Article
Mathematics, Interdisciplinary Applications
Suqi Ma
Summary: The focus is on the two-dimensional stable manifold of the Chen system. A new system with nonlinear control is proposed based on the Chen system, revealing new dynamical phenomena like the coexistence of heteroclinic orbits with periodic solutions and two new attractors. The emergence of the manifold surface is discovered through tangential self-intersection. The two-dimensional unstable manifold of the saddle-focus tends to another equilibrium solution asymptotically, and the underlying heteroclinic bifurcation and twin unstable manifolds are built from two saddle-focus points respectively.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2021)
Article
Engineering, Mechanical
Yi Zhong, Fengjuan Chen
Summary: This work establishes chaotic heteroclinic tangles when the Melnikov function is degenerate, and an explicit formula of the second-order Melnikov function is derived. It demonstrates an efficient method to characterize chaotic heteroclinic tangle.
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, Applied
L. M. Lerman, K. N. Trifonov
Summary: This study focuses on an analytic reversible Hamiltonian system with two degrees of freedom near its symmetric heteroclinic connection. The research proves chaotic behavior of the system and various types of orbits in its unfoldings, including homoclinic orbits, heteroclinic connections, and periodic orbits.
Article
Mathematics, Applied
Armel Viquit Sonna, David Yemele
Summary: The impact of asymmetric parameter on the dynamics of forced Duffing van der Pol (DVdP) and forced generalized Bonhoeffer-van der Pol (BVdP) systems is investigated. The study reveals that the asymmetric parameter significantly affects the system's chaotic behavior and can be interpreted through an effective energy potential. Additionally, the conditions for the existence of transverse intersection of stable and unstable orbits or dynamics chaos are derived using Melnikov theory. The presence of asymmetric parameter induces two types of domains, with one being more favorable for chaotic behavior.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Qiu Huang, Yongjian Liu, Chunbiao Li, Aimin Liu
Summary: The existence of homoclinic orbits is analytically discussed for a class of four-dimensional manifold piecewise linear systems with one switching manifold. An interesting phenomenon is found, where homoclinic orbits and chaos appear simultaneously in the system under the same parameter setting. Additionally, homoclinic chaos can be converted to a periodic orbit by adding a nonlinear control switch with memory. These theoretical results are illustrated with numerical simulations.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
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
Guo Wei Yu
Summary: For a monotone twist map, we showed the existence of infinitely many homoclinic and heteroclinic orbits between two periodic neighboring minimal orbits with the same rotation number, indicating chaotic dynamics. Our results also apply to geodesics of smooth Riemannian metrics on the two-dimension torus.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2022)
Article
Mathematics, Interdisciplinary Applications
Ting Yang
Summary: This paper investigates multistability in a 3D autonomous system with different types of chaotic attractors, including infinitely many equilibria and chaotic attractors when the equilibria are weak saddle-foci. Additionally, the system also exhibits a curve of equilibria and infinitely many singular degenerated heteroclinic orbits, leading to infinitely many chaotic attractors in the system.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2021)
Article
Mathematics, Applied
Bo-Wei Qin, Kwok-Wai Chung, Antonio Algaba, Alejandro J. Rodriguez-Luis
Summary: This article studies the phenomenon of canard explosion in singularly perturbed systems. By analyzing the limit cycle related to a degenerate center, an approximation of the critical parameter for canard explosion is provided and compared with numerical results, showing excellent agreement. Additionally, a good approximation of the homoclinic curve in the parameter plane is obtained.
APPLIED MATHEMATICS LETTERS
(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
X. I. A. O. T. I. N. G. LU, Y. O. N. G. J. I. A. N. LIU, A. I. M. I. N. LIU, C. H. U. N. S. H. E. N. G. FENG
Summary: This paper presents new geometric viewpoints to the Chen chaotic system. It rigorously proves the existence of two nontransverse homoclinic orbits in the system and gives a new geometric viewpoint to explore the chaos mechanism. By introducing the geometric viewpoints of the second-order system governed by the Lie-Poisson equation, geometric invariants of the Chen system can be obtained, including the torsion tensor. The obtained results show that the change in the torsion tensor leads the Chen system from periodic to chaotic, which is not found in the Lorenz system.
MISKOLC MATHEMATICAL NOTES
(2022)
Article
Engineering, Mechanical
Kadierdan Kaheman, Jason J. Bramburger, J. Nathan Kutz, Steven L. Brunton
Summary: This paper investigates the existence and significance of the double pendulum, exploring the codimension-1 invariant manifolds that separate the global phase space transport by originating from unstable periodic orbits around saddle equilibria. By comparing the dynamics of the double pendulum with transport in the solar system, this study provides a table-top benchmark for chaotic, saddle-mediated transport. The analytical results of this work have broader implications in a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum.
NONLINEAR DYNAMICS
(2023)
Article
Physics, Mathematical
Yongjian Liu, Zhouchao Wei, Chunbiao Li, Aimin Liu, Lijie Li
INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
(2019)
Article
Engineering, Mechanical
Chunbiao Li, Yujie Xu, Guanrong Chen, Yongjian Liu, Jincun Zheng
NONLINEAR DYNAMICS
(2019)
Article
Thermodynamics
Jiabing Huang, Nantian Huang, Yuming Wei, Yongjian Liu
ADVANCES IN MECHANICAL ENGINEERING
(2019)
Article
Mathematics, Interdisciplinary Applications
Zhouchao Wei, Yingying Li, Bo Sang, Yongjian Liu, Wei Zhang
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2019)
Article
Mathematics, Interdisciplinary Applications
Qiujian Huang, Aimin Liu, Yongjian Liu
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2019)
Article
Physics, Mathematical
Chunsheng Feng, Qiujian Huang, Yongjian Liu
INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
(2020)
Article
Mathematics, Interdisciplinary Applications
Chunsheng Feng, Lijie Li, Yongjian Liu, Zhouchao Wei
Article
Mathematics, Applied
Yongjian Liu, Zhenhai Liu, Dumitru Motreanu
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2020)
Article
Mathematics, Applied
Biyu Chen, Yongjian Liu, Zhouchao Wei, Chunsheng Feng
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2020)
Article
Mathematics, Interdisciplinary Applications
Yongjian Liu, Chunbiao Li, Aimin Liu
Summary: This paper discusses three types of static bifurcations in 2D differential systems based on KCC theory, and characterizes the dynamics in the nonequilibrium region. The results suggest that the dynamics in this area exhibit node-like structures for the three typical static bifurcations.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2021)
Article
Physics, Multidisciplinary
Ting Lai, Chunsheng Feng, Yongjian Liu, Aimin Liu
Summary: This paper investigates the Jacobi stability of a resonant nonlinear Schrodinger system using the KCC theory and differential geometric method. By analyzing the Lyapunov stability of equilibrium points and constructing geometric invariants, the results show that the zero point remains stable, while the stability of other nonzero equilibrium points depends on parameter values. The dynamical behavior of deviation vector is also studied to understand the focusing tendency of trajectories around equilibrium points, with numerical results indicating quasi-periodic and chaotic phenomena under periodic disturbances.
EUROPEAN PHYSICAL JOURNAL-SPECIAL TOPICS
(2022)
Article
Mathematics, Applied
Yongjian Liu, Biyu Chen, Xiezhen Huang, Li Ye, Zhouchao Wei
Summary: This paper focuses on the qualitative geometric analysis of traveling wave solutions of the MEW-Burgers wave equation. It transforms the MEW-Burgers equation into an equivalent planar dynamical system using the traveling wave transformation. The global structure of the planar system is presented, and solitary waves, kink waves, and periodic waves are found. The paper then studies the Jacobi stability of the planar system based on KCC theory, analyzing the dynamical behavior near equilibrium points and comparing Lyapunov stability and Jacobi stability. It also transforms the planar system with periodic disturbance into a six-dimensional nonlinear system and numerically simulates the periodic, quasi-periodic, and chaotic dynamical behaviors of the system.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Interdisciplinary Applications
Qiu Huang, Yongjian Liu, Chunbiao Li, Aimin Liu
Summary: The existence of homoclinic orbits is analytically discussed for a class of four-dimensional manifold piecewise linear systems with one switching manifold. An interesting phenomenon is found, where homoclinic orbits and chaos appear simultaneously in the system under the same parameter setting. Additionally, homoclinic chaos can be converted to a periodic orbit by adding a nonlinear control switch with memory. These theoretical results are illustrated with numerical simulations.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2022)
Article
Mathematics, Applied
Christopher C. Tisdell, Yongjian Liu, Zhenhai Liu
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
(2019)
Article
Mathematics, Applied
Yongjian Liu, Zhenhai Liu, Ching-Feng Wen
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2019)
Article
Mathematics, Applied
Torsten Lindstrom
Summary: This paper aims to analyze the mechanism for the interplay of deterministic and stochastic models in contagious diseases. Deterministic models usually predict global stability, while stochastic models exhibit oscillatory patterns. The study found that evolution maximizes the infectiousness of diseases and discussed the relationship between herd immunity concept and vaccination programs.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Dong Deng, Hongxun Wei
Summary: This paper investigates the existence and nonexistence of time-periodic traveling waves for a diffusive influenza model with treatment and seasonality. By utilizing the next generation operator theory and Schauder's fixed point theorem, the conditions for the existence of time-periodic traveling wave solutions are obtained, along with the proof of nonexistence in certain cases and exponential decay for waves with critical speed.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Xuan Ma, Yating Wang
Summary: In this paper, the dynamics of a rarefied gas in a finite channel is studied, specifically focusing on the phenomenon of Couette flow. The authors demonstrate that the unsteady Couette flow for the Boltzmann equation converges to a 1D steady state and derive the exponential time decay rate. The analysis holds for all hard potentials.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Meng Zhao
Summary: In this paper, a reaction-diffusion waterborne pathogen model with free boundary is studied. The existence of a unique global solution is proved, and the longtime behavior is analyzed through a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are obtained, which differs from the previous results by Zhou et al. (2018) stating that the epidemic will spread when the basic reproduction number is larger than 1.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Gulsemay Yigit, Wakil Sarfaraz, Raquel Barreira, Anotida Madzvamuse
Summary: This study presents theoretical considerations and analysis of the effects of circular geometry on the stability of reaction-diffusion systems with linear cross-diffusion on circular domains. The highlights include deriving necessary and sufficient conditions for cross-diffusion driven instability and computing parameter spaces for pattern formation. Finite element simulations are also conducted to support the theoretical findings. The study suggests that linear cross-diffusion coupled with reaction-diffusion theory is a promising mechanism for pattern formation.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Miaoqing Tian, Lili Han, Xiao He, Sining Zheng
Summary: This paper studies the attraction-repulsion chemotaxis system of two-species with two chemical substances. The behavior of solutions is determined by the interactions among diffusion, attraction, repulsion, logistic sources, and nonlinear productions in the system. The paper provides conditions for the global boundedness of solutions.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Michal Borowski, Iwona Chlebicka, Blazej Miasojedow
Summary: This article provides a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff-Havin-Maz'ya type. It characterizes the potentials that are bounded between rearrangement invariant spaces via a one-dimensional inequality of Hardy-type. By controlling very weak solutions to a broad class of quasilinear elliptic PDEs of non-standard growth, the special case of the mentioned potential infers the local regularity properties of solutions in rearrangement invariant spaces for prescribed classes of data.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Young-Pil Choi, Jinwook Jung
Summary: This study investigates the global-in-time well-posedness of the pressureless Euler-alignment system with singular communication weights. A global-in-time bounded solution is constructed using the method of characteristics, and uniqueness is obtained via optimal transport techniques.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Chuangxia Huang, Xiaodan Ding
Summary: In this paper, a diffusive Mackey-Glass model with distinct diapause and developmental delays is proposed based on the diapause effect. Some sufficient conditions for the existence of traveling wave fronts are obtained by constructing appropriate upper and lower solutions and employing inequality techniques. Two numerical examples are provided to demonstrate the reliability and feasibility of the proposed model.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Hongxing Zhao
Summary: This paper investigates the flow of fluid through a thin corrugated domain saturated with porous medium, governed by the Navier-Stokes model. Asymptotic models are derived by comparing the relation between a and the size of the periodic cylinders. The homogenization technique based on the generalized Poincare inequality is used to prove the main results.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Evgenii S. Baranovskii, Roman V. Brizitskii, Zhanna Yu. Saritskaia
Summary: This paper proves the solvability of optimal control problems for both weak and strong solutions of a boundary value problem associated with the nonlinear reaction-diffusion-convection equation with variable coefficients. In the case of strong solutions, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of solvability of the corresponding boundary value problems and the qualitative analysis of their solution properties. The paper establishes existence results for weak solutions with large data, the maximum principle, and local existence and uniqueness of a strong solution. Furthermore, an optimal feedback control problem is considered, and sufficient conditions for its solvability in the class of weak solutions are obtained using methods of the theory of topological degree for set-valued perturbations.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Antonia Chinni, Beatrice Di Bella, Petru Jebelean, Calin Serban
Summary: This article focuses on the multiplicity of solutions for differential inclusions involving the p-biharmonic operator, applying a variational approach and relying on non-smooth critical point theory.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhong Tan, Saiguo Xu
Summary: This paper investigates the Rayleigh-Taylor instability of three-dimensional inhomogeneous incompressible Euler equations with damping in a horizontal slab. It is shown that the Euler system with damping is nonlinearly unstable around the given steady state if the steady density profile is non-monotonous along the height. A new variational structure is developed to construct the growing mode solution, and the difficulty in proving the sharp exponential growth rate is overcome by exploiting the structures in linearized Euler equations. Combined with error estimates and a standard bootstrapping argument, the nonlinear instability is established.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Samuele Ricco, Andrea Torricelli
Summary: This paper presents a solution method for the autonomous obstacle problem, finding a necessary condition for the extremality of the unique solution using a primal-dual formulation. The proof is based on classical arguments of Convex Analysis and Calculus of Variations' techniques.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
Article
Mathematics, Applied
Shuxin Ge, Rong Yuan, Xiaofeng Zhang
Summary: This paper studies an initial boundary value problem for a nonlocal parabolic equation with a diffusion term and convex-concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, the existence of global solutions is proven and some blow-up conditions are obtained. Using the variational structure of the problem, the Mountain-pass theorem is utilized to demonstrate the existence of nontrivial steady-state solutions. The dynamical behavior of global solutions with relatively compact trajectories in H01 (Ω) is also established, showing uniform convergence to a non-zero steady state after a long time due to the energy functional satisfying the P.S. condition. Finally, an unstable steady states sequence is derived using another minimax theorem.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2024)
