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Applied Mathematics 2024年第8期15卷 543-567页

作者： Dahe Feng Jibin Li Airen Zhou School of Mathematics and Statistics Guizhou University of Finance and Economics Guiyang China School of Mathematical Sciences Huaqiao University Quanzhou China

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

关键词： (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation bifurcations Phase Portrait Analytical Periodic Wave Solution Periodic Cusp Wave Solution

bifurcations OF PERIODIC ORBITS IN THREE-WELL DUFFING SYSTEM WITH A PHASE SHIFT

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Journal of Systems Science & Complexity 2011年第3期24卷 519-531页

作者： Jicai HUANG Han ZHANG School of Mathematics and Statistics Central China Normal University Wuhan 430079 China College of Information Technical Science Nankai University Tianjin 300071 China.

bifurcations of periodic orbits of three-well Duffing system with a phase shift are investigated in detail. The conditions of the existence and bifurcations for harmonics, subharmonics （2-order, 3- order and m-order） and superharmonics under small perturbations are given by using second-order averaging method and Melnikov＇s method. The influence of the phase shift on the dynamics is also obtained.

关键词： bifurcations Melnikov's method periodic orbits second-order averaging three-well Duffing system.

bifurcations of Resonant Solutions and Chaos in Physical Pendulum Equation with Suspension Axis Vibrations

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Acta Mathematicae Applicatae Sinica 2010年第4期26卷 677-704页

作者： Xiang-ling Fu Jin Deng Jian-ping Yang School of Mathematics Hunan University of Science and TechnologyHunan 411201China College of Mathematics and Computer Sciences Hunan Normal UniversityHunan 410081China Department of Mathematics and Physics Hunan Institute of EngineeringHunan 411104China College of Science China Agricultural UniversityBeijing 100083China

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical *** numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic *** numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic *** exhibits many invariant torus behaviors under the resonant *** find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic ***,we did not find the cascades of period-doubling bifurcation.

关键词： Pendulum equation suspension axis vibrations averaging method Melnikov's method bifurcations chaos three frequencies resonances resonant solution

bifurcations and Dynamics of Traveling Wave Solutions to a Fujimoto-Watanabe Equation

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Communications in Theoretical Physics 2018年第6期69卷 631-636页

作者： Li-Juan Shi Zhen-Shu Wen School of Mathematical Sciences Huaqiao UniversityQuanzhou 362021China

In this paper, we study the bifurcations and dynamics of traveling wave solutions to a Fujimoto-Watanabe equation by using the method of dynamical systems. We obtain a11 possible bifurcations of phase portraits of the system in different regions of the parametric space. Then we show the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, eompactions and kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions and periodic wave solutions are implicitly given, while the expressions of kink-like and antikink-like wave solutions are explicitly shown. The dynamics of these new traveling wave solutions will greatly enrich the previews results and further help us understand the physical structures and analyze the propagation of the nonlinear wave.

关键词： Fujimoto-Watanabe equation traveling wave solutions bifurcations dynamics

bifurcations of Periodic Solutions and Chaos in Josephson System with Parametric Excitation

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Acta Mathematicae Applicatae Sinica 2015年第2期31卷 335-368页

作者： Shao-liang YUAN Zhu-jun JING College of Mathematics and Computer Sciences Yichun University The Center for Dynamical Systems Academy of Mathematics and Systems Science Chinese Academy of Sciences

Josephson system with parametric excitation is investigated. Using second-order averaging method and Melnikov function, we analyze the existence and bifurcations for harmonic,（2, 3, n-order） subharmonics and（2, 3-order） superharmonics and the heterocilinic and homoclinic bifurcations for chaos under periodic perturbation. Using numerical simulation, we check our theoretical analysis and further study the effect of the parameters on dynamics. We find the complex dynamics, including the jumping behaviors, symmetrybreaking, chaos converting to periodic orbits, interior crisis, non-attracting chaotic set, interlocking（reverse）period-doubling bifurcations from periodic orbits, the processes from interlocking period-doubling bifurcations of periodic orbits to chaos after strange non-chaotic motions when the parameter β increases, etc.

关键词： Josephson system bifurcations chaos second-order averaging method Melnikov method

bifurcations and Chaos in Duffing Equation

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Acta Mathematicae Applicatae Sinica 2007年第4期23卷 665-684页

作者： Meng Zhang Jiang-ping Yang Oriential Scientific and Technologic Institute Hunan Agriculture University Changsha 410081 China College of Science China Agricultural University Beijing 100083 China

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcing is investigated. The conditions of existence of primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using the second-averaging method, the Melnikov method and bifurcation theory. Numerical simulations including bifurcation diagram, bifurcation surfaces and phase portraits show the consistence with the theoretical analysis. The numerical results also exhibit new dynamical behaviors including onset of chaos, chaos suddenly disappearing to periodic orbit, cascades of inverse period-doubling bifurcations, period-doubling bifurcation, symmetry period-doubling bifurcations of period-3 orbit, symmetrybreaking of periodic orbits, interleaving occurrence of chaotic behaviors and period-one orbit, a great abundance of periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaotic attractors. Our results show that many dynamical behaviors are strictly departure from the behaviors of the Duffing equation with odd-nonlinear restoring force.

关键词： Duffing equation Melnikov's method second-order averaging method bifurcations chaos

bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

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Acta Mathematicae Applicatae Sinica 2005年第1期21卷 157-176页

作者： Ji-cai Huang Department of Applied Mathematics, College of Science, China Agriculture University Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China Department of Applied Mathematics College of Science China Agriculture University Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China

A discrete predator-prey system with Holling type-IV functional responseobtained by the Euler method is first investigated. The conditions of existence for foldbifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theoremand bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-twobifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximateexpressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov-Takensbifurcation point. We also show the existence of degenerated fixed point with codimension three atleast. The numerical simulations, including bifurcation diagrams, phase portraits, and computationof maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but alsoexhibit the rich and complex dynamical behaviors such as the attracting invariant circle,period-doubling bifurcation from period-2,3,4 orbits, interior crisis, intermittency mechanic, andsudden disappearance of chaotic dynamic.

关键词： predator-prey system bifurcations Bogdanov-Takens bifurcation chaos

bifurcations and Stability Boundary of a Power System

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Acta Mathematicae Applicatae Sinica 2004年第3期20卷 513-532页

作者： Ying-huiGao AcademyofMathematicsandSystemsScience ChineseAcademyofSciencesBeijing100080China

A single-axis flux decay model including an excitation control model proposed in [12,14,16] is studied. As the bifurcation parameter P m (input power to the generator) varies, the system exhibits dynamics emerging from static and dynamic bifurcations which link with system collapse. We show that the equilibrium point of the system undergoes three bifurcations: one saddle-node bifurcation and two Hopf bifurcations. The state variables dominating system collapse are different for different critical points, and the excitative control may play an important role in delaying system from collapsing. Simulations are presented to illustrate the dynamical behavior associated with the power system stability and collapse. Moreover, by computing the local quadratic approximation of the 5-dimensional stable manifold at an order 5 saddle point, an analytical expression for the approximate stability boundary is worked out.

关键词： Excitation control bifurcations stability boundary system collapse

Memristor initial-boosted coexisting plane bifurcations and its extreme multi-stability reconstitution in two-memristor-based dynamical system

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Science China(Technological Sciences) 2020年第4期63卷 603-613页

作者： BAO Han CHEN Mo WU HuaGan BAO BoCheng School of Information Science and Engineering Changzhou UniversityChangzhou 213164China

Initial-dependent extreme multi-stability and offset-boosted coexisting attractors have been significantly concerned *** paper constructs a novel five-dimensional(5-D)two-memristor-based dynamical system by introducing two memristors with cosine memductance into a three-dimensional(3-D)linear autonomous dissipative *** theoretical analyses and numerical plots,the memristor initial-boosted coexisting plane bifurcations are found and the memristor initial-dependent extreme multi-stability is revealed in such a two-memristor-based dynamical system with plane ***,a dimensionality reduction model with the determined equilibrium is established via an integral transformation method,upon which the memristor initial-dependent extreme multi-stability is reconstituted theoretically and expounded ***,physically circuit-implemented PSIM(power simulation)simulations are carried out to validate the plane offset-boosted coexisting behaviors.

关键词： memristor-based system memristor initial coexisting plane bifurcations extreme multi-stability

bifurcations FROM HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM

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Systems Science and Mathematical Sciences 1998年第1期11卷 47-52页

作者： WU Shaoping YANG Haitao(Department of Mathematics, Zhejiang University, Hangchou 310027, China)

An existence theorem of homoclinic orbit is given for second order Hamiltoniansystem-x(t) + a(t)x(t) - Wx(t, x(t)) = x(t)when = 1, where 1 is the first eigenvalue of operator Lx = x - a(t)x, and Wx(t, x) issublinear growth in x Rn. When Wx(t, x) is odd in x,. infinitely many distinct pairs ofhomoclinic orbits are obtained and the bifurcations occur for each 1.

关键词： bifurcations homoclinic orbit Hamiltonian system variational method.