Article
Mathematics, Applied
Xiaoyan Hu, Bo Sang, Ning Wang
Summary: In this work, a five-parameter jerk system with a hyperbolic sine nonlinearity is analyzed. The symmetrical and asymmetrical cases are studied, and the bifurcations are determined using analytical methods. The discovery of chaotic motion mechanisms in jerk systems is the main contribution of this work. Circuit simulations are used to validate the numerical results.
Article
Mathematics, Interdisciplinary Applications
Forwah Amstrong Tah, Conrad Bertrand Tabi, Timoleon Crepin Kofane
Summary: The study examines the spatiotemporal dynamics of the Fitzhugh-nagumo neuron, identifying Hopf bifurcations induced by relaxation and Pitchfork bifurcations due to diffusion. The analysis shows that the system allows for finite speeds of propagation for non-negligible values of relaxation time.
CHAOS SOLITONS & FRACTALS
(2021)
Article
Mathematics, Interdisciplinary Applications
Qiuyan Zhang, Yuqian Zhou, Jibin Li
Summary: The nonlinear Schrodinger equation with nonlinear dispersion is investigated, and the bifurcation-theoretic method of planar dynamical systems is used to obtain various exact solutions for this system.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Mathematics, Applied
Qiuyan Zhang
Summary: In this study, the optical soliton model in metamaterials, dominated by anti-cubic nonlinearity, was investigated using the method of dynamical systems. By applying a travelling wave transformation, the model was converted into a singular integrable travelling wave system. The dynamical behavior of the associated regular system was discussed, and all bounded exact solutions of the model could be calculated due to its integrability. Furthermore, twenty exact explicit parametric representations were derived.
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2022)
Article
Engineering, Mechanical
Andres Amador, Emilio Freire, Enrique Ponce
Summary: This paper investigates the dynamical behavior of a family of 3D memristor oscillators, deriving a new bifurcation set and obtaining analytical approximations of bifurcation curves to explore phase plane characteristics in different parameter regions. The study reveals the existence of closed surfaces completely foliated by periodic orbits in specific parameter regions, clarifying misconceptions from numerical simulations and highlighting the significance of invariant manifolds associated with the first integral.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Yijun Zhu, Huilin Shang
Summary: This paper investigates the global bifurcation behaviors of MEMS resonators and proposes two types of delayed feedback control methods to suppress pull-in instability and chaos. Pull-in instability and chaos are found to be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. Under a positive gain coefficient, delayed position feedback and delayed velocity feedback are effective in reducing pull-in instability while only the former can suppress chaos.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Aikan Shykhmamedov, Efrosiniia Karatetskaia, Alexey Kazakov, Nataliya Stankevich
Summary: We study the bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, where the attractors possess two positive Lyapunov exponents in numerical experiments. Specifically, the attractor should have two-dimensional unstable invariant manifolds for its periodic orbits. We discuss various bifurcation scenarios, including cascades of supercritical period-doubling bifurcations and supercritical Neimark-Sacker bifurcations, as well as their combinations, in order to create these periodic orbits within the attractor. These scenarios are demonstrated using the example of the three-dimensional Mira map.
Article
Mathematics, Applied
Alessandro Arsie, Chanaka Kottegoda, Chunhua Shan
Summary: This paper investigates the high codimension bifurcations of a classical predator-prey system with Allee effects and a generalized Holling type III functional response. The authors develop unfoldings of a cusp singularity and a nilpotent saddle with a fixed invariant line. The dependence of the codimension of degenerate Hopf bifurcation on a certain parameter is thoroughly examined. The paper also proves the existence of homoclinic and heteroclinic loops for certain parameter ranges, as well as multiple limit cycles. These findings complement the analysis of classical predator-prey systems with Allee effects and four types of Holling functional responses. Furthermore, simple formulas are derived to characterize the order of the nilpotent saddle, providing an easy way to determine the existence and order of heteroclinic loops in predator-prey systems with any smooth functional response.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
Huijian Zhu, Lijie Li
Summary: This study investigates the influence of delayed feedback on the unified chaotic system from the Sprott C system and Yang system. By using the central manifold theorem and bifurcation theory, the Hopf bifurcation and dynamic behavior of the system are fully studied. The explicit formula, bifurcation direction, and stability of the periodic solution of bifurcation are given correspondingly. The correctness of the theory is proved by analyzing the Hopf bifurcation diagram and chaotic phenomenon through numerical simulation.
Article
Mathematics, Interdisciplinary Applications
Uttam Ghosh, Swadesh Pal, Malay Banerjee
Summary: This paper discusses a system of two fractional order differential equations for prey-predator interaction with intra-specific competition among predators. It explains the derivation of the fractional order model in terms of memory effect on population growth and provides detailed mathematical results on the positiveness, existence-uniqueness, and boundedness of solutions. The study also demonstrates the impact of memory on system dynamics and global bifurcation threshold through extensive numerical simulations.
CHAOS SOLITONS & FRACTALS
(2021)
Article
Engineering, Mechanical
Tiantian Wu, Songmei Huan, Xiaojuan Liu
Summary: This paper presents a method to ensure the existence of sliding homoclinic orbits of three-dimensional piecewise affine systems. Additionally, sliding cycles are obtained by bifurcations of the systems with sliding homoclinic orbits to saddles. Two simulation examples of sliding homoclinic orbits and sliding cycles are provided to illustrate the effectiveness of the results.
NONLINEAR DYNAMICS
(2023)
Article
Engineering, Mechanical
Juhong Ge
Summary: This article mainly focuses on the impacts of multiple time delays on bifurcation dynamics in two-neuron neural networks. It investigates the existence of periodic oscillation and pitchfork-Hopf singularity close to the trivial equilibrium point. Furthermore, a methodology-based perturbation is introduced to tackle the zero-Hopf bifurcation singularity. The study reveals the complex and rich dynamics displayed by the two-neuron neural system in the vicinity of the pitchfork-Hopf bifurcation.
NONLINEAR DYNAMICS
(2022)
Article
Engineering, Mechanical
Zhaoxia Wang, Hebai Chen, Yilei Tang
Summary: This paper investigates the global dynamics of a nonsmooth Rayleigh-Duffing equation in the focus case, revealing complexities such as the presence of five limit cycles and gluing bifurcations. The study includes bifurcation diagrams showing various types of bifurcation curves and numerical phase portraits to illustrate theoretical results.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics
Mengke Yu, Cailiang Chen, Qiuyan Zhang
Summary: In this paper, we investigate the generalized Radhakrishnan-Kundu-Lakshmanan equation with polynomial law using the method of dynamical systems. By using traveling-wave transformation, the model can be converted into a singular integrable traveling-wave system. We discuss the dynamical behavior of the associated regular system and obtain bifurcations of the phase portraits of the traveling-wave system under different parameter conditions. Furthermore, we obtain the exact periodic solutions, as well as the peakon, homoclinic, and heteroclinic solutions under different parameter conditions.
Article
Mathematics, Interdisciplinary Applications
Ruimin Liu, Minghao Liu, Tiantian Wu
Summary: This paper focuses on the existence and bifurcations of homoclinic orbits to a saddle in a class of three-dimensional one-parameter three-zone symmetric piecewise affine systems. Based on the analysis of the Poincare maps, the systems have two types of limit cycles and do not have chaotic invariant sets near the homoclinic orbits. In addition, the paper provides a constant D to study the homoclinic bifurcations to limit cycles for the case |?(1)| = ?(3). Two examples with simulations of the homoclinic orbits and the limit cycles are given to illustrate the effectiveness of the results.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2023)
