Article
Mathematics
Danjin Zhang, Youhua Qian
Summary: This paper investigates the dynamic behavior of the van der Pol-Rayleigh system using fast-slow analysis and transformation phase portrait methods. It examines the stability and bifurcation behavior of the equilibrium point, identifying the presence of Hopf bifurcation without fold bifurcation. The system's vibration behavior in different modes is analyzed, showing that it is influenced by both fast and slow varying processes. The transformation phase portrait method reveals the mechanisms of different vibration modes, as the system trajectory encounters different attractors in the fast subsystem.
Article
Mathematics, Interdisciplinary Applications
Xindong Ma, Daixian Xia, Wenan Jiang, Mao Liu, Qinsheng Bi
Summary: The mechanism of compound bursting behaviors of a forced Mathieu-van der Pol-Duffing system is explored in this study, revealing different bursting patterns arising from different parameter values, providing more possible routes to compound bursting dynamics and strengthening the understanding of compound bursting behaviors.
CHAOS SOLITONS & FRACTALS
(2021)
Article
Mathematics, Applied
S. A. Kashchenko
Summary: A study of a ring chain of the coupled Van der Pol equations with two types of unidirectional advective couplings was conducted. The research focused on stability and dynamic behavior, revealing differences when using various couplings, and constructing special nonlinear partial differential equations models.
Article
Mathematics
Sergey Kashchenko
Summary: This article investigates the well-known Van der Pol equation with delayed feedback. Assuming a sufficiently large delay factor, critical cases in the stability problem of the zero equilibrium state are identified and found to have infinite dimension. Special local analysis methods are developed for these critical cases, resulting in the construction of nonlinear evolutionary boundary value problems that serve as normal forms. These boundary value problems can be equations of the Ginzburg-Landau type, as well as equations with delay and special nonlinearity. The nonlocal dynamics of the constructed equations determine the local behavior of the solutions to the original equation. Similar normalized boundary value problems also arise for the Van der Pol equation with a large coefficient of the delay equation. The important problem of a small perturbation containing a large delay is considered separately, as well as the Van der Pol equation with cubic nonlinearity containing a large delay. In conclusion, the dynamics of the Van der Pol equation with a large delay is complex and diverse, fundamentally differing from the dynamics of the classical Van der Pol equation.
Article
Mathematics, Applied
Bin Zhang, Xiaofang Zhang, Wenan Jiang, Hu Ding, Liqun Chen, Qinsheng Bi
Summary: This paper proposes a novel three-dimensional modified van der Pol-Duffing circuit with a quintic nonlinear resistor, and multiple stable attractors are observed. The complex bursting patterns are investigated, and various patterns of bursting oscillations are obtained under slowly changing frequency, revealing the transition mechanism.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Physics, Multidisciplinary
T. Bhagyaraj, S. Sabarathinam, A. Ishaq Ahamed, K. Thamilmaran
Summary: In this paper, the phenomenon of super-extreme events in the forced BVP oscillator is reported, and their existence is verified. This has important implications for understanding and applying extreme events in different systems.
PRAMANA-JOURNAL OF PHYSICS
(2023)
Article
Mathematics
Mohsan Raza, Hari Mohan Srivastava, Qin Xin, Fairouz Tchier, Sarfraz Nawaz Malik, Muhammad Arif
Summary: In this paper, a subclass of starlike functions related to the Van der Pol numbers is defined. Structural formula, radius of starlikeness of order a, strong starlikeness, and some inclusion results are derived for this class. Radii problems for various classes of analytic functions are also studied. Furthermore, coefficient-related problems including sharp initial coefficient bounds and sharp bounds on Hankel determinants of order two and three are investigated.
Article
Multidisciplinary Sciences
M. A. Elfouly, M. A. Sohaly
Summary: The Van der Pol equation, a second-order ordinary differential equation with cubic nonlinearity, has been studied with added time delays in this paper. The derived delay differential equations from the original Van der Pol model and RLC circuit allow the re-use of applications in the suggested equation, with numerical simulations showing different cases expressible by delay differential equations.
SCIENTIFIC REPORTS
(2022)
Article
Engineering, Multidisciplinary
Alvaro Salas, H. Lorenzo J. Martinez, R. David L. Ocampo
Summary: In this paper, we demonstrate the application of analytical and numerical techniques to solve the forced Van der Pol oscillator. The obtained results are illustrated through examples and compared with the Runge-Kutta numerical method to assess the accuracy of the approximated analytical solution.
MATHEMATICAL PROBLEMS IN ENGINEERING
(2022)
Article
Engineering, Mechanical
Fatemeh Afzali, Ehsan Kharazmi, Brian F. Feeny
Summary: This work analyzes secondary resonances in the parametrically damped van der Pol equation, both with and without external excitation. It focuses on a potential application in vertical-axis wind-turbine blades, which experience cyclic damping, aeroelastic self-excitation, and direct excitation. The system is studied using the method of multiple scales and numerical solutions. The analysis reveals various responses, including nonresonant phase drift, subharmonic resonance, and potential phase locking.
NONLINEAR DYNAMICS
(2023)
Article
Mathematics, Applied
Xindong Ma, Qinsheng Bi, Lifeng Wang
Summary: This paper theoretically presents complex bursting patterns caused by the coupling effect of different frequency scales in the RVDPDO driven by the external excitation term. Seven different kinds of bursting are studied using various analytical techniques, and the sensitivity of dynamical characteristics of RVDPDO to parameter variation is shown. The research validity is tested and verified through numerical simulations.
JOURNAL OF NONLINEAR SCIENCE
(2022)
Article
Engineering, Mechanical
A. Bochkarev, A. Zemlyanukhin
Summary: The study on active particles coupled by the Morse potential with Van der Pol dissipation reveals the existence of soliton-like perturbations and two types of kink, slow and fast. The parameters of the kinks are determined by different mathematical equations and the propagation modes of the perturbations in different boundary conditions are investigated.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Niazy Hady Hussein, Azad Ibrahim Amen
Summary: The integrability of the three-dimensional Van der Pol-Duffing system was studied, revealing that under certain conditions the system has no analytic and Darboux first integrals at the neighborhood of the origin. The stability and instability of the singular points were used to investigate the C-1 integrability of this type of system.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
Sergey Kashchenko
Summary: The study focuses on the local dynamics of a chain of coupled Van der Pol equation systems near the zero equilibrium state. Transition to a system with continuous spatial variable, identifying critical cases corresponding to Turing bifurcations, and proposing special nonlinear parabolic equations are key aspects. It is found that for systems with a large number of elements, dynamics significantly change with a slight change in the number of elements.
Article
Computer Science, Interdisciplinary Applications
N. Ramroodi, H. Ahsani Tehrani, M. H. Noori Skandari
Summary: In this article, the behavior of a Van der Pol oscillator based on variable-order Caputo fractional derivatives is investigated. The article proposes a modeling approach, a discretization method, and an algorithm for solving the variable-order Caputo fractional Van der Pol equation. Numerical simulations demonstrate the applicability of the suggested method.
JOURNAL OF COMPUTATIONAL SCIENCE
(2023)
