraan Mekanikal dan Pembuatan, Universiti Teknikal Malaysia Melaka, Durian Tunggal, Melaka 76100, Malaysia * Correspondence: denys.dutykh@univ-smb.fr Abstract: Nonlinear dynamics have become a new perspective on model human movement vari- ability; however, it is still a debate whether chaotic behavior is indeed possible to present during a rhythmic movement. This paper reports on the nonlinear dynamical behavior of coupled and synchronization models of a planar rhythmic arm movement. Two coupling schemes between a planar arm and an extended Duffing-Van der Pol (DVP) oscillator are investigated. Chaos tools, namely phase space, Poincare section, Lyapunov Exponent (LE), and heuristic approach are applied to observe the dynamical behavior of orbit solutions. For the synchronization, an orientation angle is modeled as a single well DVP oscillator implementing a Proportional Derivative (PD)-scheme. The extended DVP oscillator is used as a drive system, while the orientation angle of the planar arm is a response system. The results show that the coupled system exhibits very rich dynamical behavior where a variety of solutions from periodic, quasi-periodic, to chaotic orbits exist. An advanced coupling scheme is necessary to yield the route to chaos. By modeling the orientation angle as the single well DVP oscillator, which can synchronize with other dynamical systems, the synchronization can be achieved through the PD-scheme approach. Keywords: nonlinear dynamics; chaotic behavior; coupled system; synchronization; biomechanics modeling; rhythmic movement; healthy movement system 1. Introduction Bodily rhythm is an essential part of human life [1]. In terms of physical movement, everyday performances, such as walking, running, swimming, dancing, sports, and other life activities, frequently employ the repetition of motion. When a person loses the ability to conduct a rhythmic performance, it is not only a sign of an unhealthy phase of the human body system, but it will also significantly disturb the quality of the person’s life. For example, in the case of post-stroke injury, the person most likely will lose the ability to perform complex body movements that involve repetitive motion. The history of movement variability in biomechanics research can be traced to Bern- stein’s report in 1967. When humans perform two identical movements, the trajectories of the first movement are never repeated in the second movement [2]. This simple phe- nomenon is evidence of the variability in human movement, which Bernstein used the term “repetition without repetition” to express it. Since this Bernstein report, the study of the variability in human movement, which is commonly referred to as Bernstein’s problem [2–4], has become a principal research interest in the field of human movement science, biomechanics, and human gait. Movement variability reflects that there are many possible solutions to achieve an identical movement. Then, how the human brain chooses Bioengineering 2022, 9, 385. https://doi.org/10.3390/bioengineering9080385 https://www.mdpi.com/journal/bioengineering